CaltechAUTHORS: Combined
https://feeds.library.caltech.edu/people/Hou-T-Y/combined.rss
A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenThu, 14 Nov 2024 13:06:10 -0800Second-Order Convergence of a Projection Scheme for the Incompressible Navier–Stokes Equations with Boundaries
https://resolver.caltech.edu/CaltechAUTHORS:HOUsiamjna93
Year: 1993
DOI: 10.1137/0730030
A rigorous convergence result is given for a projection scheme for the Navies–Stokes equations in the presence of boundaries. The numerical scheme is based on a finite-difference approximation, and the pressure is chosen so that the computed velocity satisfies a discrete divergence-free condition. This choice for the pressure and the particular way that the discrete divergence is calculated near the boundary permit the error in the pressure to be controlled and the second-order convergence in the velocity and the pressure to the exact solution to be shown. Some simplifications in the calculation of the pressure in the case without boundaries are also discussed.https://resolver.caltech.edu/CaltechAUTHORS:HOUsiamjna93Convergence of a Boundary Integral Method for Water Waves
https://resolver.caltech.edu/CaltechAUTHORS:BEAsiamjna96
Year: 1996
DOI: 10.1137/S0036142993245750
We prove nonlinear stability and convergence of certain boundary integral methods for time-dependent water waves in a two-dimensional, inviscid, irrotational, incompressible fluid, with or without surface tension. The methods are convergent as long as the underlying solution remains fairly regular (and a sign condition holds in the case without surface tension). Thus, numerical instabilities are ruled out even in a fully nonlinear regime. The analysis is based on delicate energy estimates, following a framework previously developed in the continuous case [Beale, Hou, and Lowengrub, Comm. Pure Appl. Math., 46 (1993), pp. 1269–1301]. No analyticity assumption is made for the physical solution. Our study indicates that the numerical methods must satisfy certain compatibility conditions in order to be stable. Violation of these conditions will lead to numerical instabilities. A breaking wave is calculated as an illustration.https://resolver.caltech.edu/CaltechAUTHORS:BEAsiamjna96A Multiscale Finite Element Method for Elliptic Problems in Composite Materials and Porous Media
https://resolver.caltech.edu/CaltechAUTHORS:20170707-134035089
Year: 1997
DOI: 10.1006/jcph.1997.5682
In this paper, we study a multiscale finite element method for solving a class of elliptic problems arising from composite materials and flows in porous media, which contain many spatial scales. The method is designed to efficiently capture the large scale behavior of the solution without resolving all the small scale features. This is accomplished by constructing the multiscale finite element base functions that are adaptive to the local property of the differential operator. Our method is applicable to general multiple-scale problems without restrictive assumptions. The construction of the base functions is fully decoupled from element to element; thus, the method is perfectly parallel and is naturally adapted to massively parallel computers. For the same reason, the method has the ability to handle extremely large degrees of freedom due to highly heterogeneous media, which are intractable by conventional finite element (difference) methods. In contrast to some empirical numerical upscaling methods, the multiscale method is systematic and self- consistent, which makes it easier to analyze. We give a brief analysis of the method, with emphasis on the "resonant sampling" effect. Then, we propose an oversampling technique to remove the resonance effect. We demonstrate the accuracy and efficiency of our method through extensive numerical experiments, which include problems with random coefficients and problems with continuous scales. Parallel implementation and performance of the method are also addressed.https://resolver.caltech.edu/CaltechAUTHORS:20170707-134035089Numerical Study of Free Interface Problems Using Boundary Integral Methods
https://resolver.caltech.edu/CaltechAUTHORS:HOUdmicm98
Year: 1998
Numerical study of fluid interfaces is a difficult task due to the presence of high frequency numerical instabilities. Small perturbations even at the round-off error level may experience rapid growth. This makes it very difficult to distinguish the numerical instability from the physical one. Here, we perform a careful numerical stability analysis for both the spatial and time discretization. We found that there is a compatibility condition between the numerical discretizations of the singular integral operators and of the Lagrangian derivative operator. Violation of this compatibility condition will lead to numerical instability. We completely eliminate the numerical instability by enforcing this discrete compatibility condition. The resulting scheme is shown to be stable and convergent in both two and three dimensions. The improved method enables us to perform a careful numerical study of the stabilizing effect of surface tension for fluid interfaces. Several interesting phenomena have been observed. Numerical results will be presented.https://resolver.caltech.edu/CaltechAUTHORS:HOUdmicm98Effect of finite computational domain on turbulence scaling law in both physical and spectral spaces
https://resolver.caltech.edu/CaltechAUTHORS:HOUpre98
Year: 1998
DOI: 10.1103/PhysRevE.58.5841
The well-known translation between the power law of the energy spectrum and that of the correlation function or the second order structure function has been widely used in analyzing random data. Here, we show that the translation is valid only in proper scaling regimes. The regimes of valid translation are different for the correlation function and the structure function. Indeed, they do not overlap. Furthermore, in practice, the power laws exist only for a finite range of scales. We show that this finite range makes the translation inexact even in the proper scaling regime. The error depends on the scaling exponent. The current findings are applicable to data analysis in fluid turbulence and other stochastic systems.https://resolver.caltech.edu/CaltechAUTHORS:HOUpre98Dynamic generation of capillary waves
https://resolver.caltech.edu/CaltechAUTHORS:CENpof99a
Year: 1999
DOI: 10.1063/1.869975
We investigate the dynamic generation of capillary waves in two-dimensional, inviscid, and irrotational water waves with surface tension. It is well known that short capillary waves appear in the forward front of steep water waves. Although various experimental and analytical studies have contributed to the understanding of this physical phenomenon, the precise mechanism that generates the dynamic formation of capillary waves is still not well understood. Using a numerically stable and spectrally accurate boundary integral method, we perform a systematic study of the time evolution of breaking waves in the presence of surface tension. We find that the capillary waves originate near the crest in a neighborhood, where both the curvature and its derivative are maximum. For fixed but small surface tension, the maximum of curvature increases in time and the interface develops an oscillatory train of capillary waves in the forward front of the crest. Our numerical experiments also show that, as time increases, the interface tends to a possible formation of trapped bubbles through self-intersection. On the other hand, for a fixed time, as the surface tension coefficient tau is reduced, both the capillary wavelength and its amplitude decrease nonlinearly. The interface solutions approach the tau = 0 profile. At the onset of the capillaries, the derivative of the convection is comparable to that of the gravity term in the dynamic boundary condition and the surface tension becomes appreciable with respect to these two terms. We find that, based on the tau = 0 wave, it is possible to estimate a threshold value tau0 such that if tau <= tau0 then no capillary waves arise. On the other hand, for tau sufficiently large, breaking is inhibited and pure capillary motion is observed. The limiting behavior is very similar to that in the classical KdV equation. We also investigate the effect of viscosity on the generation of capillary waves. We find that the capillary waves still persist as long as the viscosity is not significantly greater than surface tension.https://resolver.caltech.edu/CaltechAUTHORS:CENpof99aNumerical study of Hele-Shaw flow with suction
https://resolver.caltech.edu/CaltechAUTHORS:CENpof99
Year: 1999
DOI: 10.1063/1.870112
We investigate numerically the effects of surface tension on the evolution of an initially circular blob of viscous fluid in a Hele-Shaw cell. The blob is surrounded by less viscous fluid and is drawn into an eccentric point sink. In the absence of surface tension, these flows are known to form cusp singularities in finite time. Our study focuses on identifying how these cusped flows are regularized by the presence of small surface tension, and what the limiting form of the regularization is as surface tension tends to zero. The two-phase Hele-Shaw flow, known as the Muskat problem, is considered. We find that, for nonzero surface tension, the motion continues beyond the zero-surface-tension cusp time, and generically breaks down only when the interface touches the sink. When the viscosity of the surrounding fluid is small or negligible, the interface develops a finger that bulges and later evolves into a wedge as it approaches the sink. A neck is formed at the top of the finger. Our computations reveal an asymptotic shape of the wedge in the limit as surface tension tends to zero. Moreover, we find evidence that, for a fixed time past the zero-surface-tension cusp time, the vanishing surface tension solution is singular at the finger neck. The zero-surface-tension cusp splits into two corner singularities in the limiting solution. Larger viscosity in the exterior fluid prevents the formation of the neck and leads to the development of thinner fingers. It is observed that the asymptotic wedge angle of the fingers decreases as the viscosity ratio is reduced, apparently towards the zero angle (cusp) of the zero-viscosity-ratio solution.https://resolver.caltech.edu/CaltechAUTHORS:CENpof99Convergence of a nonconforming multiscale finite element method
https://resolver.caltech.edu/CaltechAUTHORS:EFEsiam00
Year: 2000
DOI: 10.1137/S0036142997330329
The multiscale finite element method (MsFEM) [T. Y. Hou, X. H. Wu, and Z. Cai, Math. Comp., 1998, to appear; T. Y. Hou and X. H. Wu, J. Comput. Phys., 134 (1997), pp. 169-189] has been introduced to capture the large scale solutions of elliptic equations with highly oscillatory coefficients. This is accomplished by constructing the multiscale base functions from the local solutions of the elliptic operator. Our previous study reveals that the leading order error in this approach is caused by the "resonant sampling," which leads to large error when the mesh size is close to the small scale of the continuous problem. Similar difficulty also arises in numerical upscaling methods. An oversampling technique has been introduced to alleviate this difficulty [T. Y. Hou and X. H. Wu, J. Comput. Phys., 134 (1997), pp. 169-189]. A consequence of the oversampling method is that the resulting finite element method is no longer conforming. Here we give a detailed analysis of the nonconforming error. Our analysis also reveals a new cell resonance error which is caused by the mismatch between the mesh size and the wavelength of the small scale. We show that the cell resonance error is of lower order. Our numerical experiments demonstrate that the cell resonance error is generically small and is difficult to observe in practice.https://resolver.caltech.edu/CaltechAUTHORS:EFEsiam00The singular perturbation of surface tension in Hele-Shaw flows
https://resolver.caltech.edu/CaltechAUTHORS:CENjfm00
Year: 2000
DOI: 10.1017/S0022112099007703
Morphological instabilities are common to pattern formation problems such as the non-equilibrium growth of crystals and directional solidification. Very small perturbations caused by noise originate convoluted interfacial patterns when surface tension is small. The generic mechanisms in the formation of these complex patterns are present in the simpler problem of a Hele-Shaw interface. Amid this extreme noise sensitivity, what is then the role played by small surface tension in the dynamic formation and selection of these patterns? What is the asymptotic behaviour of the interface in the limit as surface tension tends to zero? The ill-posedness of the zero-surface-tension problem and the singular nature of surface tension pose challenging difficulties in the investigation of these questions. Here, we design a novel numerical method that greatly reduces the impact of noise, and allows us to accurately capture and identify the singular contributions of extremely small surface tensions. The numerical method combines the use of a compact interface parametrization, a rescaling of the governing equations, and very high precision. Our numerical results demonstrate clearly that the zero-surface-tension limit is indeed singular. The impact of a surface-tension-induced complex singularity is revealed in detail. The singular effects of surface tension are first felt at the tip of the interface and subsequently spread around it. The numerical simulations also indicate that surface tension defines a length scale in the fingers developing in a later stage of the interface evolution.https://resolver.caltech.edu/CaltechAUTHORS:CENjfm00Coastal hydrodynamics of ocean waves on beach
https://resolver.caltech.edu/CaltechAUTHORS:20200226-133732456
Year: 2001
DOI: 10.1016/s0065-2156(00)80005-8
This chapter describes the coastal hydrodynamics of ocean waves on beach. A comprehensive study on modeling three-dimensional ocean waves coming from an open ocean of uniform depth and obliquely incident on beach with arbitrary offshore slope distribution, while evolving under balanced effects of nonlinearity and dispersion is presented. A family of beach configurations that is uniform in the long-shore direction as a first approximation for beaches with negligible long-shore curvature is considered. The beach slope variation is assumed to have such distributions that the ocean waves will evolve on beach without breaking. The overall approach adopted begins with development of a three-dimensional linear shallow-water wave theory, followed by taking, step by step, the nonlinear and dispersive effects into account. The linear theory is shown to provide a fundamental solution involving a central function, called the beach-wave function that delineates the evolution of the incoming train of simple waves during interaction with any beach belonging to this broad family of beach configurations. This linear theory can easily afford to cover such factors as oblique wave incidence, arbitrary distribution of offshore beach slope, and wavelength variations with respect to beach breadth.https://resolver.caltech.edu/CaltechAUTHORS:20200226-133732456Multiscale Domain Decomposition Methods for Elliptic Problems with High Aspect Ratios
https://resolver.caltech.edu/CaltechAUTHORS:20200221-160350446
Year: 2002
DOI: 10.1007/s102550200004
In this paper we study some nonoverlapping domain decomposition methods for solving a class of elliptic problems arising from composite materials and flows in porous media which contain many spatial scales. Our preconditioner differs from traditional domain decomposition preconditioners by using a coarse solver which is adaptive to small scale heterogeneous features. While the convergence rate of traditional domain decomposition algorithms using coarse solvers based on linear or polynomial interpolations may deteriorate in the presence of rapid small scale oscillations or high aspect ratios, our preconditioner is applicable to multiple-scale problems without restrictive assumptions and seems to have a convergence rate nearly independent of the aspect ratio within the substructures. A rigorous convergence analysis based on the Schwarz framework is carried out, and we demonstrate the efficiency and robustness of the proposed preconditioner through numerical experiments which include problems with multiple-scale coefficients, as well problems with continuous scales.https://resolver.caltech.edu/CaltechAUTHORS:20200221-160350446A mixed multiscale finite element method for elliptic problems with oscillating coefficients
https://resolver.caltech.edu/CaltechAUTHORS:CHEmc03
Year: 2002
The recently introduced multiscale finite element method for solving elliptic equations with oscillating coefficients is designed to capture the large-scale structure of the solutions without resolving all the fine-scale structures. Motivated by the numerical simulation of flow transport in highly heterogeneous porous media, we propose a mixed multiscale finite element method with an over-sampling technique for solving second order elliptic equations with rapidly oscillating coefficients. The multiscale finite element bases are constructed by locally solving Neumann boundary value problems. We provide a detailed convergence analysis of the method under the assumption that the oscillating coefficients are locally periodic. While such a simplifying assumption is not required by our method, it allows us to use homogenization theory to obtain the asymptotic structure of the solutions. Numerical experiments are carried out for flow transport in a porous medium with a random log-normal relative permeability to demonstrate the efficiency and accuracy of the proposed method.https://resolver.caltech.edu/CaltechAUTHORS:CHEmc03Singularity formation in three-dimensional vortex sheets
https://resolver.caltech.edu/CaltechAUTHORS:HOUpof03
Year: 2003
DOI: 10.1063/1.1526100
We study singularity formation of three-dimensional (3-D) vortex sheets without surface tension using a new approach. First, we derive a leading order approximation to the boundary integral equation governing the 3-D vortex sheet. This leading order equation captures the most singular contributions of the integral equation. By introducing an appropriate change of variables, we show that the leading order vortex sheet equation degenerates to a two-dimensional vortex sheet equation in the direction of the tangential velocity jump. This change of variables is guided by a careful analysis based on properties of certain singular integral operators, and is crucial in identifying the leading order singular behavior. Our result confirms that the tangential velocity jump is the physical driving force of the vortex sheet singularities. We also show that the singularity type of the three-dimensional problem is similar to that of the two-dimensional problem. Moreover, we introduce a model equation for 3-D vortex sheets. This model equation captures the leading order singularity structure of the full 3-D vortex sheet equation, and it can be computed efficiently using fast Fourier transform. This enables us to perform well-resolved calculations to study the generic type of 3-D vortex sheet singularities. We will provide detailed numerical results to support the analytic prediction, and to reveal the generic form of the vortex sheet singularity.https://resolver.caltech.edu/CaltechAUTHORS:HOUpof03Homogenization of incompressible Euler equations
https://resolver.caltech.edu/CaltechAUTHORS:20110819-082640450
Year: 2004
In this paper, we perform a nonlinear multiscale analysis for incompressible Euler equations with rapidly oscillating initial data. The initial condition for velocity field is assumed to have two scales. The fast scale velocity component is periodic and is of order one. One of the important questions is how the two-scale velocity structure propagates in time and whether nonlinear interaction will generate more scales dynamically. By using a Lagrangian
framework to describe the propagation of small scale solution, we show that the two-scale structure is preserved dynamically. Moreover, we derive a well-posed homogenized
equation for the incompressible Euler equations. Preliminary numerical experiments are presented to demonstrate that the homogenized equation captures the correct averaged solution of the incompressible Euler equation.https://resolver.caltech.edu/CaltechAUTHORS:20110819-082640450Removing the cell resonance error in the multiscale finite element method via a Petrov-Galerkin formulation
https://resolver.caltech.edu/CaltechAUTHORS:20111014-070953819
Year: 2004
We continue the study of the nonconforming multiscale finite element method (Ms- FEM) introduced in 17, 14 for second order elliptic equations with highly oscillatory coefficients. The main difficulty in MsFEM, as well as other numerical upscaling methods, is the scale resonance effect. It has been show that the leading order resonance error can be effectively removed by using an over-sampling technique. Nonetheless, there is still a secondary cell resonance error of O(Є^2/h^2). Here, we introduce a Petrov-Galerkin MsFEM formulation with nonconforming multiscale trial functions and linear test functions. We show that the cell resonance error is eliminated in this formulation and hence the convergence rate is greatly improved. Moreover, we show that a similar formulation can be used to enhance the convergence of an immersed-interface finite element method for elliptic interface problems.https://resolver.caltech.edu/CaltechAUTHORS:20111014-070953819Multiscale Finite Element Methods for Nonlinear Problems and their Applications
https://resolver.caltech.edu/CaltechAUTHORS:20111012-093928630
Year: 2004
In this paper we propose a generalization of multiscale finite element methods (Ms-FEM) to nonlinear problems. We study the convergence of the proposed method for nonlinear elliptic equations and propose an oversampling technique. Numerical examples demonstrate that the over-sampling technique greatly reduces the error. The application of MsFEM to porous media flows is considered. Finally, we describe further generalizations of MsFEM to nonlinear time-dependent equations and discuss the convergence of the method for various kinds of heterogeneities.https://resolver.caltech.edu/CaltechAUTHORS:20111012-093928630A modified particle method for semilinear hyperbolic systems with oscillatory solutions
https://resolver.caltech.edu/CaltechAUTHORS:20111013-071805594
Year: 2004
We introduce a modified particle method for semi-linear hyperbolic systems with highly oscillatory solutions. The main feature of this modified particle method is that we do not require different families of characteristics to meet at one point. In the modified particle method, we update the ith component of the solution along its own characteristics, and interpolate the other components of the solution from their own characteristic points to the ith characteristic point. We prove the convergence of the modified particle method essentially independent of the small scale for the variable coefficient Carleman model. The same result also applies to the non-resonant Broadwell model. Numerical evidence suggests that the modified particle method also converges essentially independent of the small scale for the original Broadwell model if a cubic spline interpolation is used.https://resolver.caltech.edu/CaltechAUTHORS:20111013-071805594Geometric Properties and Nonblowup of 3D Incompressible Euler Flow
https://resolver.caltech.edu/CaltechAUTHORS:20160322-075428588
Year: 2005
DOI: 10.1081/PDE-200044488
By exploring a local geometric property of the vorticity field along a vortex filament, we establish a sharp relationship between the geometric properties of the vorticity field and the maximum vortex stretching. This new understanding leads to an improved result of the global existence of the 3D Euler equation under mild assumptions.https://resolver.caltech.edu/CaltechAUTHORS:20160322-075428588Multiscale modelling and computation of fluid flow
https://resolver.caltech.edu/CaltechAUTHORS:20180404-083725024
Year: 2005
DOI: 10.1002/fld.866
Many problems of fundamental and practical importance have multiscale solutions. Direct numerical simulation of these multiscale problems is difficult due to the range of length scales in the underlying physical problems. Here, we describe two multiscale methods for computing nonlinear partial differential equations with multiscale solutions. The first method relies on constructing local multiscale bases for diffusion‐dominated problems. We demonstrate that such an approach can be used to upscale two‐phase flow in heterogeneous porous media. The second method is to construct semi‐analytic multiscale solutions local in space and time. We use these solutions to approximate the large‐scale solution for convection‐dominated transport. This approach overcomes the common difficulty due to the memory effect in deriving the averaged equations for convection‐dominated transport. Our multiscale analysis provides a useful guideline for designing effective numerical methods for incompressible flow.https://resolver.caltech.edu/CaltechAUTHORS:20180404-083725024Multiscale analysis in Lagrangian formulation for the 2-D incompressible Euler equation
https://resolver.caltech.edu/CaltechAUTHORS:20170408-161459048
Year: 2005
DOI: 10.3934/dcds.2005.13.1153
We perform a systematic multiscale analysis for the 2-D incompressible Euler equation with rapidly oscillating initial data using a Lagrangian approach. The Lagrangian formulation enables us to capture the propagation of the multiscale solution in a natural way. By making an appropriate multiscale expansion in the vorticity-stream function formulation, we derive a well-posed homogenized equation for the Euler equation. Based on the multiscale analysis in the Lagrangian formulation, we also derive the corresponding multiscale analysis in the Eulerian formulation. Moreover, our multiscale analysis reveals some interesting structure for the Reynolds stress term, which provides a theoretical base for establishing systematic multiscale modeling of 2-D incompressible flow.https://resolver.caltech.edu/CaltechAUTHORS:20170408-161459048Preconditioning Markov Chain Monte Carlo Simulations Using Coarse-Scale Models
https://resolver.caltech.edu/CaltechAUTHORS:EFEsiamjsc06
Year: 2006
DOI: 10.1137/050628568
We study the preconditioning of Markov chain Monte Carlo (MCMC) methods using coarse-scale models with applications to subsurface characterization. The purpose of preconditioning is to reduce the fine-scale computational cost and increase the acceptance rate in the MCMC sampling. This goal is achieved by generating Markov chains based on two-stage computations. In the first stage, a new proposal is first tested by the coarse-scale model based on multiscale finite volume methods. The full fine-scale computation will be conducted only if the proposal passes the coarse-scale screening. For more efficient simulations, an approximation of the full fine-scale computation using precomputed multiscale basis functions can also be used. Comparing with the regular MCMC method, the preconditioned MCMC method generates a modified Markov chain by incorporating the coarse-scale information of the problem. The conditions under which the modified Markov chain will converge to the correct posterior distribution are stated in the paper. The validity of these assumptions for our application and the conditions which would guarantee a high acceptance rate are also discussed. We would like to note that coarse-scale models used in the simulations need to be inexpensive but not necessarily very accurate, as our analysis and numerical simulations demonstrate. We present numerical examples for sampling permeability fields using two-point geostatistics. The Karhunen--Loève expansion is used to represent the realizations of the permeability field conditioned to the dynamic data, such as production data, as well as some static data. Our numerical examples show that the acceptance rate can be increased by more than 10 times if MCMC simulations are preconditioned using coarse-scale models.https://resolver.caltech.edu/CaltechAUTHORS:EFEsiamjsc06Multiscale computation of isotropic homogeneous turbulent
flow
https://resolver.caltech.edu/CaltechAUTHORS:20110602-094821767
Year: 2006
In this article we perform a systematic multi-scale analysis and
computation for incompressible Euler equations and Navier-Stokes Equations
in both 2D and 3D. The initial condition for velocity field has multiple length
scales. By reparameterizing them in the Fourier space, we can formally organize
the initial condition into two scales with the fast scale component being
periodic. By making an appropriate multiscale expansion for the velocity field,
we show that the two-scale structure is preserved dynamically. Moreover, we
derive a well-posed homogenized equation for the incompressible Euler equations
in the Eulerian formulations. Numerical experiments are presented to
demonstrate that the homogenized equations indeed capture the correct averaged
solution of the incompressible Euler and Navier Stokes equations. Moreover,
our multiscale analysis reveals some interesting structure for the Reynolds
stress terms, which provides a theoretical base for establishing an effective LES
type of model for incompressible fluid flows.https://resolver.caltech.edu/CaltechAUTHORS:20110602-094821767Improved Geometric Conditions for Non-Blowup of the 3D Incompressible Euler Equation
https://resolver.caltech.edu/CaltechAUTHORS:20170408-133541409
Year: 2006
DOI: 10.1080/03605300500358152
This is a follow-up of our recent article Deng et al. (2004 Deng, J.,Hou, T. Y., Yu, X. (2004). ). In Deng et al. (2004), we derive some local geometric conditions on vortex filaments which can prevent finite time blowup of the 3D incompressible Euler equation. In this article, we derive improved geometric conditions which can be applied to the scenario when velocity blows up at the same time as vorticity and the rate of blowup of velocity is proportional to the square root of vorticity. This scenario is in some sense the worst possible blow-up scenario for velocity field due to Kelvin's circulation theorem. The improved conditions can be checked by numerical computations. This provides a sharper local geometric constraint on the finite time blowup of the 3D incompressible Euler equation.https://resolver.caltech.edu/CaltechAUTHORS:20170408-133541409A 3D Numerical Method for Studying Vortex Formation Behind a Moving Plate
https://resolver.caltech.edu/CaltechAUTHORS:HOUccp06
Year: 2006
In this paper, we introduce a three-dimensional numerical method for computing the wake behind a flat plate advancing perpendicular to the flow. Our numerical method is inspired by the panel method of J. Katz and A. Plotkin [J. Katz and A. Plotkin, Low-speed Aerodynamics, 2001] and the 2D vortex blob method of Krasny [R. Krasny, Lectures in Appl. Math., 28 (1991), pp. 385--402]. The accuracy of the method will be demonstrated by comparing the 3D computation at the center section of a very high aspect ratio plate with the corresponding two-dimensional computation. Furthermore, we compare the numerical results obtained by our 3D numerical method with the corresponding experimental results obtained recently by Ringuette [M. J. Ringuette, Ph.D. Thesis, 2004] in the towing tank. Our numerical results are shown to be in excellent agreement with the experimental results up to the so-called formation time.https://resolver.caltech.edu/CaltechAUTHORS:HOUccp06On Global Well-Posedness of the Lagrangian Averaged Euler Equations
https://resolver.caltech.edu/CaltechAUTHORS:HOUsiamjma06
Year: 2006
DOI: 10.1137/050625783
We study the global well-posedness of the Lagrangian averaged Euler equations in three dimensions. We show that a necessary and sufficient condition for the global existence is that the bounded mean oscillation of the stream function is integrable in time. We also derive a sufficient condition in terms of the total variation of certain level set functions, which guarantees the global existence. Furthermore, we obtain the global existence of the averaged two-dimensional (2D) Boussinesq equations and the Lagrangian averaged 2D quasi-geostrophic equations in finite Sobolev space in the absence of viscosity or dissipation.https://resolver.caltech.edu/CaltechAUTHORS:HOUsiamjma06Dynamic Depletion of Vortex Stretching and Non-Blowup of the 3-D Incompressible Euler Equations
https://resolver.caltech.edu/CaltechAUTHORS:20160322-090700507
Year: 2006
DOI: 10.1007/s00332-006-0800-3
We study the interplay between the local geometric properties and the non-blowup of the 3D incompressible Euler equations. We consider the interaction of two perturbed antiparallel vortex tubes using Kerr's initial condition [15] [Phys. Fluids 5 (1993), 1725]. We use a pseudo-spectral method with resolution up to 1536 × 1024 × 3072 to resolve the nearly singular behavior of the Euler equations. Our numerical results demonstrate that the maximum vorticity does not grow faster than doubly exponential in time, up to t = 19, beyond the singularity time t = 18.7 predicted by Kerr's computations [15], [22]. The velocity, the enstrophy, and the enstrophy production rate remain bounded throughout the computations. As the flow evolves, the vortex tubes are flattened severely and turned into thin vortex sheets, which roll up subsequently. The vortex lines near the region of the maximum vorticity are relatively straight. This local geometric regularity of vortex lines seems to be responsible for the dynamic depletion of vortex stretching.https://resolver.caltech.edu/CaltechAUTHORS:20160322-090700507A Framework for Modeling Subgrid Effects for Two-Phase Flows in Porous Media
https://resolver.caltech.edu/CaltechAUTHORS:HOUmms06
Year: 2006
DOI: 10.1137/050646020
In this paper, we study upscaling for two-phase flows in strongly heterogeneous porous media. Upscaling a hyperbolic convection equation is known to be very difficult due to the presence of nonlocal memory effects. Even for a linear hyperbolic equation with a shear velocity field, the upscaled equation involves a nonlocal history dependent diffusion term, which is not amenable to computation. By performing a systematic multiscale analysis, we derive coupled equations for the average and the fluctuations for the two-phase flow. The homogenized equations for the coupled system are obtained by projecting the fluctuations onto a suitable subspace. This projection corresponds exactly to averaging along streamlines of the flow. Convergence of the multiscale analysis is verified numerically. Moreover, we show how to apply this multiscale analysis to upscale two-phase flows in practical applications.https://resolver.caltech.edu/CaltechAUTHORS:HOUmms06A Relay-Zone Technique for Computing Dynamic Dislocations
https://resolver.caltech.edu/CaltechAUTHORS:20190820-155926497
Year: 2007
DOI: 10.1007/978-3-540-75999-7_28
We propose a multiscale method for simulating solids with moving dislocations. Away from atomistic subdomains where the atomistic dynamics are fully resolved, a dislocation is represented by a localized jump profile, superposed on a defect-free field. We assign a thin relay zone around an atomistic subdomain to detect the dislocation profile and its propagation speed at a selected relay time. The detection technique utilizes a lattice time history integral treatment. After the relay, an atomistic computation is performed only for the defect-free field. The method allows one to effectively absorb the fine scale fluctuations and the dynamic dislocations at the interface between the atomistic and continuum domains. In the surrounding region, a coarse grid computation is adequate.
We illustrate the algorithm for a 1D Frenkel-Kontorova model at finite temperature. By comparison of the numerical results in the following figure, the reflection is absorbed by the proposed relay-zone technique.https://resolver.caltech.edu/CaltechAUTHORS:20190820-155926497Bridging Atomistic/Continuum Scales in Solids with Moving Dislocations
https://resolver.caltech.edu/CaltechAUTHORS:TANcpl07
Year: 2007
DOI: 10.1088/0256-307X/24/1/044
We propose a multiscale method for simulating solids with moving dislocations. Away from atomistic subdomains where the atomistic dynamics are fully resolved, a dislocation is represented by a localized jump profile, superposed on a defect-free field. We assign a thin relay zone around an atomistic subdomain to detect the dislocation profile and its propagation speed at a selected relay time. The detection technique utilizes a lattice time history integral treatment. After the relay, an atomistic computation is performed only for the defect-free field. The method allows one to effectively absorb the fine scale fluctuations and the dynamic dislocations at the interface between the atomistic and continuum domains. In the surrounding region, a coarse grid computation is adequate.https://resolver.caltech.edu/CaltechAUTHORS:TANcpl07Organized structures, memory, and the decay of turbulence
https://resolver.caltech.edu/CaltechAUTHORS:HOUpnas07
Year: 2007
DOI: 10.1073/pnas.0700639104
PMCID: PMC1871813
The rapid increase in computational power has led to an unprecedented enhancement of our ability to study the behavior of complex systems in the physical, biological, and social sciences. However, there are still many systems that are too complex to tackle. A turbulent fluid is the archetypal example of such a complex system. Its complexity is manifested as the appearance of organized structures across all of the scales available to a turbulent fluid. Thus, the task that a numerical analyst working on turbulence faces is to reduce the complexity of the problem into something manageable, which at the same time preserves the essential features of the problem. Although much knowledge about the Euler and Navier–Stokes equations has accumulated over the years (1–8), it has proven very difficult to incorporate this knowledge in the construction of effective models. The work of Hald and Stinis (9) in this issue of PNAS is an attempt toward the construction of an effective model that utilizes qualitative information about the structure of a turbulent flow. The work in ref. 9 rests on the idea that the organization of a fluid flow in vortices leads to "long memory" effects, i.e., the motion of a vortex at one scale is influenced by the past history of the motion of vortices in other scales. This line of thought first appeared in the work of Alder and Wainwright (ref. 10; see also ref. 11 for a recent review on memory and problem reduction).https://resolver.caltech.edu/CaltechAUTHORS:HOUpnas07Multiscale finite element methods for porous media flows and their applications
https://resolver.caltech.edu/CaltechAUTHORS:20100826-114814124
Year: 2007
DOI: 10.1016/j.apnum.2006.07.009
In this paper, we discuss some applications of multiscale finite element methods to two-phase immiscible flow simulations in heterogeneous porous media. We discuss some extensions of multiscale finite element methods which take into account limited global information. These methods are well suited for channelized porous media, where the long-range effects are important. This is typical for some recent benchmark tests, such as the SPE comparative solution project [M. Christie, M. Blunt, Tenth SPE comparative solution project: A comparison of upscaling techniques, SPE Reser. Eval. Engrg. 4 (2001) 308–317], where porous media has a channelized structure. The applications of multiscale finite element methods to inverse problems arisen in subsurface characterization are also discussed in the paper.https://resolver.caltech.edu/CaltechAUTHORS:20100826-114814124Nonexistence of locally self-similar blow-up for the 3D incompressible Navier-Stokes equations
https://resolver.caltech.edu/CaltechAUTHORS:20160322-091323469
Year: 2007
DOI: 10.3934/dcds.2007.18.637
We study locally self-similar solutions of the three dimensional incompressible Navier-Stokes equations. The locally self-similar solutions we consider here are different from the global self-similar solutions. The self-similar scaling is only valid in an inner core region that shrinks to a point dynamically as the time, t, approaches a possible singularity time, T. The solution outside the inner core region is assumed to be regular, but it does not satisfy self-similar scaling. Under the assumption that the dynamically rescaled velocity profile converges to a limiting profile as t → T in L^p for some p ϵ (3,∞), we prove that such a locally self-similar blow-up is not possible. We also obtain a simple but useful non-blowup criterion for the 3D Euler equations.https://resolver.caltech.edu/CaltechAUTHORS:20160322-091323469Mathematical modeling and simulation of aquatic and aerial animal locomotion
https://resolver.caltech.edu/CaltechAUTHORS:20200310-145803950
Year: 2007
DOI: 10.1016/j.jcp.2007.02.015
In this paper, we investigate the locomotion of fish and birds by applying a new unsteady, flexible wing theory that takes into account the strong nonlinear dynamics semi-analytically. We also make extensive comparative study between the new approach and the modified vortex blob method inspired from Chorin's and Krasny's work. We first implement the modified vortex blob method for two examples and then discuss the numerical implementation of the nonlinear analytical mathematical model of Wu. We will demonstrate that Wu's method can capture the nonlinear effects very well by applying it to some specific cases and by comparing with the experiments available. In particular, we apply Wu's method to analyze Wagner's result for a wing abruptly undergoing an increase in incidence angle. Moreover, we study the vorticity generated by a wing in heaving, pitching and bending motion. In both cases, we show that the new method can accurately represent the vortex structure behind a flying wing and its influence on the bound vortex sheet on the wing.https://resolver.caltech.edu/CaltechAUTHORS:20200310-145803950Computing nearly singular solutions using pseudo-spectral methods
https://resolver.caltech.edu/CaltechAUTHORS:20160322-083351617
Year: 2007
DOI: 10.1016/j.jcp.2007.04.014
In this paper, we investigate the performance of pseudo-spectral methods in computing nearly singular solutions of fluid dynamics equations. We consider two different ways of removing the aliasing errors in a pseudo-spectral method. The first one is the traditional 2/3 dealiasing rule. The second one is a high (36th) order Fourier smoothing which keeps a significant portion of the Fourier modes beyond the 2/3 cut-off point in the Fourier spectrum for the 2/3 dealiasing method. Both the 1D Burgers equation and the 3D incompressible Euler equations are considered. We demonstrate that the pseudo-spectral method with the high order Fourier smoothing gives a much better performance than the pseudo-spectral method with the 2/3 dealiasing rule. Moreover, we show that the high order Fourier smoothing method captures about 12–15% more effective Fourier modes in each dimension than the 2/3 dealiasing method. For the 3D Euler equations, the gain in the effective Fourier codes for the high order Fourier smoothing method can be as large as 20% over the 2/3 dealiasing method. Another interesting observation is that the error produced by the high order Fourier smoothing method is highly localized near the region where the solution is most singular, while the 2/3 dealiasing method tends to produce oscillations in the entire domain. The high order Fourier smoothing method is also found be very stable dynamically. No high frequency instability has been observed. In the case of the 3D Euler equations, the energy is conserved up to at least six digits of accuracy throughout the computations.https://resolver.caltech.edu/CaltechAUTHORS:20160322-083351617Numerical Study of Nearly Singular Solutions of the 3-D Incompressible Euler Equations
https://resolver.caltech.edu/CaltechAUTHORS:20200127-092428769
Year: 2008
DOI: 10.1007/978-3-540-68850-1_3
In this paper, we perform a careful numerical study of nearly singular solutions of the 3D incompressible Euler equations with smooth initial data. We consider the interaction of two perturbed antiparallel vortex tubes which was previously investigated by Kerr in [16, 19]. In our numerical study, we use both the pseudo-spectral method with the 2/3 dealiasing rule and the pseudo-spectral method with a high order Fourier smoothing. Moreover, we perform a careful resolution study with grid points as large as 1,536 × 1,024 × 3,072 to demonstrate the convergence of both numerical methods. Our computational results show that the maximum vorticity does not grow faster than doubly exponential in time while the velocity field remains bounded up to T = 19, beyond the singularity time T = 18.7 reported by Kerr in [16, 19]. The local geometric regularity of vortex lines near the region of maximum vorticity seems to play an important role in depleting the nonlinear vortex stretching dynamically.https://resolver.caltech.edu/CaltechAUTHORS:20200127-092428769Multiscale Analysis and Computation for the Three-Dimensional Incompressible Navier–Stokes Equations
https://resolver.caltech.edu/CaltechAUTHORS:HOUmms08
Year: 2008
DOI: 10.1137/070682046
In this paper, we perform a systematic multiscale analysis for the three-dimensional incompressible Navier–Stokes equations with multiscale initial data. There are two main ingredients in our multiscale method. The first one is that we reparameterize the initial data in the Fourier space into a formal two-scale structure. The second one is the use of a nested multiscale expansion together with a multiscale phase function to characterize the propagation of the small-scale solution dynamically. By using these two techniques and performing a systematic multiscale analysis, we derive a multiscale model which couples the dynamics of the small-scale subgrid problem to the large-scale solution without a closure assumption or unknown parameters. Furthermore, we propose an adaptive multiscale computational method which has a complexity comparable to a dynamic Smagorinsky model. We demonstrate the accuracy of the multiscale model by comparing with direct numerical simulations for both two- and three-dimensional problems. In the two-dimensional case we consider decaying turbulence, while in the three-dimensional case we consider forced turbulence. Our numerical results show that our multiscale model not only captures the energy spectrum very accurately, it can also reproduce some of the important statistical properties that have been observed in experimental studies for fully developed turbulent flows.https://resolver.caltech.edu/CaltechAUTHORS:HOUmms08Dynamic stability of the three-dimensional axisymmetric Navier-Stokes equations with swirl
https://resolver.caltech.edu/CaltechAUTHORS:20160322-093035560
Year: 2008
DOI: 10.1002/cpa.20212
In this paper, we study the dynamic stability of the three-dimensional axisymmetric Navier-Stokes Equations with swirl. To this purpose, we propose a new one-dimensional model that approximates the Navier-Stokes equations along the symmetry axis. An important property of this one-dimensional model is that one can construct from its solutions a family of exact solutions of the three-dimensionaFinal Navier-Stokes equations. The nonlinear structure of the one-dimensional model has some very interesting properties. On one hand, it can lead to tremendous dynamic growth of the solution within a short time. On the other hand, it has a surprising dynamic depletion mechanism that prevents the solution from blowing up in finite time. By exploiting this special nonlinear structure, we prove the global regularity of the three-dimensional Navier-Stokes equations for a family of initial data, whose solutions can lead to large dynamic growth, but yet have global smooth solutions.https://resolver.caltech.edu/CaltechAUTHORS:20160322-093035560Multiscale Computations for Flow and Transport in Heterogeneous Media
https://resolver.caltech.edu/CaltechAUTHORS:EFElnm08
Year: 2008
DOI: 10.1007/978-3-540-79574-2_4
Many problems of fundamental and practical importance have multiple scale solutions. The direct numerical solution of multiple scale problems is difficult to obtain even with modern supercomputers. The major difficulty of direct solutions is due to disparity of scales. From an engineering perspective, it is often sufficient to predict macroscopic properties of the multiple-scale systems, such as the effective conductivity, elastic moduli, permeability, and eddy diffusivity. Therefore, it is desirable to develop a method that captures the small scale effect on the large scales, but does not require resolving all the small scale features. The purpose of this lecture note is to review some recent advances in developing multiscale finite element (finite volume) methods for flow and transport in strongly heterogeneous porous media. Extra effort is made in developing a multiscale computational method that can be potentially used for practical multiscale for problems with a large range of nonseparable scales. Some recent theoretical and computational developments in designing global upscaling methods will be reviewed. The lectures can be roughly divided into four parts. In part 1, we review some homogenization theory for elliptic and hyperbolic equations. This homogenization theory provides a guideline for designing effective multiscale methods. In part 2, we review some recent developments of multiscale finite element (finite volume) methods. We also discuss the issue of upscaling one-phase, two-phase flows through heterogeneous porous media and the use of limited global information in multiscale finite element (volume) methods. In part 4, we will consider multiscale simulations of two-phase flow immiscible flows using a flow-based adaptive coordinate, and introduce a theoretical framework which enables us to perform global upscaling for heterogeneous media with long range connectivity.https://resolver.caltech.edu/CaltechAUTHORS:EFElnm08Multiscale finite element methods for stochastic porous media flow equations and application to uncertainty quantification
https://resolver.caltech.edu/CaltechAUTHORS:DOScmame08
Year: 2008
DOI: 10.1016/j.cma.2008.02.030
In this paper, we study multiscale finite element methods for stochastic porous media flow equations as well as applications to uncertainty quantification. We assume that the permeability field (the diffusion coefficient) is stochastic and can be described in a finite dimensional stochastic space. This is common in applications where the coefficients are expanded using chaos approximations. The proposed multiscale method constructs multiscale basis functions corresponding to sparse realizations, and these basis functions are used to approximate the solution on the coarse-grid for any realization. Furthermore, we apply our coarse-scale model to uncertainty quantification problem where the goal is to sample the porous media properties given an integrated response such as production data. Our algorithm employs pre-computed posterior response surface obtained via the proposed coarse-scale model. Using fast analytical computations of the gradients of this posterior, we propose approximate Langevin samples. These samples are further screened through the coarse-scale simulation and, finally, used as a proposal in Metropolis–Hasting Markov chain Monte Carlo method. Numerical results are presented which demonstrate the efficiency of the proposed approach.https://resolver.caltech.edu/CaltechAUTHORS:DOScmame08Multiscale simulations of porous media flows in flow-based coordinate system
https://resolver.caltech.edu/CaltechAUTHORS:EFEcg08
Year: 2008
DOI: 10.1007/s10596-007-9073-7
In this paper, we propose a multiscale technique for the simulation of porous media flows in a flow-based coordinate system. A flow-based coordinate system allows us to simplify the scale interaction and derive the upscaled equations for purely hyperbolic transport equations. We discuss the applications of the method to two-phase flows in heterogeneous porous media. For two-phase flow simulations, the use of a flow-based coordinate system requires limited global information, such as the solution of single-phase flow. Numerical results show that one can achieve accurate upscaling results using a flow-based coordinate system.https://resolver.caltech.edu/CaltechAUTHORS:EFEcg08Global Regularity of the 3D Axi-symmetric
Navier-Stokes Equations with Anisotropic Data
https://resolver.caltech.edu/CaltechAUTHORS:20091014-100827184
Year: 2008
DOI: 10.1080/03605300802108057
In this paper, we study the 3D axisymmetric Navier-Stokes Equations with swirl.
We prove the global regularity of the 3D Navier-Stokes equations for a family of large
anisotropic initial data. Moreover, we obtain a global bound of the solution in terms
of its initial data in some Lp norm. Our results also reveal some interesting dynamic
growth behavior of the solution due to the interaction between the angular velocity
and the angular vorticity fields.https://resolver.caltech.edu/CaltechAUTHORS:20091014-100827184An efficient semi-implicit immersed boundary method
for the Navier–Stokes equations
https://resolver.caltech.edu/CaltechAUTHORS:HOUjcp08a
Year: 2008
DOI: 10.1016/j.jcp.2008.07.005
The immersed boundary method is one of the most useful computational methods in studying fluid structure interaction. On the other hand, the Immersed Boundary method
is also known to require small time steps to maintain stability when solved with an explicit method. Many implicit or approximately implicit methods have been proposed in the literature to remove this severe time step stability constraint, but none of them give satisfactory performance. In this paper, we propose an efficient semi-implicit scheme to remove this stiffness from the immersed boundary method for the Navier–Stokes equations. The construction of our semi-implicit scheme consists of two steps. First, we obtain a semi-implicit discretization which is proved to be unconditionally stable. This unconditionally stable semi-implicit scheme is still quite expensive to implement in practice. Next, we apply the small scale decomposition to the unconditionally stable semi-implicit scheme to construct our efficient semi-implicit scheme. Unlike other implicit or semi-implicit schemes proposed in the literature, our semi-implicit scheme can be solved explicitly in the spectral space. Thus the computational cost of our semi-implicit schemes is comparable to that of an explicit scheme. Our extensive numerical experiments show that our semi-implicit scheme has much better stability property than an explicit scheme. This offers a substantial computational saving in using the immersed boundary method.https://resolver.caltech.edu/CaltechAUTHORS:HOUjcp08aRemoving the Stiffness of Elastic Force from the Immersed Boundary Method for the 2D Stokes Equations
https://resolver.caltech.edu/CaltechAUTHORS:HOUjcp08b
Year: 2008
DOI: 10.1016/j.jcp.2008.03.002
The immersed boundary method has evolved into one of the most useful computational methods in studying fluid structure interaction. On the other hand, the immersed boundary method is also known to suffer from a severe timestep stability restriction when using an explicit time discretization. In this paper, we propose several efficient semi-implicit schemes to remove this stiffness from the immersed boundary method for the two-dimensional Stokes flow. First, we obtain a novel unconditionally stable semi-implicit discretization for the immersed boundary problem. Using this unconditionally stable discretization as a building block, we derive several efficient semi-implicit schemes for the immersed boundary problem by applying the small scale decomposition to this unconditionally stable discretization. Our stability analysis and extensive numerical experiments show that our semi-implicit schemes offer much better stability property than the explicit scheme. Unlike other implicit or semi-implicit schemes proposed in the literature, our semi-implicit schemes can be solved explicitly in the spectral space. Thus the computational cost of our semi-implicit schemes is comparable to that of an explicit scheme, but with a much better stability property.https://resolver.caltech.edu/CaltechAUTHORS:HOUjcp08bMultiscale computations for flow and transport in porous media
https://resolver.caltech.edu/CaltechAUTHORS:20100622-112319775
Year: 2009
Many problems of fundamental and practical importance have multiple scale solutions. The direct numerical solution of multiple scale problems is difficult to obtain even with modern supercomputers. The major difficulty of direct solutions is the scale of computation. The ratio between the largest scale and the smallest scale could be as large as 10^5 in each space dimension. From an engineering perspective, it is often sufficient to predict the macroscopic properties of the multiple-scale systems, such as the effective conductivity, elastic moduli, permeability, and eddy diffusivity. Therefore, it is desirable to develop a method that captures the small scale features. This paper reviews some of the recent advances in developing systematic multiscale methods with particular emphasis on multiscale finite element methods with applications to flow and transport in heterogeneous porous media. This manuscript is not intended to be a general survey paper on this topic. The discussion is limited by the scope of the lectures and expertise of the author.https://resolver.caltech.edu/CaltechAUTHORS:20100622-112319775Multiscale Finite Element Methods: Theory and Applications
https://resolver.caltech.edu/CaltechAUTHORS:20180810-134356708
Year: 2009
DOI: 10.1007/978-0-387-09496-0
This expository book surveys the main concepts and recent advances in multiscale finite element methods. This monograph is intended for the broader audiences including engineers, applied scientists and those who are interested in multiscale simulations. The book is self-contained, starts from the basic concepts and proceeds to the latest developments in the field.
Each chapter of the book starts with a simple introduction and the description of the proposed methods as well as with motivating examples. Numerical examples demonstrating the significance of the proposed methods are presented in each chapter. The book addresses mathematical and numerical issues in multiscale finite element methods and connects them to real-world applications. Narrative introduction provides a key to the book's organization and its scope. To make the presentation accessible to a broader audience, the analyses of the methods are given in the last chapter.https://resolver.caltech.edu/CaltechAUTHORS:20180810-134356708Blow-up or no blow-up? A unified computational and analytic approach to 3D incompressible Euler and Navier–Stokes equations
https://resolver.caltech.edu/CaltechAUTHORS:20110321-155230291
Year: 2009
DOI: 10.1017/S0962492906420018
Whether the 3D incompressible Euler and Navier–Stokes equations can develop a finite-time singularity from smooth initial data with finite energy has been one of the most long-standing open questions. We review some recent theoretical and computational studies which show that there is a subtle dynamic depletion of nonlinear vortex stretching due to local geometric regularity of vortex filaments. We also investigate the dynamic stability of the 3D Navier–Stokes equations and the stabilizing effect of convection. A unique feature of our approach is the interplay between computation and analysis. Guided by our local non-blow-up theory, we have performed large-scale computations of the 3D Euler equations using a novel pseudo-spectral method on some of the most promising blow-up candidates. Our results show that there is tremendous dynamic depletion of vortex stretching. Moreover, we observe that the support of maximum vorticity becomes severely flattened as the maximum vorticity increases and the direction of the vortex filaments near the support of maximum vorticity is very regular. Our numerical observations in turn provide valuable insight, which leads to further theoretical breakthrough. Finally, we present a new class of solutions for the 3D Euler and Navier–Stokes equations, which exhibit very interesting dynamic growth properties. By exploiting the special nonlinear structure of the equations, we prove nonlinear stability and the global regularity of this class of solutions.https://resolver.caltech.edu/CaltechAUTHORS:20110321-155230291Stable Fourth Order Stream-Function Methods for Incompressible Flows with Boundaries
https://resolver.caltech.edu/CaltechAUTHORS:20090819-164116745
Year: 2009
DOI: 10.4208/jcm.2009.27.4.012
Fourth-order stream-function methods are proposed for the time dependent, incompressible Navier-Stokes and Boussinesq equations. Wide difference stencils are used instead of compact ones and the boundary terms are handled by extrapolating the stream-function values inside the computational domain to grid points outside, up to fourth-order in the noslip condition. Formal error analysis is done for a simple model problem, showing that this extrapolation introduces numerical boundary layers at fifth-order in the stream-function. The fourth-order convergence in velocity of the proposed method for the full problem is shown numerically.https://resolver.caltech.edu/CaltechAUTHORS:20090819-164116745Multiscale analysis for convection dominated transport equations
https://resolver.caltech.edu/CaltechAUTHORS:HOUdcds09
Year: 2009
DOI: 10.3934/dcds.2009.23.281
In this paper, we perform a systematic multiscale analysis for convection dominated transport equations with a weak diffusion and a highly oscillatory velocity field. The paper primarily focuses on upscaling linear transport equations. But we also discuss briefly how to upscale two-phase miscible flows, in which case the concentration equation is coupled to the pressure equation in a nonlinear fashion. For the problem we consider here, the local Peclet number is of O(ε^(-m+1)) with m is an element of [2, infinity] being any integer, where ε characterizes the small scale in the heterogeneous media. Due to the presence of the nonlocal memory effect, upscaling a convection dominated transport equation is known to be very difficult. One of the key ideas in deriving a well-posed homogenized equation for the convection dominated transport equation is to introduce a projection operator which projects the fluctuation onto a suitable subspace. This projection operator corresponds to averaging along the streamlines of the flow. In the case of linear convection dominated transport equations, we prove the well-posedness of the homogenized equations and establish rigorous error estimates for our multiscale expansion.https://resolver.caltech.edu/CaltechAUTHORS:HOUdcds09On the Partial Regularity of a 3D Model of the Navier-Stokes Equations
https://resolver.caltech.edu/CaltechAUTHORS:20090624-105309162
Year: 2009
DOI: 10.1007/s00220-008-0689-9
We study the partial regularity of a 3D model of the incompressible Navier-Stokes equations which was recently introduced by the authors in [11]. This model is derived for axisymmetric flows with swirl using a set of new variables. It preserves almost all the properties of the full 3D Euler or Navier-Stokes equations except for the convection term which is neglected in the model. If we add the convection term back to our model, we would recover the full Navier-Stokes equations. In [11], we presented numerical evidence which seems to support that the 3D model develops finite time singularities while the corresponding solution of the 3D Navier-Stokes equations remains smooth. This suggests that the convection term play an essential role in stabilizing the nonlinear vortex stretching term. In this paper, we prove that for any suitable weak solution of the 3D model in an open set in space-time, the one-dimensional Hausdorff measure of the associated singular set is zero. The partial regularity result of this paper is an analogue of the Caffarelli-Kohn-Nirenberg theory for the 3D Navier-Stokes equations.https://resolver.caltech.edu/CaltechAUTHORS:20090624-105309162On the stabilizing effect of convection in three-dimensional incompressible flows
https://resolver.caltech.edu/CaltechAUTHORS:20090528-090507806
Year: 2009
DOI: 10.1002/cpa.20254
We investigate the stabilizing effect of convection in three-dimensional incompressible Euler and Navier-Stokes equations. The convection term is the main source of nonlinearity for these equations. It is often considered destabilizing although it conserves energy due to the incompressibility condition. In this paper, we show that the convection term together with the incompressibility condition actually has a surprising stabilizing effect. We demonstrate this by constructing a new three-dimensional model that is derived for axisymmetric flows with swirl using a set of new variables. This model preserves almost all the properties of the full three-dimensional Euler or Navier-Stokes equations except for the convection term, which is neglected in our model. If we added the convection term back to our model, we would recover the full Navier-Stokes equations. We will present numerical evidence that seems to support that the three-dimensional model may develop a potential finite time singularity. We will also analyze the mechanism that leads to these singular events in the new three-dimensional model and how the convection term in the full Euler and Navier-Stokes equations destroys such a mechanism, thus preventing the singularity from forming in a finite time.https://resolver.caltech.edu/CaltechAUTHORS:20090528-090507806Introduction to the Theory of Incompressible Inviscid Flows
https://resolver.caltech.edu/CaltechAUTHORS:20180808-083855888
Year: 2009
DOI: 10.1142/9789814273282_0001
In this chapter, we consider the 3D incompressible Euler equations. We present classical and recent results on the issue of global existence/finite time singularity. We also introduce the theories of lower dimensional model equations of the 3D Euler equations and the vortex patch problem.https://resolver.caltech.edu/CaltechAUTHORS:20180808-083855888Multi-Scale Phenomena in Complex Fluids: Modeling, Analysis and Numerical Simulation
https://resolver.caltech.edu/CaltechAUTHORS:20180808-083034185
Year: 2009
DOI: 10.1142/7291
Multi-Scale Phenomena in Complex Fluids is a collection of lecture notes delivered during the first two series of mini-courses from "Shanghai Summer School on Analysis and Numerics in Modern Sciences", which was held in 2004 and 2006 at Fudan University, Shanghai, China.
This review volume of 5 chapters, covering various fields in complex fluids, places emphasis on multi-scale modeling, analyses and simulations. It will be of special interest to researchers and graduate students who want to work in the field of complex fluids.https://resolver.caltech.edu/CaltechAUTHORS:20180808-083034185A new multiscale finite element method for high-contrast elliptic interface problems
https://resolver.caltech.edu/CaltechAUTHORS:20101012-100934632
Year: 2010
We introduce a new multiscale finite element method which is
able to accurately capture solutions of elliptic interface problems with high
contrast coefficients by using only coarse quasiuniform meshes, and without
resolving the interfaces. A typical application would be the modelling of flow
in a porous medium containing a number of inclusions of low (or high) permeability
embedded in a matrix of high (respectively low) permeability. Our
method is H^1- conforming, with degrees of freedom at the nodes of a triangular
mesh and requiring the solution of subgrid problems for the basis functions on
elements which straddle the coefficient interface but which use standard linear
approximation otherwise. A key point is the introduction of novel coefficientdependent
boundary conditions for the subgrid problems. Under moderate
assumptions, we prove that our methods have (optimal) convergence rate of
O(h) in the energy norm and O(h^2) in the L_2 norm where h is the (coarse)
mesh diameter and the hidden constants in these estimates are independent
of the "contrast" (i.e. ratio of largest to smallest value) of the PDE coefficient.
For standard elements the best estimate in the energy norm would be
O(h^(1/2−ε)) with a hidden constant which in general depends on the contrast.
The new interior boundary conditions depend not only on the contrast of the
coefficients, but also on the angles of intersection of the interface with the
element edges.https://resolver.caltech.edu/CaltechAUTHORS:20101012-100934632Introduction
https://resolver.caltech.edu/CaltechAUTHORS:20120322-080850518
Year: 2011
DOI: 10.1142/S1793536911000672
This special issue of AADA is devoted to the research topics presented in the
highly stimulating international workshop on "Sparse Representation of Multiscale
Data and Images: Theory and Applications", which was held in the Institute for
Advanced Studies at the Nanyang Technological University (NTU) at Singapore
from December 14 to 17, 2009. This workshop was coorganized by Thomas Hou
(Caltech, USA) and Xue-Cheng Tai (NTU, Singapore).https://resolver.caltech.edu/CaltechAUTHORS:20120322-080850518On Singularity Formation of a Nonlinear Nonlocal System
https://resolver.caltech.edu/CaltechAUTHORS:20110228-113453092
Year: 2011
DOI: 10.1007/s00205-010-0319-5
We investigate the singularity formation of a nonlinear nonlocal system. This nonlocal system is a simplified one-dimensional system of the 3D model that was recently proposed by Hou and Lei (Comm Pure Appl Math 62(4):501–564, 2009) for axisymmetric 3D incompressible Navier–Stokes equations with swirl. The main difference between the 3D model of Hou and Lei and the reformulated 3D Navier–Stokes equations is that the convection term is neglected in the 3D model. In the nonlocal system we consider in this paper, we replace the Riesz operator in the 3D model by the Hilbert transform. One of the main results of this paper is that we prove rigorously the finite time singularity formation of the nonlocal system for a large class of smooth initial data with finite energy. We also prove global regularity for a class of smooth initial data. Numerical results will be presented to demonstrate the asymptotically self-similar blow-up of the solution. The blowup rate of the self-similar singularity of the nonlocal system is similar to that of the 3D model.https://resolver.caltech.edu/CaltechAUTHORS:20110228-113453092Introduction
https://resolver.caltech.edu/CaltechAUTHORS:20180904-093329630
Year: 2011
DOI: 10.1142/S1793536911000672
This special issue of AADA is devoted to the research topics presented in the highly stimulating international workshop on "Sparse Representation of Multiscale Data and Images: Theory and Applications", which was held in the Institute for Advanced Studies at the Nanyang Technological University (NTU) at Singapore from December 14 to 17, 2009. This workshop was co organized by Thomas Hou (Caltech, USA) and Xue-Cheng Tai (NTU, Singarpore).https://resolver.caltech.edu/CaltechAUTHORS:20180904-093329630Adaptive data analysis via sparse time-frequency representation
https://resolver.caltech.edu/CaltechAUTHORS:20120314-131024684
Year: 2011
DOI: 10.1142/S1793536911000647
We introduce a new adaptive method for analyzing nonlinear and nonstationary data. This method is inspired by the empirical mode decomposition (EMD) method and the recently developed compressed sensing theory. The main idea is to look for the sparsest representation of multiscale data within the largest possible dictionary consisting of intrinsic mode functions of the form {α(t) cos(θ(t))}, where α ≥ 0 is assumed to be smoother than cos(θ(t)) and θ is a piecewise smooth increasing function. We formulate this problem as a nonlinear L^1 optimization problem. Further, we propose an iterative algorithm to solve this nonlinear optimization problem recursively. We also introduce an adaptive filter method to decompose data with noise. Numerical examples are given to demonstrate the robustness of our method and comparison is made with the EMD method. One advantage of performing such a decomposition is to preserve some intrinsic physical property of the signal, such as trend and instantaneous frequency. Our method shares many important properties of the original EMD method. Because our method is based on a solid mathematical formulation, its performance does not depend on numerical parameters such as the number of shifting or stop criterion, which seem to have a major effect on the original EMD method. Our method is also less sensitive to noise perturbation and the end effect compared with the original EMD method.https://resolver.caltech.edu/CaltechAUTHORS:20120314-131024684Numerical Analysis of Multiscale Problems
https://resolver.caltech.edu/CaltechAUTHORS:20200203-134704241
Year: 2012
DOI: 10.1007/978-3-642-22061-6
The 91st London Mathematical Society Durham Symposium took place from July 5th to 15th 2010, with more than 100 international participants attending. The Symposium focused on Numerical Analysis of Multiscale Problems and this book contains 10 invited articles from some of the meeting's key speakers, covering a range of topics of contemporary interest in this area. Articles cover the analysis of forward and inverse PDE problems in heterogeneous media, high-frequency wave propagation, atomistic-continuum modeling and high-dimensional problems arising in modeling uncertainty. Novel upscaling and preconditioning techniques, as well as applications to turbulent multi-phase flow, and to problems of current interest in materials science are all addressed. As such this book presents the current state-of-the-art in the numerical analysis of multiscale problems and will be of interest to both practitioners and mathematicians working in those fields.https://resolver.caltech.edu/CaltechAUTHORS:20200203-134704241Dynamic growth estimates of maximum vorticity for 3D incompressible Euler equations and the SQG model
https://resolver.caltech.edu/CaltechAUTHORS:20120326-132813054
Year: 2012
DOI: 10.3934/dcds.2012.32.1449
By performing estimates on the integral of the absolute value of vorticity along a local vortex line segment, we establish a relatively sharp dynamic growth estimate of maximum vorticity under some assumptions on the local geometric regularity of the vorticity vector. Our analysis applies to both the 3D incompressible Euler equations and the surface quasi-geostrophic model (SQG). As an application of our vorticity growth estimate, we apply our result to the 3D Euler equation with the two anti-parallel vortex tubes initial data considered by Hou-Li [12]. Under some additional assumption on the vorticity field, which seems to be consistent with the computational results of [12], we show that the maximum vorticity can not grow faster than double exponential in time. Our analysis extends the earlier results by Cordoba-Fefferman [6, 7] and Deng-Hou-Yu [8, 9].https://resolver.caltech.edu/CaltechAUTHORS:20120326-132813054On singularity formation of a 3D model for incompressible Navier–Stokes equations
https://resolver.caltech.edu/CaltechAUTHORS:20120523-112912673
Year: 2012
DOI: 10.1016/j.aim.2012.02.015
We investigate the singularity formation of a 3D model that was recently proposed by Hou and Lei (2009) in [15] for axisymmetric 3D incompressible Navier–Stokes equations with swirl. The main difference between the 3D model of Hou and Lei and the reformulated 3D Navier–Stokes equations is that the convection term is neglected in the 3D model. This model shares many properties of the 3D incompressible Navier–Stokes equations. One of the main results of this paper is that we prove rigorously the finite time singularity formation of the 3D inviscid model for a class of initial boundary value problems with smooth initial data of finite energy. We also prove the global regularity of the 3D inviscid model for a class of small smooth initial data.https://resolver.caltech.edu/CaltechAUTHORS:20120523-112912673Numerical simulation of water resources problems: Models, methods, and trends
https://resolver.caltech.edu/CaltechAUTHORS:20130305-095227632
Year: 2013
DOI: 10.1016/j.advwatres.2012.05.008
Mechanistic modeling of water resources systems is a broad field with abundant challenges. We consider classes of model formulations that are considered routine, the focus of current work, and the foundation of foreseeable work over the coming decade. These model formulations are used to assess the current and evolving state of solution algorithms, discretization methods, nonlinear and linear algebraic solution methods, computational environments, and hardware trends and implications. The goal of this work is to provide guidance to enable modelers of water resources systems to make sensible choices when developing solution methods based upon the current state of knowledge and to focus future collaborative work among water resources scientists, applied mathematicians, and computational scientists on productive areas.https://resolver.caltech.edu/CaltechAUTHORS:20130305-095227632Multiscale modeling of incompressible turbulent flows
https://resolver.caltech.edu/CaltechAUTHORS:20130104-102700776
Year: 2013
DOI: 10.1016/j.jcp.2012.08.029
Developing an effective turbulence model is important for engineering applications as well as for fundamental understanding of the flow physics. We present a mathematical derivation of a closure relating the Reynolds stress to the mean strain rate for incompressible flows. A systematic multiscale analysis expresses the Reynolds stress in terms of the solutions of local periodic cell problems. We reveal an asymptotic structure of the Reynolds stress by invoking the frame invariant property of the cell problems and an iterative dynamic homogenization of large- and small-scale solutions. The recovery of the Smagorinsky model for homogeneous turbulence validates our derivation. Another example is the channel flow, where we derive a simplified turbulence model using the asymptotic structure near the wall. Numerical simulations at two Reynolds numbers (Re's) using our model agrees well with both experiments and Direct Numerical Simulations of turbulent channel flow.https://resolver.caltech.edu/CaltechAUTHORS:20130104-102700776A dynamically bi-orthogonal method for time-dependent
stochastic partial differential equations II: Adaptivity and
generalizations
https://resolver.caltech.edu/CaltechAUTHORS:20130701-154101667
Year: 2013
DOI: 10.1016/j.jcp.2013.02.020
This is part II of our paper in which we propose and develop a dynamically bi-orthogonal method (DyBO) to study a class of time-dependent stochastic partial differential equations (SPDEs) whose solutions enjoy a low-dimensional structure. In part I of our paper [9], we derived the DyBO formulation and proposed numerical algorithms based on this formulation. Some important theoretical results regarding consistency and bi-orthogonality preservation were also established in the first part along with a range of numerical examples to illustrate the effectiveness of the DyBO method. In this paper, we focus on the computational complexity analysis and develop an effective adaptivity strategy to add or remove modes dynamically. Our complexity analysis shows that the ratio of computational complexities between the DyBO method and a generalized polynomial chaos method (gPC) is roughly of order O((m/N_p)^3) for a quadratic nonlinear SPDE, where m is the number of mode pairs used in the DyBO method and N_p is the number of elements in the polynomial basis in gPC. The effective dimensions of the stochastic solutions have been found to be small in many applications, so we can expect m is much smaller than N_p and computational savings of our DyBO method against gPC are dramatic. The adaptive strategy plays an essential role for the DyBO method to be effective in solving some challenging problems. Another important contribution of this paper is the generalization of the DyBO formulation for a system of time-dependent SPDEs. Several numerical examples are provided to demonstrate the effectiveness of our method, including the Navier–Stokes equations and the Boussinesq approximation with Brownian forcing.https://resolver.caltech.edu/CaltechAUTHORS:20130701-154101667A dynamically bi-orthogonal method for time-dependent
stochastic partial differential equations I: Derivation and
algorithms
https://resolver.caltech.edu/CaltechAUTHORS:20130702-134527463
Year: 2013
DOI: 10.1016/j.jcp.2013.02.033
We propose a dynamically bi-orthogonal method (DyBO) to solve time dependent stochastic partial differential equations (SPDEs). The objective of our method is to exploit some intrinsic sparse structure in the stochastic solution by constructing the sparsest representation of the stochastic solution via a bi-orthogonal basis. It is well-known that the Karhunen–Loeve expansion (KLE) minimizes the total mean squared error and gives the sparsest representation of stochastic solutions. However, the computation of the KL expansion could be quite expensive since we need to form a covariance matrix and solve a large-scale eigenvalue problem. The main contribution of this paper is that we derive an equivalent system that governs the evolution of the spatial and stochastic basis in the KL expansion. Unlike other reduced model methods, our method constructs the reduced basis on-the-fly without the need to form the covariance matrix or to compute its eigendecomposition. In the first part of our paper, we introduce the derivation of the dynamically bi-orthogonal formulation for SPDEs, discuss several theoretical issues, such as the dynamic bi-orthogonality preservation and some preliminary error analysis of the DyBO method. We also give some numerical implementation details of the DyBO methods, including the representation of stochastic basis and techniques to deal with eigenvalue crossing. In the second part of our paper [11], we will present an adaptive strategy to dynamically remove or add modes, perform a detailed complexity analysis, and discuss various generalizations of this approach. An extensive range of numerical experiments will be provided in both parts to demonstrate the effectiveness of the DyBO method.https://resolver.caltech.edu/CaltechAUTHORS:20130702-134527463A decadal microwave record of tropical air temperature from AMSU-A/aqua observations
https://resolver.caltech.edu/CaltechAUTHORS:20140822-131136037
Year: 2013
DOI: 10.1007/s00382-013-1696-x
Atmospheric temperature is one of the most important climate variables. This observational study presents detailed descriptions of the temperature variability imprinted in the 9-year brightness temperature data acquired by the Advanced Microwave Sounding Unit-Instrument A (AMSU-A) aboard Aqua since September 2002 over tropical oceans. A non-linear, adaptive method called the Ensemble Joint Multiple Extraction has been employed to extract the principal modes of variability in the AMSU-A/Aqua data. The semi-annual, annual, quasi-biennial oscillation (QBO) modes and QBO–annual beat in the troposphere and the stratosphere have been successfully recovered. The modulation by the El Niño/Southern oscillation (ENSO) in the troposphere was found and correlates well with the Multivariate ENSO Index. The long-term variations during 2002–2011 reveal a cooling trend (−0.5 K/decade at 10 hPa) in the tropical stratosphere; the trend below the tropical tropopause is not statistically significant due to the length of our data. A new tropospheric near-annual mode (period ~1.6 years) was also revealed in the troposphere, whose existence was confirmed using National Centers for Environmental Prediction Reanalysis air temperature data. The near-annual mode in the troposphere is found to prevail in the eastern Pacific region and is coherent with a near-annual mode in the observed sea surface temperature over the Warm Pool region that has previously been reported. It remains a challenge for climate models to simulate the trends and principal modes of natural variability reported in this work.https://resolver.caltech.edu/CaltechAUTHORS:20140822-131136037Data-driven time–frequency analysis
https://resolver.caltech.edu/CaltechAUTHORS:20130725-100732399
Year: 2013
DOI: 10.1016/j.acha.2012.10.001
In this paper, we introduce a new adaptive data analysis method to study trend and instantaneous frequency of nonlinear and nonstationary data. This method is inspired by the Empirical Mode Decomposition method (EMD) and the recently developed compressed (compressive) sensing theory. The main idea is to look for the sparsest representation of multiscale data within the largest possible dictionary consisting of intrinsic mode functions of the form {a(t)cos(θ(t))}, where a∈V(θ), V(θ) consists of the functions smoother than cos(θ(t)) and θ′⩾0. This problem can be formulated as a nonlinear l^0 optimization problem. In order to solve this optimization problem, we propose a nonlinear matching pursuit method by generalizing the classical matching pursuit for the l^0 optimization problem. One important advantage of this nonlinear matching pursuit method is it can be implemented very efficiently and is very stable to noise. Further, we provide an error analysis of our nonlinear matching pursuit method under certain scale separation assumptions. Extensive numerical examples will be given to demonstrate the robustness of our method and comparison will be made with the state-of-the-art methods. We also apply our method to study data without scale separation, and data with incomplete or under-sampled data.https://resolver.caltech.edu/CaltechAUTHORS:20130725-100732399Generalized multiscale finite element methods (GMsFEM)
https://resolver.caltech.edu/CaltechAUTHORS:20130830-131942975
Year: 2013
DOI: 10.1016/j.jcp.2013.04.045
In this paper, we propose a general approach called Generalized Multiscale Finite Element Method (GMsFEM) for performing multiscale simulations for problems without scale separation over a complex input space. As in multiscale finite element methods (MsFEMs), the main idea of the proposed approach is to construct a small dimensional local solution space that can be used to generate an efficient and accurate approximation to the multiscale solution with a potentially high dimensional input parameter space. In the proposed approach, we present a general procedure to construct the offline space that is used for a systematic enrichment of the coarse solution space in the online stage. The enrichment in the online stage is performed based on a spectral decomposition of the offline space. In the online stage, for any input parameter, a multiscale space is constructed to solve the global problem on a coarse grid. The online space is constructed via a spectral decomposition of the offline space and by choosing the eigenvectors corresponding to the largest eigenvalues. The computational saving is due to the fact that the construction of the online multiscale space for any input parameter is fast and this space can be re-used for solving the forward problem with any forcing and boundary condition. Compared with the other approaches where global snapshots are used, the local approach that we present in this paper allows us to eliminate unnecessary degrees of freedom on a coarse-grid level. We present various examples in the paper and some numerical results to demonstrate the effectiveness of our method.https://resolver.caltech.edu/CaltechAUTHORS:20130830-131942975Sparse time-frequency representation of nonlinear and nonstationary data
https://resolver.caltech.edu/CaltechAUTHORS:20140113-074436396
Year: 2013
DOI: 10.1007/s11425-013-4733-7
Adaptive data analysis provides an important tool in extracting hidden physical information from multiscale data that arise from various applications. In this paper, we review two data-driven time-frequency analysis methods that we introduced recently to study trend and instantaneous frequency of nonlinear and nonstationary data. These methods are inspired by the empirical mode decomposition method (EMD) and the recently developed compressed (compressive) sensing theory. The main idea is to look for the sparsest representation of multiscale data within the largest possible dictionary consisting of intrinsic mode functions of the form {ɑ(t) cos(θ(t))}, where a is assumed to be less oscillatory than cos(θ(t)) and θ′ ⩾ 0. This problem can be formulated as a nonlinear l^0 optimization problem. We have proposed two methods to solve this nonlinear optimization problem. The first one is based on nonlinear basis pursuit and the second one is based on nonlinear matching pursuit. Convergence analysis has been carried out for the nonlinear matching pursuit method. Some numerical experiments are given to demonstrate the effectiveness of the proposed methodshttps://resolver.caltech.edu/CaltechAUTHORS:20140113-074436396A Multiscale Model Reduction Method for Partial Differential Equations
https://resolver.caltech.edu/CaltechAUTHORS:20140407-110251743
Year: 2014
DOI: 10.1051/m2an/2013115
We propose a multiscale model reduction method for partial differential equations. The main purpose of this method is to derive an effective equation for multiscale problems without scale separation. An essential ingredient of our method is to decompose the harmonic coordinates into a smooth part and a highly oscillatory part so that the smooth part is invertible and the highly oscillatory part is small. Such a decomposition plays a key role in our construction of the effective equation. We show that the solution to the effective equation is in H^2, and can be approximated by a regular coarse mesh. When the multiscale problem has scale separation and a periodic structure, our method recovers the traditional homogenized equation. Furthermore, we provide error analysis for our method and show that the solution to the effective equation is close to the original multiscale solution in the H^1 norm. Numerical results are presented to demonstrate the accuracy and robustness of the proposed method for several multiscale problems without scale separation, including a problem with a high contrast coefficient.https://resolver.caltech.edu/CaltechAUTHORS:20140407-110251743Toward the Finite-Time Blowup of the 3D Axisymmetric Euler Equations: A Numerical Investigation
https://resolver.caltech.edu/CaltechAUTHORS:20150202-082208889
Year: 2014
DOI: 10.1137/140966411
Whether the three-dimensional incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. This work attempts to provide an affirmative answer to this long-standing open question from a numerical point of view by presenting a class of potentially singular solutions to the Euler equations computed in axisymmetric geometries. The solutions satisfy a periodic boundary condition along the axial direction and a no-flow boundary condition on the solid wall. The equations are discretized in space using a hybrid 6th-order Galerkin and 6th-order finite difference method on specially designed adaptive (moving) meshes that are dynamically adjusted to the evolving solutions. With a maximum effective resolution of over (3 x 10^(12))^2 near the point of the singularity, we are able to advance the solution up to tau_2 = 0.003505 and predict a singularity time of t(s) approximate to 0.0035056, while achieving a pointwise relative error of O(10^(-4)) in the vorticity vector. and observing a (3 x 10^8)-fold increase in the maximum vorticity parallel to omega parallel to(infinity). The numerical data are checked against all major blowup/non-blowup criteria, including Beale-Kato-Majda, Constantin-Fefferman-Majda, and Deng-Hou-Yu, to confirm the validity of the singularity. A local analysis near the point of the singularity also suggests the existence of a self-similar blowup in the meridian plane.https://resolver.caltech.edu/CaltechAUTHORS:20150202-082208889On Finite Time Singularity and Global Regularity of an Axisymmetric Model for the 3D Euler Equations
https://resolver.caltech.edu/CaltechAUTHORS:20140422-134241808
Year: 2014
DOI: 10.1007/s00205-013-0717-6
In (Comm Pure Appl Math 62(4):502–564, 2009), Hou and Lei proposed a 3D model for the axisymmetric incompressible Euler and Navier–Stokes equations with swirl. This model shares a number of properties of the 3D incompressible Euler and Navier–Stokes equations. In this paper, we prove that the 3D inviscid model with an appropriate Neumann-Robin or Dirichlet-Robin boundary condition will develop a finite time singularity in an axisymmetric domain. We also provide numerical confirmation for our finite time blowup results. We further demonstrate that the energy of the blowup solution is bounded up to the singularity time, and the blowup mechanism for the mixed Dirichlet-Robin boundary condition is essentially the same as that for the energy conserving homogeneous Dirichlet boundary condition. Finally, we prove that the 3D inviscid model has globally smooth solutions for a class of large smooth initial data with some appropriate boundary condition. Both the analysis and the results we obtain here improve the previous work in a rectangular domain by Hou et al. (Adv Math 230:607–641, 2012) in several respects.https://resolver.caltech.edu/CaltechAUTHORS:20140422-134241808Intrinsic frequency for a systems approach to haemodynamic waveform analysis with clinical applications
https://resolver.caltech.edu/CaltechAUTHORS:20140825-134136758
Year: 2014
DOI: 10.1098/rsif.2014.0617
PMCID: PMC4233710
The reductionist approach has dominated the fields of biology and medicine for nearly a century. Here, we present a systems science approach to the analysis of physiological waveforms in the context of a specific case, cardiovascular physiology. Our goal in this study is to introduce a methodology that allows for novel insight into cardiovascular physiology and to show proof of concept for a new index for the evaluation of the cardiovascular system through pressure wave analysis. This methodology uses a modified version of sparse time–frequency representation (STFR) to extract two dominant frequencies we refer to as intrinsic frequencies (IFs; ω_1 and ω_2). The IFs are the dominant frequencies of the instantaneous frequency of the coupled heart + aorta system before the closure of the aortic valve and the decoupled aorta after valve closure. In this study, we extract the IFs from a series of aortic pressure waves obtained from both clinical data and a computational model. Our results demonstrate that at the heart rate at which the left ventricular pulsatile workload is minimized the two IFs are equal (ω_1 = ω_2). Extracted IFs from clinical data indicate that at young ages the total frequency variation (Δω = ω_1 − ω_2) is close to zero and that Δω increases with age or disease (e.g. heart failure and hypertension). While the focus of this paper is the cardiovascular system, this approach can easily be extended to other physiological systems or any biological signal.https://resolver.caltech.edu/CaltechAUTHORS:20140825-134136758Adaptive ANOVA-Based Data-Driven Stochastic Method for Elliptic PDEs with Random Coefficient
https://resolver.caltech.edu/CaltechAUTHORS:20150112-104708436
Year: 2014
DOI: 10.4208/cicp.270913.020414a
In this paper, we present an adaptive, analysis of variance (ANOVA)-based data-driven stochastic method (ANOVA-DSM) to study the stochastic partial differential equations (SPDEs) in the multi-query setting. Our new method integrates the advantages of both the adaptive ANOVA decomposition technique and the data-driven stochastic method. To handle high-dimensional stochastic problems, we investigate the use of adaptive ANOVA decomposition in the stochastic space as an effective dimension-reduction technique. To improve the slow convergence of the generalized polynomial chaos (gPC) method or stochastic collocation (SC) method, we adopt the data-driven stochastic method (DSM) for speed up. An essential ingredient of the DSM is to construct a set of stochastic basis under which the stochastic solutions enjoy a compact representation for a broad range of forcing functions and/or boundary conditions. Our ANOVA-DSM consists of offline and online stages. In the offline stage, the original high-dimensional stochastic problem is decomposed into a series of low-dimensional stochastic subproblems, according to the ANOVA decomposition technique. Then, for each subproblem, a data-driven stochastic basis is computed using the Karhunen-Loeve expansion (KLE) and a two-level preconditioning optimization approach. Multiple trial functions are used to enrich the stochastic basis and improve the accuracy. In the online stage, we solve each stochastic subproblem for any given forcing function by projecting the stochastic solution into the data-driven stochastic basis constructed offline. In our ANOVA-DSM framework, solving the original high-dimensional stochastic problem is reduced to solving a series of ANOVA-decomposed stochastic subproblems using the DSM. An adaptive ANOVA strategy is also provided to further reduce the number of the stochastic subproblems and speed up our method. To demonstrate the accuracy and efficiency of our method, numerical examples are presented for one- and two-dimensional elliptic PDEs with random coefficients.https://resolver.caltech.edu/CaltechAUTHORS:20150112-104708436Convergence of a data-driven time–frequency analysis method
https://resolver.caltech.edu/CaltechAUTHORS:20140731-103641810
Year: 2014
DOI: 10.1016/j.acha.2013.12.004
In a recent paper [11], Hou and Shi introduced a new adaptive data analysis method to analyze nonlinear and non-stationary data. The main idea is to look for the sparsest representation of multiscale data within the largest possible dictionary consisting of intrinsic mode functions of the form {a(t)cos(θ(t))}, where a∈V(θ),V(θ) consists of the functions that are less oscillatory than cos(θ(t)) and θ′⩾0. This problem was formulated as a nonlinear L^0 optimization problem and an iterative nonlinear matching pursuit method was proposed to solve this nonlinear optimization problem. In this paper, we prove the convergence of this nonlinear matching pursuit method under some scale separation assumptions on the signal. We consider both well-resolved and poorly sampled signals, as well as signals with noise. In the case without noise, we prove that our method gives exact recovery of the original signal.https://resolver.caltech.edu/CaltechAUTHORS:20140731-103641810Potentially singular solutions of the 3D axisymmetric Euler equations
https://resolver.caltech.edu/CaltechAUTHORS:20140825-230506377
Year: 2014
DOI: 10.1073/pnas.1405238111
PMCID: PMC4246962
The question of finite-time blowup of the 3D incompressible Euler equations is numerically investigated in a periodic cylinder with solid boundaries. Using rotational symmetry, the equations are discretized in the (2D) meridian plane on an adaptive (moving) mesh and is integrated in time with adaptively chosen time steps. The vorticity is observed to develop a ring-singularity on the solid boundary with a growth proportional to ∼(t_s − t)^(−2.46), where t_s ∼ 0.0035056 is the estimated singularity time. A local analysis also suggests the existence of a self-similar blowup. The simulations stop at τ_2 = 0.003505 at which time the vorticity amplifies by more than (3 × 10^8)-fold and the maximum mesh resolution exceeds (3 × 10^(12))^2. The vorticity vector is observed to maintain four significant digits throughout the computations.https://resolver.caltech.edu/CaltechAUTHORS:20140825-230506377Extraction of Intrawave Signals Using the Sparse Time-Frequency Representation Method
https://resolver.caltech.edu/CaltechAUTHORS:20150202-093859868
Year: 2014
DOI: 10.1137/140957767
Analysis and extraction of strongly frequency modulated signals have been a challenging problem for adaptive data analysis methods, e.g., empirical mode decomposition [N.E. Huang et al., R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 454 (1998), pp. 903--995]. In fact, many of the Newtonian dynamical systems, including conservative mechanical systems, are sources of signals with low to strong levels of frequency modulation. Analysis of such signals is an important issue in system identification problems. In this paper, we present a novel method to accurately extract intrawave signals. This method is a descendant of sparse time-frequency representation methods [T.Y. Hou and Z. Shi, Appl. Comput. Harmon. Anal., 35 (2013), pp. 284--308, T.Y. Hou and Z. Shi, Adv. Adapt. Data Anal., 3 (2011), pp. 1--28]. We will present numerical examples to show the performance of this new algorithm. Theoretical analysis of convergence of the algorithm is also presented as a support for the method. We will show that the algorithm is stable to noise perturbation as well.https://resolver.caltech.edu/CaltechAUTHORS:20150202-093859868Sparse Time Frequency Representations and Dynamical Systems
https://resolver.caltech.edu/CaltechAUTHORS:20150519-083002339
Year: 2015
DOI: 10.4310/CMS.2015.v13.n3.a4
In this paper, we establish a connection between the recently developed data-driven time-frequency analysis [T.Y. Hou and Z. Shi, Advances in Adaptive Data Analysis, 3, 1–28, 2011], [T.Y. Hou and Z. Shi, Applied and Comput. Harmonic Analysis, 35, 284–308, 2013] and the classical second order differential equations. The main idea of the data-driven time-frequency analysis is to decompose a multiscale signal into the sparsest collection of Intrinsic Mode Functions (IMFs) over the largest possible dictionary via nonlinear optimization. These IMFs are of the form a(t)cos(θ(t)), where the amplitude a(t) is positive and slowly varying. The non-decreasing phase function θ(t) is determined by the data and in general depends on the signal in a nonlinear fashion. One of the main results of this paper is that we show that each IMF can be associated with a solution of a second order ordinary differential equation of the form x+p(x,t)x+q(x,t)=0. Further, we propose a localized variational formulation for this problem and develop an effective l1-based optimization method to recover p(x,t) and q(x,t) by looking for a sparse representation of p and q in terms of the polynomial basis. Depending on the form of nonlinearity in p(x,t) and q(x,t), we can define the order of nonlinearity for the associated IMF. This generalizes a concept recently introduced by Prof. N. E. Huang et al. [N.E. Huang, M.-T. Lo, Z. Wu, and Xianyao Chen, US Patent filling number 12/241.565, Sept. 2011]. Numerical examples will be provided to illustrate the robustness and stability of the proposed method for data with or without noise. This manuscript should be considered as a proof of concept.https://resolver.caltech.edu/CaltechAUTHORS:20150519-083002339A heterogeneous stochastic FEM framework for elliptic PDEs
https://resolver.caltech.edu/CaltechAUTHORS:20150115-154548810
Year: 2015
DOI: 10.1016/j.jcp.2014.10.020
We introduce a new concept of sparsity for the stochastic elliptic operator - div(ɑ(x,ω)∇(•)), which reflects the compactness of its inverse operator in the stochastic direction and allows for spatially heterogeneous stochastic structure. This new concept of sparsity motivates a heterogeneous stochastic finite element method (HSFEM) framework for linear elliptic equations, which discretizes the equations using the heterogeneous coupling of spatial basis with local stochastic basis to exploit the local stochastic structure of the solution space. We also provide a sampling method to construct the local stochastic basis for this framework using the randomized range finding techniques. The resulting HSFEM involves two stages and suits the multi-query setting: in the offline stage, the local stochastic structure of the solution space is identified; in the online stage, the equation can be efficiently solved for multiple forcing functions. An online error estimation and correction procedure through Monte Carlo sampling is given. Numerical results for several problems with high dimensional stochastic input are presented to demonstrate the efficiency of the HSFEM in the online stage.https://resolver.caltech.edu/CaltechAUTHORS:20150115-154548810A Multiscale Data-Driven Stochastic Method for Elliptic PDEs with Random Coefficients
https://resolver.caltech.edu/CaltechAUTHORS:20150501-105112539
Year: 2015
DOI: 10.1137/130948136
In this paper, we propose a multiscale data-driven stochastic method (MsDSM) to study stochastic partial differential equations (SPDEs) in the multiquery setting. This method combines the advantages of the recently developed multiscale model reduction method [M. L. Ci, T. Y. Hou, and Z. Shi, ESAIM Math. Model. Numer. Anal., 48 (2014), pp. 449--474] and the data-driven stochastic method (DSM) [M. L. Cheng et al., SIAM/ASA J. Uncertain. Quantif., 1 (2013), pp. 452--493]. Our method consists of offline and online stages. In the offline stage, we decompose the harmonic coordinate into a smooth part and a highly oscillatory part so that the smooth part is invertible and the highly oscillatory part is small. Based on the Karhunen--Loève (KL) expansion of the smooth parts and oscillatory parts of the harmonic coordinates, we can derive an effective stochastic equation that can be well-resolved on a coarse grid. We then apply the DSM to the effective stochastic equation to construct a data-driven stochastic basis under which the stochastic solutions enjoy a compact representation for a broad range of forcing functions. In the online stage, we expand the SPDE solution using the data-driven stochastic basis and solve a small number of coupled deterministic partial differential equations (PDEs) to obtain the expansion coefficients. The MsDSM reduces both the stochastic and the physical dimensions of the solution. We have performed complexity analysis which shows that the MsDSM offers considerable savings over not only traditional methods but also DSM in solving multiscale SPDEs. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method for several multiscale stochastic problems without scale separation.https://resolver.caltech.edu/CaltechAUTHORS:20150501-105112539On the Uniqueness of Sparse Time-Frequency Representation of Multiscale Data
https://resolver.caltech.edu/CaltechAUTHORS:20151023-103928257
Year: 2015
DOI: 10.1137/141002098
In this paper, we analyze the uniqueness of the sparse time-frequency decomposition and investigate the efficiency of the nonlinear matching pursuit method. Under the assumption of scale separation, we show that the sparse time-frequency decomposition is unique up to an error that is determined by the scale separation property of the signal. We further show that the unique decomposition can be obtained approximately by the sparse time-frequency decomposition using nonlinear matching pursuit.https://resolver.caltech.edu/CaltechAUTHORS:20151023-103928257On the convergence and accuracy of the cardiovascular intrinsic frequency method
https://resolver.caltech.edu/CaltechAUTHORS:20160404-090937429
Year: 2015
DOI: 10.1098/rsos.150475
PMCID: PMC4807454
In this paper, we analyse the convergence, accuracy and
stability of the intrinsic frequency (IF) method. The IF method
is a descendant of the sparse time frequency representation
methods. These methods are designed for analysing nonlinear
and non-stationary signals. Specifically, the IF method is
created to address the cardiovascular system that by nature is a
nonlinear and non-stationary dynamical system. The IF method
is capable of handling specific nonlinear and non-stationary
signals with less mathematical regularity. In previous works,
we showed the clinical importance of the IF method. There,
we showed that the IF method can be used to evaluate
cardiovascular performance. In this article, we will present
further details of the mathematical background of the IF
method by discussing the convergence and the accuracy of
the method with and without noise. It will be shown that the
waveform fit extracted from the signal is accurate even in the
presence of noise.https://resolver.caltech.edu/CaltechAUTHORS:20160404-090937429A Model Reduction Method for Elliptic PDEs with Random Input Using the Heterogeneous Stochastic FEM Framework
https://resolver.caltech.edu/CaltechAUTHORS:20161014-130127343
Year: 2016
We introduce a model reduction method for elliptic PDEs with random input, which follows the heterogeneous stochastic finite element method framework and exploits the compactness of the solution operator in the stochastic direction on local regions of the spatial domain. This method consists of two stages and suits the multi-query setting. In the offline stage, we adaptively construct local stochastic basis functions that can capture the stochastic structure of the solution space in local regions of the domain. This is achieved through local Hilbert-Karhunen-Loève expansions of sampled stochastic solutions with randomly chosen forcing functions. In the online stage, for given forcing functions, we discretize the equation using the heterogeneous coupling of spatial basis with the constructed local stochastic basis, and obtain the numerical solutions through Galerkin projection. Convergence of the online numerical solutions is proved based on the thresholding in the offline stage. Numerical results are presented to demonstrate the effectiveness of this model reduction method.https://resolver.caltech.edu/CaltechAUTHORS:20161014-130127343Stable Self-Similar Profiles for Two 1D Models of the 3D Axisymmetric Euler Equations
https://resolver.caltech.edu/CaltechAUTHORS:20170621-112302718
Year: 2016
DOI: 10.1007/978-3-319-10151-4_17-1
Global regularity of the Euler equations in the three-dimensional (3D) setting is regarded as one of the most important open questions in mathematical fluid mechanics. In this work we consider two one-dimensional (1D) models approximating the dynamics of the 3D axisymmetric Euler equations on the solid boundary of a periodic cylinder, which are motivated by a potential finite-time singularity formation scenario proposed recently by Luo and Hou (PNAS 111(36):12968–12973, 2014), and numerically investigate the stability of the self-similar profiles in their singular solutions. We first review some recent existence results about the self-similar profiles for one model, and then derive the evolution equations of the spatial profiles in the singular solutions for both models through a dynamic rescaling formulation. We demonstrate the stability of the self-similar profiles by analyzing their discretized dynamics using linearization, and it is hoped that these computations can help to understand the potential singularity formation mechanism of the 3D Euler equations.https://resolver.caltech.edu/CaltechAUTHORS:20170621-112302718Extracting a shape function for a signal with intra-wave frequency modulation
https://resolver.caltech.edu/CaltechAUTHORS:20160315-095910709
Year: 2016
DOI: 10.1098/rsta.2015.0194
PMCID: PMC4792404
In this paper, we develop an effective and robust adaptive time-frequency analysis method for signals with intra-wave frequency modulation. To handle this kind of signals effectively, we generalize our data-driven time-frequency analysis by using a shape function to describe the intra-wave frequency modulation. The idea of using a shape function in time-frequency analysis was first proposed by Wu (Wu 2013 Appl. Comput. Harmon. Anal. 35, 181–199. (doi:10.1016/j.acha.2012.08.008)). A shape function could be any smooth 2π-periodic function. Based on this model, we propose to solve an optimization problem to extract the shape function. By exploring the fact that the shape function is a periodic function with respect to its phase function, we can identify certain low-rank structure of the signal. This low-rank structure enables us to extract the shape function from the signal. Once the shape function is obtained, the instantaneous frequency with intra-wave modulation can be recovered from the shape function. We demonstrate the robustness and efficiency of our method by applying it to several synthetic and real signals. One important observation is that this approach is very stable to noise perturbation. By using the shape function approach, we can capture the intra-wave frequency modulation very well even for noise-polluted signals. In comparison, existing methods such as empirical mode decomposition/ensemble empirical mode decomposition seem to have difficulty in capturing the intra-wave modulation when the signal is polluted by noise.https://resolver.caltech.edu/CaltechAUTHORS:20160315-095910709Sparse time-frequency decomposition based on dictionary adaptation
https://resolver.caltech.edu/CaltechAUTHORS:20160315-095503958
Year: 2016
DOI: 10.1098/rsta.2015.0192
PMCID: PMC4792402
In this paper, we propose a time-frequency analysis method to obtain instantaneous frequencies and the corresponding decomposition by solving an optimization problem. In this optimization problem, the basis that is used to decompose the signal is not known a priori. Instead, it is adapted to the signal and is determined as part of the optimization problem. In this sense, this optimization problem can be seen as a dictionary adaptation problem, in which the dictionary is adaptive to one signal rather than a training set in dictionary learning. This dictionary adaptation problem is solved by using the augmented Lagrangian multiplier (ALM) method iteratively. We further accelerate the ALM method in each iteration by using the fast wavelet transform. We apply our method to decompose several signals, including signals with poor scale separation, signals with outliers and polluted by noise and a real signal. The results show that this method can give accurate recovery of both the instantaneous frequencies and the intrinsic mode functions.https://resolver.caltech.edu/CaltechAUTHORS:20160315-095503958Adaptive data analysis: theory and applications
https://resolver.caltech.edu/CaltechAUTHORS:20160315-092552095
Year: 2016
DOI: 10.1098/rsta.2015.0207
In many applications in science, engineering and mathematics, it is useful to understand functions depending on time and/or space from many different points of view. Accordingly, a wide range of transformations and analysis tools have been developed over time. Fourier series and the Fourier transform, first proposed almost 200 years ago, provided one of the first mechanisms to write a complex waveform as the linear combination of elementary wave functions; many more would follow. Nearly 100 years ago, it became clear that for some applications it is especially useful that the elementary 'building block functions', into which more complex signals are decomposed, have a limited spread in both time and frequency—transformations or representations that used such simultaneous time–frequency (or space/spatial frequency) localization have been important tools in micro-local arguments in mathematics, quantum mechanics and semi-classical approximations, and many types of signal and data analysis. Typically, the tools used to compute such transforms or representations are linear in the input—making them (fairly) easy to implement, and versatile instruments in the data analyst's toolbox. Yet, in some cases, the very versatility of these linear tools makes them come up short, and to obtain a more detailed, precise analysis, it becomes necessary to adapt parameters and procedures to (often local) behaviour changes of the data or signal. Examples of this abound. With the advances of sensor technology, we are dealing with vast increases in not only the volume of data to be analysed, but also in their quality—leading to the ubiquitous discussions of what to do with all these 'big data'. Big data provide not only a challenge, but also an opportunity, especially because computation and storage have likewise become much more powerful. In response to these needs and opportunities, adaptive data analysis methods are being developed and explored for many different scientific and engineering frameworks.https://resolver.caltech.edu/CaltechAUTHORS:20160315-092552095Sparse Time-Frequency decomposition for multiple signals with same frequencies
https://resolver.caltech.edu/CaltechAUTHORS:20160315-152129723
Year: 2016
DOI: 10.48550/arXiv.1507.02037
In this paper, we consider multiple signals sharing same instantaneous frequencies. This kind of data is very common in scientific and engineering problems. To take advantage of this special structure, we modify our data-driven time-frequency analysis by updating the instantaneous frequencies simultaneously. Moreover, based on the simultaneously sparsity approximation and fast Fourier transform, some efficient algorithms is developed. Since the information of multiple signals is used, this method is very robust to the perturbation of noise. And it is applicable to the general nonperiodic signals even with missing samples or outliers. Several synthetic and real signals are used to test this method. The performances of this method are very promising.https://resolver.caltech.edu/CaltechAUTHORS:20160315-152129723On the Local Well-posedness of a 3D Model for Incompressible Navier-Stokes Equations with Partial Viscosity
https://resolver.caltech.edu/CaltechAUTHORS:20160315-133702384
Year: 2016
DOI: 10.48550/arXiv.1107.1823
In this short note, we study the local well-posedness of a 3D model for incompressible Navier-Stokes equations with partial viscosity. This model was originally proposed by Hou-Lei in \cite{HouLei09a}. In a recent paper, we prove that this 3D model with partial viscosity will develop a finite time singularity for a class of initial condition using a mixed Dirichlet Robin boundary condition. The local well-posedness analysis of this initial boundary value problem is more subtle than the corresponding well-posedness analysis using a standard boundary condition because the Robin boundary condition we consider is non-dissipative. We establish the local well-posedness of this initial boundary value problem by designing a Picard iteration in a Banach space and proving the convergence of the Picard iteration by studying the well-posedness property of the heat equation with the same Dirichlet Robin boundary condition.https://resolver.caltech.edu/CaltechAUTHORS:20160315-133702384On the Finite-Time Blowup of a 1D Model for the 3D Incompressible Euler Equations
https://resolver.caltech.edu/CaltechAUTHORS:20160315-134409579
Year: 2016
DOI: 10.48550/arXiv.1311.2613
We study a 1D model for the 3D incompressible Euler equations in axisymmetric geometries,
which can be viewed as a local approximation to the Euler equations near the solid boundary of a
cylindrical domain. We prove the local well-posedness of the model in spaces of zero-mean functions,
and study the potential formation of a finite-time singularity under certain convexity conditions
for the velocity field. It is hoped that the results obtained on the 1D model will be useful in the
analysis of the full 3D problem, whose loss of regularity in finite time has been observed in a recent
numerical study (Luo and Hou, 2013).https://resolver.caltech.edu/CaltechAUTHORS:20160315-134409579On the Finite-Time Blowup of a 1D Model for the 3D Axisymmetric Euler Equations
https://resolver.caltech.edu/CaltechAUTHORS:20160315-123826115
Year: 2016
DOI: 10.48550/arXiv.1407.4776
In connection with the recent proposal for possible singularity formation at the boundary for solutions of 3d axi-symmetric incompressible Euler's equations (Luo and Hou, 2013), we study models for the dynamics at the boundary and show that they exhibit a finite-time blow-up from smooth data.https://resolver.caltech.edu/CaltechAUTHORS:20160315-123826115Self-similar Singularity of a 1D Model for the 3D Axisymmetric Euler Equations
https://resolver.caltech.edu/CaltechAUTHORS:20160315-133921211
Year: 2016
DOI: 10.48550/arXiv.1407.5740
We investigate the self-similar singularity of a 1D model for the 3D axisymmetric Euler equations, which approximates the dynamics of the Euler equations on the solid boundary of a cylindrical domain. We prove the existence of a discrete family of self-similar profiles for this model and analyze their far-field properties. The self-similar profiles we find are consistent with direct simulation of the model and enjoy some stability property.https://resolver.caltech.edu/CaltechAUTHORS:20160315-133921211Level Set Dynamics and the Non-blowup of the 2D Quasi-geostrophic Equation
https://resolver.caltech.edu/CaltechAUTHORS:20160322-071936317
Year: 2016
DOI: 10.48550/arXiv.0601427
In this article we apply the technique proposed in Deng-Hou-Yu [7] to study the level set dynamics of the 2D quasi-geostrophic equation. Under certain assumptions on the local geometric regularity of the level sets of θ, we obtain global regularity results with improved growth estimate on │∇^⊥θ│. We further perform numerical simulations to study the local geometric properties of the level sets near the region of maximum │∇^⊥θ│. The numerical results indicate that the assumptions on the local geometric regularity of the level sets of θ in our theorems are satisfied. Therefore these theorems provide a good explanation of the double exponential growth of │∇^⊥θ│ observed in this and past numerical simulations.https://resolver.caltech.edu/CaltechAUTHORS:20160322-071936317Optimal Local Multi-scale Basis Functions for Linear Elliptic Equations with Rough Coefficient
https://resolver.caltech.edu/CaltechAUTHORS:20160315-134934235
Year: 2016
DOI: 10.3934/dcds.2016.36.4451
This paper addresses a multi-scale finite element method for second order linear elliptic equations with rough coefficients, which is based on the compactness of the solution operator, and does not depend on any scale-separation or periodicity assumption of the coefficient. We consider a special type of basis functions, the multi-scale basis, which are harmonic on each element and show that they have optimal approximation property for fixed local boundary conditions. To build the optimal local boundary conditions, we introduce a set of interpolation basis functions, and reduce our problem to approximating the interpolation residual of the solution space on each edge of the coarse mesh. And this is achieved through the singular value decompositions of some local oversampling operators. Rigorous error control can be obtained through thresholding in constructing the basis functions. The optimal interpolation basis functions are also identified and they can be constructed by solving some local least square problems. Numerical results for several problems with rough coefficients and high contrast inclusions are presented to demonstrate the capacity of our method in identifying and exploiting the compact structure of the local solution space to achieve computational savings.https://resolver.caltech.edu/CaltechAUTHORS:20160315-134934235Adaptive multiscale model reduction with Generalized Multiscale Finite Element Methods
https://resolver.caltech.edu/CaltechAUTHORS:20160624-151830832
Year: 2016
DOI: 10.1016/j.jcp.2016.04.054
In this paper, we discuss a general multiscale model reduction framework based on multiscale finite element methods. We give a brief overview of related multiscale methods. Due to page limitations, the overview focuses on a few related methods and is not intended to be comprehensive. We present a general adaptive multiscale model reduction framework, the Generalized Multiscale Finite Element Method. Besides the method's basic outline, we discuss some important ingredients needed for the method's success. We also discuss several applications. The proposed method allows performing local model reduction in the presence of high contrast and no scale separation.https://resolver.caltech.edu/CaltechAUTHORS:20160624-151830832An Accelerated Method for Nonlinear Elliptic PDE
https://resolver.caltech.edu/CaltechAUTHORS:20161103-105632603
Year: 2016
DOI: 10.1007/s10915-016-0215-8
We propose two numerical methods for accelerating the convergence of the standard fixed point method associated with a nonlinear and/or degenerate elliptic partial differential equation. The first method is linearly stable, while the second is provably convergent in the viscosity solution sense. In practice, the methods converge at a nearly linear complexity in terms of the number of iterations required for convergence. The methods are easy to implement and do not require the construction or approximation of the Jacobian. Numerical examples are shown for Bellman's equation, Isaacs' equation, Pucci's equations, the Monge–Ampère equation, a variant of the infinity Laplacian, and a system of nonlinear equations.https://resolver.caltech.edu/CaltechAUTHORS:20161103-105632603Identification of time-varying cable tension forces based on adaptive sparse time-frequency analysis of cable vibrations
https://resolver.caltech.edu/CaltechAUTHORS:20170214-074909079
Year: 2017
DOI: 10.1002/stc.1889
For cable bridges, the cable tension force plays a crucial role in their construction, assessment and long-term structural health monitoring. Cable tension forces vary in real time with the change of the moving vehicle loads and environmental effects, and this continual variation in tension force may cause fatigue damage of a cable. Traditional vibration-based cable tension force estimation methods can only obtain the time-averaged cable tension force and not the instantaneous force. This paper proposes a new approach to identify the time-varying cable tension forces of bridges based on an adaptive sparse time-frequency analysis method. This is a recently developed method to estimate the instantaneous frequency by looking for the sparsest time-frequency representation of the signal within the largest possible time-frequency dictionary (i.e. set of expansion functions). In the proposed approach, first, the time-varying modal frequencies are identified from acceleration measurements on the cable, then, the time-varying cable tension is obtained from the relation between this force and the identified frequencies. By considering the integer ratios of the different modal frequencies to the fundamental frequency of the cable, the proposed algorithm is further improved to increase its robustness to measurement noise. A cable experiment is implemented to illustrate the validity of the proposed method. For comparison, the Hilbert–Huang transform is also employed to identify the time-varying frequencies, which are then used to calculate the time-varying cable-tension force. The results show that the adaptive sparse time-frequency analysis method produces more accurate estimates of the time-varying cable tension forces than the Hilbert–Huang transform method.https://resolver.caltech.edu/CaltechAUTHORS:20170214-074909079A sparse decomposition of low rank symmetric positive semi-definite matrices
https://resolver.caltech.edu/CaltechAUTHORS:20170413-141136299
Year: 2017
DOI: 10.1137/16M107760X
Suppose that A∈R^(N×N) is symmetric positive semidefinite with rank K ≤ N. Our goal is to decompose A into K rank-one matrices ∑^K_k=1gkg^T_k where the modes {gk}^K_(k=1) are required to be as sparse as possible. In contrast to eigendecomposition, these sparse modes are not required to be orthogonal. Such a problem arises in random field parametrization where A is the covariance function and is intractable to solve in general. In this paper, we partition the indices from 1 to N into several patches and propose to quantify the sparseness of a vector by the number of patches on which it is nonzero, which is called patchwise sparseness. Our aim is to find the decomposition which minimizes the total patchwise sparseness of the decomposed modes. We propose a domain-decomposition type method, called intrinsic sparse mode decomposition (ISMD), which follows the "local-modes-construction + patching-up" procedure. The key step in the ISMD is to construct local pieces of the intrinsic sparse modes by a joint diagonalization problem. Thereafter, a pivoted Cholesky decomposition is utilized to glue these local pieces together. Optimal sparse decomposition, consistency with different domain decomposition, and robustness to small perturbation are proved under the so-called regular-sparse assumption (see Definition 1.2). We provide simulation results to show the efficiency and robustness of the ISMD. We also compare the ISMD to other existing methods, e.g., eigendecomposition, pivoted Cholesky decomposition, and convex relaxation of sparse principal component analysis [R. Lai, J. Lu, and S. Osher, Comm. Math. Sci., to appear; V. Q. Vu, J. Cho, J. Lei, and K. Rohe, Fantope projection and selection: A near-optimal convex relaxation of sparse PCA, in Proceedings in Advances in Neural Information Processing Systems 26, 2013, pp. 2670--2678].https://resolver.caltech.edu/CaltechAUTHORS:20170413-141136299Exploring the locally low dimensional structure in solving random elliptic PDEs
https://resolver.caltech.edu/CaltechAUTHORS:20170413-142311390
Year: 2017
DOI: 10.1137/16M1077611
We propose a stochastic multiscale finite element method (StoMsFEM) to solve random elliptic partial differential equations with a high stochastic dimension. The key idea is to simultaneously upscale the stochastic solutions in the physical space for all random samples and explore the low stochastic dimensions of the stochastic solution within each local patch. We propose two effective methods for achieving this simultaneous local upscaling. The first method is a high-order interpolation method in the stochastic space that explores the high regularity of the local upscaled quantities with respect to the random variables. The second method is a reduced-order method that explores the low rank property of the multiscale basis functions within each coarse grid patch. Our complexity analysis shows that, compared with the standard FEM on a fine grid, the StoMsFEM can achieve computational savings on the order of (H/h)^d/(log(H/h))^k, where H/h is the ratio between the coarse and the fine grid sizes, d is the physical dimension, and k is the local stochastic dimension. Several numerical examples are presented to demonstrate the accuracy and effectiveness of the proposed methods. In the high contrast example, we observe a factor of 2000 speed-up.https://resolver.caltech.edu/CaltechAUTHORS:20170413-142311390An iteratively adaptive multi-scale finite element method for elliptic PDEs with rough coefficients
https://resolver.caltech.edu/CaltechAUTHORS:20170427-144329075
Year: 2017
DOI: 10.1016/j.jcp.2017.02.002
We propose an iteratively adaptive Multi-scale Finite Element Method (MsFEM) for elliptic PDEs with rough coefficients. The choice of the local boundary conditions for the multi-sale basis functions determines the accuracy of the MsFEM numerical solution, and one needs to incorporate the global information of the elliptic equation into the local boundary conditions of the multi-scale basis functions to recover the underlying fine-mesh solution of the equation. In our proposed iteratively adaptive method, we achieve this global-to-local information transfer through the combination of coarse-mesh solving using adaptive multi-scale basis functions and fine-mesh smoothing operations. In each iteration step, we first update the multi-scale basis functions based on the approximate numerical solutions of the previous iteration steps, and obtain the coarse-mesh approximate solution using a Galerkin projection. Then we apply several steps of smoothing operations to the coarse-mesh approximate solution on the underlying fine mesh to get the updated approximate numerical solution. The proposed algorithm can be viewed as a nonlinear two-level multi-grid method with the restriction and prolongation operators adapted to the approximate numerical solutions of the previous iteration steps. Convergence analysis of the proposed algorithm is carried out under the framework of two-level multi-grid method, and the harmonic coordinates are employed to establish the approximation property of the adaptive multi-scale basis functions. We demonstrate the efficiency of our proposed multi-scale methods through several numerical examples including a multi-scale coefficient problem, a high-contrast interface problem, and a convection-dominated diffusion problem.https://resolver.caltech.edu/CaltechAUTHORS:20170427-144329075A two-level method for sparse time-frequency representation of multiscale data
https://resolver.caltech.edu/CaltechAUTHORS:20170719-100127267
Year: 2017
DOI: 10.1007/s11425-016-9088-9
Based on the recently developed data-driven time-frequency analysis (Hou and Shi, 2013), we propose a two-level method to look for the sparse time-frequency decomposition of multiscale data. In the two-level method, we first run a local algorithm to get a good approximation of the instantaneous frequency. We then pass this instantaneous frequency to the global algorithm to get an accurate global intrinsic mode function (IMF) and instantaneous frequency. The two-level method alleviates the difficulty of the mode mixing to some extent. We also present a method to reduce the end effects.https://resolver.caltech.edu/CaltechAUTHORS:20170719-100127267On the Finite-Time Blowup of a One-Dimensional Model for the Three-Dimensional Axisymmetric Euler Equations
https://resolver.caltech.edu/CaltechAUTHORS:20170509-075902405
Year: 2017
DOI: 10.1002/cpa.21697
In connection with the recent proposal for possible singularity formation at the boundary for solutions of three-dimensional axisymmetric incompressible Euler's equations (Luo and Hou, Proc. Natl. Acad. Sci. USA (2014)), we study models for the dynamics at the boundary and show that they exhibit a finite-time blowup from smooth data.https://resolver.caltech.edu/CaltechAUTHORS:20170509-075902405Sparse operator compression of higher-order elliptic operators with rough coefficients
https://resolver.caltech.edu/CaltechAUTHORS:20171205-123956892
Year: 2017
DOI: 10.1186/s40687-017-0113-1
We introduce the sparse operator compression to compress a self-adjoint higher-order elliptic operator with rough coefficients and various boundary conditions. The operator compression is achieved by using localized basis functions, which are energy minimizing functions on local patches. On a regular mesh with mesh size h, the localized basis functions have supports of diameter O(hlog(1/h)) and give optimal compression rate of the solution operator. We show that by using localized basis functions with supports of diameter O(hlog(1/h)), our method achieves the optimal compression rate of the solution operator. From the perspective of the generalized finite element method to solve elliptic equations, the localized basis functions have the optimal convergence rate O(h^k)for a (2k)th-order elliptic problem in the energy norm. From the perspective of the sparse PCA, our results show that a large set of Matérn covariance functions can be approximated by a rank-n operator with a localized basis and with the optimal accuracy.https://resolver.caltech.edu/CaltechAUTHORS:20171205-123956892An Adaptive Fast Solver for a General Class of Positive Definite Matrices Via Energy Decomposition
https://resolver.caltech.edu/CaltechAUTHORS:20180718-150855095
Year: 2018
DOI: 10.1137/17M1140686
In this paper, we propose an adaptive fast solver for a general class of symmetric positive definite (SPD) matrices which include the well-known graph Laplacian. We achieve this by developing an adaptive operator compression scheme and a multiresolution matrix factorization algorithm which achieve nearly optimal performance on both complexity and well-posedness. To develop our adaptive operator compression and multiresolution matrix factorization methods, we first introduce a novel notion of energy decomposition for SPD matrix $A$ using the representation of energy elements. The interaction between these energy elements depicts the underlying topological structure of the operator. This concept of decomposition naturally reflects the hidden geometric structure of the operator which inherits the localities of the structure. By utilizing the intrinsic geometric information under this energy framework, we propose a systematic operator compression scheme for the inverse operator $A^{-1}$. In particular, with an appropriate partition of the underlying geometric structure, we can construct localized basis by using the concept of interior and closed energy. Meanwhile, two important localized quantities are introduced, namely, the error factor and the condition factor. Our error analysis results show that these two factors will be the guidelines for finding the appropriate partition of the basis functions such that prescribed compression error and acceptable condition number can be achieved. By virtue of this insight, we propose the patch pairing algorithm to realize our energy partition framework for operator compression with controllable compression error and condition number.https://resolver.caltech.edu/CaltechAUTHORS:20180718-150855095Global regularity for a family of 3D models of the axi-symmetric Navier–Stokes equations
https://resolver.caltech.edu/CaltechAUTHORS:20180405-130644881
Year: 2018
DOI: 10.1088/1361-6544/aaaa0b
We consider a family of three-dimensional models for the axi-symmetric incompressible Navier–Stokes equations. The models are derived by changing the strength of the convection terms in the axisymmetric Navier–Stokes equations written using a set of transformed variables. We prove the global regularity of the family of models in the case that the strength of convection is slightly stronger than that of the original Navier–Stokes equations, which demonstrates the potential stabilizing effect of convection.https://resolver.caltech.edu/CaltechAUTHORS:20180405-130644881Potential Singularity for a Family of Models of the Axisymmetric Incompressible Flow
https://resolver.caltech.edu/CaltechAUTHORS:20170306-105506989
Year: 2018
DOI: 10.1007/s00332-017-9370-9
We study a family of 3D models for the incompressible axisymmetric Euler and Navier–Stokes equations. The models are derived by changing the strength of the convection terms in the equations written using a set of transformed variables. The models share several regularity results with the Euler and Navier–Stokes equations, including an energy identity, the conservation of a modified circulation quantity, the BKM criterion and the Prodi–Serrin criterion. The inviscid models with weak convection are numerically observed to develop stable self-similar singularity with the singular region traveling along the symmetric axis, and such singularity scenario does not seem to persist for strong convection.https://resolver.caltech.edu/CaltechAUTHORS:20170306-105506989A Fast Hierarchically Preconditioned Eigensolver Based on Multiresolution Matrix Decomposition
https://resolver.caltech.edu/CaltechAUTHORS:20190416-073721627
Year: 2019
DOI: 10.1137/18m1180827
In this paper we propose a new iterative method to hierarchically compute a relatively large number of leftmost eigenpairs of a sparse symmetric positive matrix under the multiresolution operator compression framework. We exploit the well-conditioned property of every decomposition component by integrating the multiresolution framework into the implicitly restarted Lanczos method. We achieve this combination by proposing an extension-refinement iterative scheme, in which the intrinsic idea is to decompose the target spectrum into several segments such that the corresponding eigenproblem in each segment is well-conditioned. Theoretical analysis and numerical illustration are also reported to illustrate the efficiency and effectiveness of this algorithm.https://resolver.caltech.edu/CaltechAUTHORS:20190416-073721627A pseudo knockoff filter for correlated features
https://resolver.caltech.edu/CaltechAUTHORS:20191114-104110525
Year: 2019
DOI: 10.1093/imaiai/iay012
In Barber & Candès (2015, Ann. Statist., 43, 2055–2085), the authors introduced a new variable selection procedure called the knockoff filter to control the false discovery rate (FDR) and proved that this method achieves exact FDR control. Inspired by the work by Barber & Candès (2015, Ann. Statist., 43, 2055–2085), we propose a pseudo knockoff filter that inherits some advantages of the original knockoff filter and has more flexibility in constructing its knockoff matrix. Moreover, we perform a number of numerical experiments that seem to suggest that the pseudo knockoff filter with the half Lasso statistic has FDR control and offers more power than the original knockoff filter with the Lasso Path or the half Lasso statistic for the numerical examples that we consider in this paper. Although we cannot establish rigourous FDR control for the pseudo knockoff filter, we provide some partial analysis of the pseudo knockoff filter with the half Lasso statistic and establish a uniform false discovery proportion bound and an expectation inequality.https://resolver.caltech.edu/CaltechAUTHORS:20191114-104110525A Model Reduction Method for Multiscale Elliptic Pdes with Random Coefficients Using an Optimization Approach
https://resolver.caltech.edu/CaltechAUTHORS:20190711-100817562
Year: 2019
DOI: 10.1137/18m1205844
In this paper, we propose a model reduction method for solving multiscale elliptic PDEs with random coefficients in the multiquery setting using an optimization approach. The optimization approach enables us to construct a set of localized multiscale data-driven stochastic basis functions that give an optimal approximation property of the solution operator. Our method consists of the offline and online stages. In the offline stage, we construct the localized multiscale data-driven stochastic basis functions by solving an optimization problem. In the online stage, using our basis functions, we can efficiently solve multiscale elliptic PDEs with random coefficients with relatively small computational costs. Therefore, our method is very efficient in solving target problems with many different force functions. The convergence analysis of the proposed method is also presented and has been verified by the numerical simulations.https://resolver.caltech.edu/CaltechAUTHORS:20190711-100817562Solving Bayesian inverse problems from the perspective of deep generative networks
https://resolver.caltech.edu/CaltechAUTHORS:20190620-093003600
Year: 2019
DOI: 10.1007/s00466-019-01739-7
Deep generative networks have achieved great success in high dimensional density approximation, especially for applications in natural images and language. In this paper, we investigate their approximation capability in capturing the posterior distribution in Bayesian inverse problems by learning a transport map. Because only the unnormalized density of the posterior is available, training methods that learn from posterior samples, such as variational autoencoders and generative adversarial networks, are not applicable in our setting. We propose a class of network training methods that can be combined with sample-based Bayesian inference algorithms, such as various MCMC algorithms, ensemble Kalman filter and Stein variational gradient descent. Our experiment results show the pros and cons of deep generative networks in Bayesian inverse problems. They also reveal the potential of our proposed methodology in capturing high dimensional probability distributions.https://resolver.caltech.edu/CaltechAUTHORS:20190620-093003600Formation of Finite-Time Singularities in the 3D Axisymmetric Euler Equations: A Numerics Guided Study
https://resolver.caltech.edu/CaltechAUTHORS:20191202-133631090
Year: 2019
DOI: 10.1137/19m1288061
Whether the three-dimensional incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. This work attempts to provide an affirmative answer to this long-standing open question, by first describing a class of potentially singular solutions to the Euler equations numerically discovered in axisymmetric geometries, and then by presenting evidence from rigorous analysis that strongly supports the existence of such singular solutions. The initial data leading to these singular solutions possess certain special symmetry and monotonicity properties, and the subsequent flows are assumed to satisfy a periodic boundary condition along the axial direction and a no-flow, free-slip boundary condition on the solid wall. The numerical study employs a hybrid 6th-order Galerkin/finite difference discretization of the governing equations in space and a 4th-order Runge--Kutta discretization in time, where the emerging singularity is captured on specially designed adaptive (moving) meshes that are dynamically adjusted to the evolving solutions. With a maximum effective resolution of over (3 x 10¹²)² near the point of the singularity, the simulations are able to advance the solution to a point that is asymptotically close to the predicted singularity time, while achieving a pointwise relative error of O(10⁻⁴) in the vorticity vector and obtaining a 3 x 10⁸-fold increase in the maximum vorticity. The numerical data are checked against all major blowup/nonblowup criteria, including Beale-Kato-Majda, Constantin-Fefferman-Majda, and Deng-Hou-Yu, to confirm the validity of the singularity. A close scrutiny of the data near the point of the singularity also reveals a self-similar structure in the blowup, as well as a one-dimensional model which is seen to capture the essential features of the singular solutions along the solid wall, and for which existence of finite-time singularities can be established rigorously.https://resolver.caltech.edu/CaltechAUTHORS:20191202-133631090Analysis of Asymptotic Escape of Strict Saddle Sets in Manifold Optimization
https://resolver.caltech.edu/CaltechAUTHORS:20200122-133158689
Year: 2020
DOI: 10.48550/arXiv.1911.12518
In this paper, we provide some analysis on the asymptotic escape of strict saddles in manifold optimization using the projected gradient descent (PGD) algorithm. One of our main contributions is that we extend the current analysis to include non-isolated and possibly continuous saddle sets with complicated geometry. We prove that the PGD is able to escape strict critical submanifolds under certain conditions on the geometry and the distribution of the saddle point sets. We also show that the PGD may fail to escape strict saddles under weaker assumptions even if the saddle point set has zero measure and there is a uniform escape direction. We provide a counterexample to illustrate this important point. We apply this saddle analysis to the phase retrieval problem on the low-rank matrix manifold, prove that there are only a finite number of saddles, and they are strict saddles with high probability. We also show the potential application of our analysis for a broader range of manifold optimization problems.https://resolver.caltech.edu/CaltechAUTHORS:20200122-133158689A class of robust numerical methods for solving dynamical systems with multiple time scales
https://resolver.caltech.edu/CaltechAUTHORS:20200122-143531561
Year: 2020
DOI: 10.48550/arXiv.1909.04289
In this paper, we develop a class of robust numerical methods for solving dynamical systems with multiple time scales. We first represent the solution of a multiscale dynamical system as a transformation of a slowly varying solution. Then, under the scale separation assumption, we provide a systematic way to construct the transformation map and derive the dynamic equation for the slowly varying solution. We also provide the convergence analysis of the proposed method. Finally, we present several numerical examples, including ODE system with three and four separated time scales to demonstrate the accuracy and efficiency of the proposed method. Numerical results verify that our method is robust in solving ODE systems with multiple time scale, where the time step does not depend on the multiscale parameters.https://resolver.caltech.edu/CaltechAUTHORS:20200122-143531561A Minimal Mechanosensing Model Predicts Keratocyte Evolution on Flexible Substrates
https://resolver.caltech.edu/CaltechAUTHORS:20200123-103312153
Year: 2020
DOI: 10.1098/rsif.2020.0175
PMCID: PMC7276546
A mathematical model is proposed for shape evolution and locomotion of fish epidermal keratocytes on elastic substrates. The model is based on mechanosensing concepts: cells apply contractile forces onto the elastic substrate, while cell shape evolution depends locally on the substrate stress generated by themselves or external mechanical stimuli acting on the substrate. We use the level set method to study the behaviour of the model numerically, and predict a number of distinct phenomena observed in experiments, such as (i) symmetry breaking from the stationary centrosymmetric to the well-known steadily propagating crescent shape, (ii) asymmetric bipedal oscillations and travelling waves in the lamellipodium leading edge, (iii) response to remote mechanical stress externally applied to the substrate (tensotaxis) and (iv) changing direction of motion towards an interface with a rigid substrate (durotaxis).https://resolver.caltech.edu/CaltechAUTHORS:20200123-103312153A prototype knockoff filter for group selection with FDR control
https://resolver.caltech.edu/CaltechAUTHORS:20200124-145020991
Year: 2020
DOI: 10.1093/imaiai/iaz012
In many applications, we need to study a linear regression model that consists of a response variable and a large number of potential explanatory variables, and determine which variables are truly associated with the response. In Foygel Barber & Candès (2015, Ann. Statist., 43, 2055–2085), the authors introduced a new variable selection procedure called the knockoff filter to control the false discovery rate (FDR) and proved that this method achieves exact FDR control. In this paper, we propose a prototype knockoff filter for group selection by extending the Reid–Tibshirani (2016, Biostatistics, 17, 364–376) prototype method. Our prototype knockoff filter improves the computational efficiency and statistical power of the Reid–Tibshirani prototype method when it is applied for group selection. In some cases when the group features are spanned by one or a few hidden factors, we demonstrate that the Principal Component Analysis (PCA) prototype knockoff filter outperforms the Dai–Foygel Barber (2016, 33rd International Conference on Machine Learning (ICML 2016)) group knockoff filter. We present several numerical experiments to compare our prototype knockoff filter with the Reid–Tibshirani prototype method and the group knockoff filter. We have also conducted some analysis of the knockoff filter. Our analysis reveals that some knockoff path method statistics, including the Lasso path statistic, may lead to loss of power for certain design matrices and a specially designed response even if their signal strengths are still relatively strong.https://resolver.caltech.edu/CaltechAUTHORS:20200124-145020991Function approximation via the subsampled Poincaré inequality
https://resolver.caltech.edu/CaltechAUTHORS:20200122-075925100
Year: 2021
DOI: 10.3934/dcds.2020296
Function approximation and recovery via some sampled data have long been studied in a wide array of applied mathematics and statistics fields. Analytic tools, such as the Poincaré inequality, have been handy for estimating the approximation errors in different scales. The purpose of this paper is to study a generalized Poincaré inequality, where the measurement function is of subsampled type, with a small but non-zero lengthscale that will be made precise. Our analysis identifies this inequality as a basic tool for function recovery problems. We discuss and demonstrate the optimality of the inequality concerning the subsampled lengthscale, connecting it to existing results in the literature. In application to function approximation problems, the approximation accuracy using different basis functions and under different regularity assumptions is established by using the subsampled Poincaré inequality. We observe that the error bound blows up as the subsampled lengthscale approaches zero, due to the fact that the underlying function is not regular enough to have well-defined pointwise values. A weighted version of the Poincaré inequality is proposed to address this problem; its optimality is also discussed.https://resolver.caltech.edu/CaltechAUTHORS:20200122-075925100Finite Time Blowup of 2D Boussinesq and 3D Euler Equations with C^(1,α) Velocity and Boundary
https://resolver.caltech.edu/CaltechAUTHORS:20200122-142818756
Year: 2021
DOI: 10.1007/s00220-021-04067-1
Inspired by the numerical evidence of a potential 3D Euler singularity by Luo-Hou [30, 31] and the recent breakthrough by Elgindi [11] on the singularity formation of the 3D Euler equation without swirl with C^(1,α) initial data for the velocity, we prove the finite time singularity for the 2D Boussinesq and the 3D axisymmetric Euler equations in the presence of boundary with C^(1,α) initial data for the velocity (and density in the case of Boussinesq equations). Our finite time blowup solution for the 3D Euler equations and the singular solution considered in [30, 31] share many essential features, including the symmetry properties of the solution, the flow structure, and the sign of the solution in each quadrant, except that we use C^(1,α) initial data for the velocity field. We use a dynamic rescaling formulation and follow the general framework of analysis developed by Elgindi in [11]. We also use some strategy proposed in our recent joint work with Huang in [7] and adopt several methods of analysis in [11] to establish the linear and nonlinear stability of an approximate self-similar profile. The nonlinear stability enables us to prove that the solution of the 3D Euler equations or the 2D Boussinesq equations with C^(1,α) initial data will develop a finite time singularity. Moreover, the velocity field has finite energy before the singularity time.https://resolver.caltech.edu/CaltechAUTHORS:20200122-142818756On the Finite Time Blowup of the De Gregorio Model for the 3D Euler Equations
https://resolver.caltech.edu/CaltechAUTHORS:20200123-101534089
Year: 2021
DOI: 10.1002/cpa.21991
We present a novel method of analysis and prove finite time asymptotically self‐similar blowup of the De Gregorio model [13, 14] for some smooth initial data on the real line with compact support. We also prove self‐similar blowup results for the generalized De Gregorio model [41] for the entire range of parameter on ℝ or S¹ for Hölder‐continuous initial data with compact support. Our strategy is to reformulate the problem of proving finite time asymptotically self‐similar singularity into the problem of establishing the nonlinear stability of an approximate self‐similar profile with a small residual error using the dynamic rescaling equation. We use the energy method with appropriate singular weight functions to extract the damping effect from the linearized operator around the approximate self‐similar profile and take into account cancellation among various nonlocal terms to establish stability analysis. We remark that our analysis does not rule out the possibility that the original De Gregorio model is well‐posed for smooth initial data on a circle. The method of analysis presented in this paper provides a promising new framework to analyze finite time singularity of nonlinear nonlocal systems of partial differential equations.https://resolver.caltech.edu/CaltechAUTHORS:20200123-101534089Exponential Convergence for Multiscale Linear Elliptic PDEs via Adaptive Edge Basis Functions
https://resolver.caltech.edu/CaltechAUTHORS:20210922-170252834
Year: 2021
DOI: 10.1137/20m1352922
In this paper, we introduce a multiscale framework based on adaptive edge basis functions to solve second-order linear elliptic PDEs with rough coefficients. One of the main results is that we prove that the proposed multiscale method achieves nearly exponential convergence in the approximation error with respect to the computational degrees of freedom. Our strategy is to perform an energy orthogonal decomposition of the solution space into a coarse scale component comprising a-harmonic functions in each element of the mesh, and a fine scale component named the bubble part that can be computed locally and efficiently. The coarse scale component depends entirely on function values on edges. Our approximation on each edge is made in the Lions--Magenes space H_₀₀^(1/2)(e), which we will demonstrate to be a natural and powerful choice. We construct edge basis functions using local oversampling and singular value decomposition. When local information of the right-hand side is adaptively incorporated into the edge basis functions, we prove a nearly exponential convergence rate of the approximation error. Numerical experiments validate and extend our theoretical analysis; in particular, we observe no obvious degradation in accuracy for high-contrast media problems.https://resolver.caltech.edu/CaltechAUTHORS:20210922-170252834Multiscale Invertible Generative Networks for High-Dimensional Bayesian Inference
https://resolver.caltech.edu/CaltechAUTHORS:20220622-203755587
Year: 2021
DOI: 10.48550/arXiv.2105.05489
We propose a Multiscale Invertible Generative Network (MsIGN) and associated training algorithm that leverages multiscale structure to solve high-dimensional Bayesian inference. To address the curse of dimensionality, MsIGN exploits the low-dimensional nature of the posterior, and generates samples from coarse to fine scale (low to high dimension) by iteratively upsampling and refining samples. MsIGN is trained in a multi-stage manner to minimize the Jeffreys divergence, which avoids mode dropping in high-dimensional cases. On two high-dimensional Bayesian inverse problems, we show superior performance of MsIGN over previous approaches in posterior approximation and multiple mode capture. On the natural image synthesis task, MsIGN achieves superior performance in bits-per-dimension over baseline models and yields great interpret-ability of its neurons in intermediate layers.https://resolver.caltech.edu/CaltechAUTHORS:20220622-203755587Multiscale Elliptic PDE Upscaling and Function Approximation via Subsampled Data
https://resolver.caltech.edu/CaltechAUTHORS:20220801-490095000
Year: 2022
DOI: 10.1137/20m1372214
There is an intimate connection between numerical upscaling of multiscale PDEs and scattered data approximation of heterogeneous functions: the coarse variables selected for deriving an upscaled equation (in the former) correspond to the sampled information used for approximation (in the latter). As such, both problems can be thought of as recovering a target function based on some coarse data that are either artificially chosen by an upscaling algorithm or determined by some physical measurement process. The purpose of this paper is then to study, under such a setup and for a specific elliptic problem, how the lengthscale of the coarse data, which we refer to as the subsampled lengthscale, influences the accuracy of recovery, given limited computational budgets. Our analysis and experiments identify that reducing the subsampling lengthscale may improve the accuracy, implying a guiding criterion for coarse-graining or data acquisition in this computationally constrained scenario, especially leading to direct insights for the implementation of the Gamblets method in the numerical homogenization literature. Moreover, reducing the lengthscale to zero may lead to a blow-up of approximation error if the target function does not have enough regularity, suggesting the need for a stronger prior assumption on the target function to be approximated. We introduce a singular weight function to deal with it, both theoretically and numerically. This work sheds light on the interplay of the lengthscale of coarse data, the computational costs, the regularity of the target function, and the accuracy of approximations and numerical simulations.https://resolver.caltech.edu/CaltechAUTHORS:20220801-490095000Asymptotically self-similar blowup of the Hou-Luo model for the 3D Euler equations
https://resolver.caltech.edu/CaltechAUTHORS:20221128-494241100.17
Year: 2022
DOI: 10.1007/s40818-022-00140-7
Inspired by the numerical evidence of a potential 3D Euler singularity [54, 55], we prove finite time singularity from smooth initial data for the HL model introduced by Hou-Luo in [54, 55] for the 3D Euler equations with boundary. Our finite time blowup solution for the HL model and the singular solution considered in [54, 55] share some essential features, including similar blowup exponents, symmetry properties of the solution, and the sign of the solution. We use a dynamical rescaling formulation and the strategy proposed in our recent work in [11] to establish the nonlinear stability of an approximate self-similar profile. The nonlinear stability enables us to prove that the solution of the HL model with smooth initial data and finite energy will develop a focusing asymptotically self-similar singularity in finite time. Moreover the self-similar profile is unique within a small energy ball and the C_γ norm of the density θ with γ ≈ 1/3 is uniformly bounded up to the singularity time.
[11] Chen, J., Hou, T.Y., Huang, D.: On the finite time blowup of the De Gregorio model for the 3D Euler equations. Communications on Pure and Applied Mathematics 74(6), 1282–1350 (2021)
[54] Luo, G., Hou, T.: Toward the finite-time blowup of the 3D incompressible Euler equations: a numerical investigation. SIAM Multiscale Modeling and Simulation 12(4), 1722–1776 (2014)
[55] Luo, G., Hou, T.Y.: Potentially singular solutions of the 3d axisymmetric euler equations. Proceedings of the National Academy of Sciences 111(36), 12968–12973 (2014)https://resolver.caltech.edu/CaltechAUTHORS:20221128-494241100.17A potential two-scale traveling wave singularity for 3D incompressible Euler equations
https://resolver.caltech.edu/CaltechAUTHORS:20220607-425321000
Year: 2022
DOI: 10.1016/j.physd.2022.133257
In this paper, we investigate a potential two-scale traveling wave singularity of the 3D incompressible axisymmetric Euler equations with smooth initial data of finite energy. The two-scale feature is characterized by the property that the center of the traveling wave approaches to the origin at a slower rate than the rate of the collapse of the singularity. The driving mechanism for this potential singularity is due to two antisymmetric vortex dipoles that generate a strong shearing layer in both the radial and axial velocity fields. Without any viscous regularization, the 3D Euler equations develop an additional small scale characterizing the thickness of the sharp front. In order to stabilize the rapidly decreasing thickness of the sharp front, we apply a vanishing first order numerical viscosity to the Euler equations. We present numerical evidence that the 3D Euler equations with this first order numerical viscosity develop a locally self-similar blowup at the origin.https://resolver.caltech.edu/CaltechAUTHORS:20220607-425321000Potentially Singular Behavior of the 3D Navier-Stokes Equations
https://resolver.caltech.edu/CaltechAUTHORS:20220919-258327700
Year: 2022
DOI: 10.1007/s10208-022-09578-4
Whether the 3D incompressible Navier-Stokes equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this paper, we present some new numerical evidence that the incompressible axisymmetric Navier-Stokes equations with smooth initial data of finite energy seem to develop potentially singular behavior at the origin. This potentially singular behavior is induced by a potential finite time singularity of the 3D Euler equations that we reported in the companion paper (arXiv:2107.05870). We present numerical evidence that the 3D Navier--Stokes equations develop nearly self-similar singular scaling properties with maximum vorticity increased by a factor of 10⁷. We have applied several blow-up criteria to study the potentially singular behavior of the Navier--Stokes equations. The Beale-Kato-Majda blow-up criterion and the blow-up criteria based on the growth of enstrophy and negative pressure seem to imply that the Navier--Stokes equations using our initial data develop a potential finite time singularity. We have also examined the Ladyzhenskaya-Prodi-Serrin regularity criteria. Our numerical results for the cases of (p,q) = (4,8), (6,4), (9,3) and (p,q) = (∞,2) provide strong evidence for the potentially singular behavior of the Navier--Stokes equations. Our numerical study shows that while the global L^3 norm of the velocity grows very slowly, the localized version of the L³ norm of the velocity experiences rapid dynamic growth relative to the localized L³ norm of the initial velocity. This provides further evidence for the potentially singular behavior of the NavieStokes equations.https://resolver.caltech.edu/CaltechAUTHORS:20220919-258327700Potential Singularity of the 3D Euler Equations in the Interior Domain
https://resolver.caltech.edu/CaltechAUTHORS:20220919-922490500
Year: 2022
DOI: 10.1007/s10208-022-09585-5
Whether the 3D incompressible Euler equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this paper, we present some new numerical evidence that the 3D axisymmetric incompressible Euler equations with smooth initial data of finite energy develop a potential finite time singularity at the origin. This potential singularity is different from the blow-up scenario revealed by Luo and Hou (111:12968–12973, 2014) and (12:1722–1776, 2014), which occurs on the boundary. Our initial condition has a simple form and shares several attractive features of a more sophisticated initial condition constructed by Hou and Huang in (arXiv:2102.06663, 2021) and (435:133257, 2022). One important difference between these two blow-up scenarios is that the solution for our initial data has a one-scale structure instead of a two-scale structure reported in Hou and Huang (arXiv:2102.06663, 2021) and (435:133257, 2022). More importantly, the solution seems to develop nearly self-similar scaling properties that are compatible with those of the 3D Navier–Stokes equations. We will present numerical evidence that the 3D Euler equations seem to develop a potential finite time singularity. Moreover, the nearly self-similar profile seems to be very stable to the small perturbation of the initial data.https://resolver.caltech.edu/CaltechAUTHORS:20220919-922490500Fourier Continuation for Exact Derivative Computation in Physics-Informed Neural Operators
https://resolver.caltech.edu/CaltechAUTHORS:20221221-004750416
Year: 2022
DOI: 10.48550/arXiv.2211.15960
The physics-informed neural operator (PINO) is a machine learning architecture that has shown promising empirical results for learning partial differential equations. PINO uses the Fourier neural operator (FNO) architecture to overcome the optimization challenges often faced by physics-informed neural networks. Since the convolution operator in PINO uses the Fourier series representation, its gradient can be computed exactly on the Fourier space. While Fourier series cannot represent nonperiodic functions, PINO and FNO still have the expressivity to learn nonperiodic problems with Fourier extension via padding. However, computing the Fourier extension in the physics-informed optimization requires solving an ill-conditioned system, resulting in inaccurate derivatives which prevent effective optimization. In this work, we present an architecture that leverages Fourier continuation (FC) to apply the exact gradient method to PINO for nonperiodic problems. This paper investigates three different ways that FC can be incorporated into PINO by testing their performance on a 1D blowup problem. Experiments show that FC-PINO outperforms padded PINO, improving equation loss by several orders of magnitude, and it can accurately capture the third order derivatives of nonsmooth solution functions.https://resolver.caltech.edu/CaltechAUTHORS:20221221-004750416Asymptotic Escape of Spurious Critical Points on the Low-rank Matrix Manifold
https://resolver.caltech.edu/CaltechAUTHORS:20221221-220346410
Year: 2022
DOI: 10.48550/arXiv.2107.09207
We show that on the manifold of fixed-rank and symmetric positive semi-definite matrices, the Riemannian gradient descent algorithm almost surely escapes some spurious critical points on the boundary of the manifold. Our result is the first to partially overcome the incompleteness of the low-rank matrix manifold without changing the vanilla Riemannian gradient descent algorithm. The spurious critical points are some rank-deficient matrices that capture only part of the eigen components of the ground truth. Unlike classical strict saddle points, they exhibit very singular behavior. We show that using the dynamical low-rank approximation and a rescaled gradient flow, some of the spurious critical points can be converted to classical strict saddle points in the parameterized domain, which leads to the desired result. Numerical experiments are provided to support our theoretical findings.https://resolver.caltech.edu/CaltechAUTHORS:20221221-220346410Fast Global Convergence for Low-rank Matrix Recovery via Riemannian Gradient Descent with Random Initialization
https://resolver.caltech.edu/CaltechAUTHORS:20221221-220354531
Year: 2022
DOI: 10.48550/arXiv.2012.15467
In this paper, we propose a new global analysis framework for a class of low-rank matrix recovery problems on the Riemannian manifold. We analyze the global behavior for the Riemannian optimization with random initialization. We use the Riemannian gradient descent algorithm to minimize a least squares loss function, and study the asymptotic behavior as well as the exact convergence rate. We reveal a previously unknown geometric property of the low-rank matrix manifold, which is the existence of spurious critical points for the simple least squares function on the manifold. We show that under some assumptions, the Riemannian gradient descent starting from a random initialization with high probability avoids these spurious critical points and only converges to the ground truth in nearly linear convergence rate, i.e. O(log(1/ϵ) + log(n)) iterations to reach an ϵ-accurate solution. We use two applications as examples for our global analysis. The first one is a rank-1 matrix recovery problem. The second one is a generalization of the Gaussian phase retrieval problem. It only satisfies the weak isometry property, but has behavior similar to that of the first one except for an extra saddle set. Our convergence guarantee is nearly optimal and almost dimension-free, which fully explains the numerical observations. The global analysis can be potentially extended to other data problems with random measurement structures and empirical least squares loss functions.https://resolver.caltech.edu/CaltechAUTHORS:20221221-220354531Stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth data
https://resolver.caltech.edu/CaltechAUTHORS:20230227-191437852
Year: 2023
DOI: 10.48550/arXiv.2210.07191
Inspired by the numerical evidence of a potential 3D Euler singularity [Luo-Hou-14a, Luo-Hou-14b], we prove finite time blowup of the 2D Boussinesq and 3D axisymmetric Euler equations with smooth initial data of finite energy and boundary. There are several essential difficulties in proving finite time blowup of 3D Euler with smooth initial data. One of the essential difficulties is to control a number of nonlocal terms that do not seem to offer any damping effect. Another essential difficulty is that the strong advection normal to the boundary introduces a large growth factor for the perturbation if we use weighted L² estimates. We overcome this difficulty by using a combination of a weighted L∞ norm and a weighted C^(1/2) norm, and develop sharp functional inequalities using the symmetry properties of the kernels and some techniques from optimal transport. Moreover we decompose the linearized operator into a leading order operator plus a finite rank operator. The leading order operator is designed in such a way that we can obtain sharp stability estimates. The contribution from the finite rank operator can be captured by an auxiliary variable and its contribution to linear stability can be estimated by constructing approximate solution in space-time. This enables us to establish nonlinear stability of the approximate self-similar profile and prove stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth initial data and boundary.https://resolver.caltech.edu/CaltechAUTHORS:20230227-191437852Potential Singularity of the Axisymmetric Euler Equations with C^α Initial Vorticity for A Large Range of α. Part I: the 3-Dimensional Case
https://resolver.caltech.edu/CaltechAUTHORS:20230227-194424192
Year: 2023
DOI: 10.48550/arXiv.2212.11912
n Part I of our sequence of 2 papers, we provide numerical evidence for a potential finite-time self-similar singularity of the 3D axisymmetric Euler equations with no swirl and with C^α initial vorticity for a large range of α. We employ an adaptive mesh method using a highly effective mesh to resolve the potential singularity sufficiently close to the potential blow-up time. Resolution study shows that our numerical method is at least second-order accurate. Scaling analysis and the dynamic rescaling formulation are presented to quantitatively study the scaling properties of the potential singularity. We demonstrate that this potential blow-up is stable with respect to the perturbation of initial data. Our study shows that the 3D Euler equations with our initial data develop finite-time blow-up when the Hölder exponent α is smaller than some critical value α^∗. By properly rescaling the initial data in the z-axis, this upper bound for potential blow-up α^∗ can asymptotically approach 1/3. Compared with Elgindi's blow-up result in a similar setting [15], our potential blow-up scenario has a different Hölder continuity property in the initial data and the scaling properties of the two initial data are also quite different.https://resolver.caltech.edu/CaltechAUTHORS:20230227-194424192Potential singularity of the 3D Euler equations in the interior domain
https://resolver.caltech.edu/CaltechAUTHORS:20230227-192413744
Year: 2023
DOI: 10.48550/arXiv.2107.05870
Whether the 3D incompressible Euler equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this paper, we present some new numerical evidence that the 3D axisymmetric incompressible Euler equations with smooth initial data of finite energy develop a potential finite time singularity at the origin. This potential singularity is different from the blow-up scenario revealed by Luo-Hou in [31, 32], which occurs on the boundary. Our initial condition has a simple form and shares several attractive features of a more sophisticated initial condition constructed by Hou-Huang in [20, 21]. One important difference between these two blow-up scenarios is that the solution for our initial data has a one-scale structure instead of a two-scale structure reported in \cite{Hou-Huang-2021,Hou-Huang-2022}. More importantly, the solution seems to develop nearly self-similar scaling properties that are compatible with those of the 3D Navier-Stokes equations. We will present numerical evidence that the 3D Euler equations seem to develop a potential finite time singularity. Moreover, the nearly self-similar profile seems to be very stable to the small perturbation of the initial data.https://resolver.caltech.edu/CaltechAUTHORS:20230227-192413744Exponentially Convergent Multiscale Finite Element Method
https://resolver.caltech.edu/CaltechAUTHORS:20230227-194420642
Year: 2023
DOI: 10.48550/arXiv.2212.00823
We provide a concise review of the exponentially convergent multiscale finite element method (ExpMsFEM) for efficient model reduction of PDEs in heterogeneous media without scale separation and in high-frequency wave propagation. ExpMsFEM is built on the non-overlapped domain decomposition in the classical MsFEM while enriching the approximation space systematically to achieve a nearly exponential convergence rate regarding the number of basis functions. Unlike most generalizations of MsFEM in the literature, ExpMsFEM does not rely on any partition of unity functions.
In general, it is necessary to use function representations dependent on the right-hand side to break the algebraic Kolmogorov n-width barrier to achieve exponential convergence. Indeed, there are online and offline parts in the function representation provided by ExpMsFEM. The online part depends on the right-hand side locally and can be computed in parallel efficiently. The offline part contains basis functions that are used in the Galerkin method to assemble the stiffness matrix; they are all independent of the right-hand side, so the stiffness matrix can be used repeatedly in multi-query scenarios.https://resolver.caltech.edu/CaltechAUTHORS:20230227-194420642On stability and instability of C^(1,α) singular solutions to the 3D Euler and 2D Boussinesq equations
https://resolver.caltech.edu/CaltechAUTHORS:20230227-194417020
Year: 2023
DOI: 10.48550/arXiv.2206.01296
Singularity formation of the 3D incompressible Euler equations is known to be extremely challenging. In [18], Elgindi proved that the 3D axisymmetric Euler equations with no swirl and C^(1,α) initial velocity develops a finite time singularity. Inspired by Elgindi's work, we proved that the 3D axisymmetric Euler and 2D Boussinesq equations with C^(1,α) initial velocity and boundary develop a stable asymptotically (or approximately) self-similar finite time singularity [8]. On the other hand, the authors of [35,52] recently showed that blowup solutions to the 3D Euler equations are hydrodynamically unstable. The instability results obtained in [35,52] require some strong regularity assumption on the initial data, which is not satisfied by the C^(1,α) velocity field. In this paper, we generalize the analysis of [8,18,35,52] to show that the blowup solutions of the 3D Euler and 2D Boussinesq equations with C^(1,α) velocity are unstable under the notion of stability introduced in [35,52]. These two seemingly contradictory results reflect the difference of the two approaches in studying the stability of 3D Euler blowup solutions. The stability analysis of the blowup solution obtained in [8,18] is based on the stability of a dynamically rescaled blowup profile in space and time, which is nonlinear in nature. The linear stability analysis in [35,52] is performed by directly linearizing the 3D Euler equations around a blowup solution in the original variables. It does not take into account the changes in the blowup time, the dynamic changes of the rescaling rate of the perturbed blowup profile and the blowup exponent of the original 3D Euler equations using a perturbed initial condition when there is an approximate self-similar blowup profile. Such information has been used in an essential way in establishing the nonlinear stability of the blowup profile in [8,18,19].https://resolver.caltech.edu/CaltechAUTHORS:20230227-194417020Potential Singularity of the Axisymmetric Euler Equations with C^α Initial Vorticity for A Large Range of α. Part II: the N-Dimensional Case
https://resolver.caltech.edu/CaltechAUTHORS:20230227-194427740
Year: 2023
DOI: 10.48550/arXiv.2212.11924
In Part II of this sequence to our previous paper for the 3-dimensional Euler equations [8], we investigate potential singularity of the n-diemnsional axisymmetric Euler equations with C^α initial vorticity for a large range of α. We use the adaptive mesh method to solve the n-dimensional axisymmetric Euler equations and use the scaling analysis and dynamic rescaling method to examine the potential blow-up and capture its self-similar profile. Our study shows that the n-dimensional axisymmetric Euler equations with our initial data develop finite-time blow-up when the Hölder exponent α < α^∗, and this upper bound α∗ can asymptotically approach 1 − 2/n. Moreover, we introduce a stretching parameter δ along the z-direction. Based on a few assumptions inspired by our numerical experiments, we obtain α^∗ = 1 − 2/n by studying the limiting case of δ→0. For the general case, we propose a relatively simple one-dimensional model and numerically verify its approximation to the n-dimensional Euler equations. This one-dimensional model sheds useful light to our understanding of the blowup mechanism for the n-dimensional Euler equations. As shown in [8], the scaling behavior and regularity properties of our initial data are quite different from those of the initial data considered by Elgindi in [6].https://resolver.caltech.edu/CaltechAUTHORS:20230227-194427740The potentially singular behavior of the 3D Navier-Stokes equations
https://resolver.caltech.edu/CaltechAUTHORS:20230227-193545252
Year: 2023
DOI: 10.48550/arXiv.2107.06509
Whether the 3D incompressible Navier-Stokes equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this paper, we present some new numerical evidence that the incompressible axisymmetric Navier-Stokes equations with smooth initial data of finite energy seem to develop potentially singular behavior at the origin. This potentially singular behavior is induced by a potential finite time singularity of the 3D Euler equations that we reported in the companion paper (arXiv:2107.05870). We present numerical evidence that the 3D Navier--Stokes equations develop nearly self-similar singular scaling properties with maximum vorticity increased by a factor of 10⁷. We have applied several blow-up criteria to study the potentially singular behavior of the Navier--Stokes equations. The Beale-Kato-Majda blow-up criterion and the blow-up criteria based on the growth of enstrophy and negative pressure seem to imply that the Navier--Stokes equations using our initial data develop a potential finite time singularity. We have also examined the Ladyzhenskaya-Prodi-Serrin regularity criteria. Our numerical results for the cases of (p,q) = (4,8), (6,4), (9,3) and (p,q) = (∞,2) provide strong evidence for the potentially singular behavior of the Navier--Stokes equations. Our numerical study shows that while the global L^3 norm of the velocity grows very slowly, the localized version of the L³ norm of the velocity experiences rapid dynamic growth relative to the localized L³ norm of the initial velocity. This provides further evidence for the potentially singular behavior of the NavieStokes equations.https://resolver.caltech.edu/CaltechAUTHORS:20230227-193545252Potential Singularity Formation of Incompressible Axisymmetric Euler Equations with Degenerate Viscosity Coefficients
https://resolver.caltech.edu/CaltechAUTHORS:20230530-441187700.9
Year: 2023
DOI: 10.1137/22m1470906
In this paper, we present strong numerical evidence that the incompressible axisymmetric Euler equations with degenerate viscosity coefficients and smooth initial data of finite energy develop a potential finite-time locally self-similar singularity at the origin. An important feature of this potential singularity is that the solution develops a two-scale traveling wave that travels toward the origin. The two-scale feature is characterized by the scaling property that the center of the traveling wave is located at a ring of radius O((T-t)½) surrounding the symmetry axis while the thickness of the ring collapses at a rate O(T-t). The driving mechanism for this potential singularity is due to an antisymmetric vortex dipole that generates a strong shearing layer in both the radial and axial velocity fields. Without the viscous regularization, the three-dimensional Euler equations develop a sharp front and some shearing instability in the far field. On the other hand, the Navier–Stokes equations with a constant viscosity coefficient regularize the two-scale solution structure and do not develop a finite-time singularity for the same initial data.https://resolver.caltech.edu/CaltechAUTHORS:20230530-441187700.9Exponentially Convergent Multiscale Finite Element Method
https://resolver.caltech.edu/CaltechAUTHORS:20230515-138520000.12
Year: 2023
DOI: 10.1007/s42967-023-00260-2
We provide a concise review of the exponentially convergent multiscale finite element method (ExpMsFEM) for efficient model reduction of PDEs in heterogeneous media without scale separation and in high-frequency wave propagation. The ExpMsFEM is built on the non-overlapped domain decomposition in the classical MsFEM while enriching the approximation space systematically to achieve a nearly exponential convergence rate regarding the number of basis functions. Unlike most generalizations of the MsFEM in the literature, the ExpMsFEM does not rely on any partition of unity functions. In general, it is necessary to use function representations dependent on the right-hand side to break the algebraic Kolmogorov n-width barrier to achieve exponential convergence. Indeed, there are online and offline parts in the function representation provided by the ExpMsFEM. The online part depends on the right-hand side locally and can be computed in parallel efficiently. The offline part contains basis functions that are used in the Galerkin method to assemble the stiffness matrix; they are all independent of the right-hand side, so the stiffness matrix can be used repeatedly in multi-query scenarios.https://resolver.caltech.edu/CaltechAUTHORS:20230515-138520000.12Blowup analysis for a quasi-exact 1D model of 3D Euler and Navier–Stokes
https://authors.library.caltech.edu/records/ca73j-gkv41
Year: 2024
DOI: 10.1088/1361-6544/ad1c2f
<div class="article-text wd-jnl-art-abstract cf">
<p>We study the singularity formation of a quasi-exact 1D model proposed by Hou and Li (2008 <em>Commun. Pure Appl. Math.</em><strong>61</strong> 661–97). This model is based on an approximation of the axisymmetric Navier–Stokes equations in the <em>r</em> direction. The solution of the 1D model can be used to construct an exact solution of the original 3D Euler and Navier–Stokes equations if the initial angular velocity, angular vorticity, and angular stream function are linear in <em>r</em>. This model shares many intrinsic properties similar to those of the 3D Euler and Navier–Stokes equations. It captures the competition between advection and vortex stretching as in the 1D De Gregorio (De Gregorio 1990 <em>J. Stat. Phys.</em><strong>59</strong> 1251–63; De Gregorio 1996 <em>Math. Methods Appl. Sci.</em><strong>19</strong> 1233–55) model. We show that the inviscid model with weakened advection and smooth initial data or the original 1D model with Hölder continuous data develops a self-similar blowup. We also show that the viscous model with weakened advection and smooth initial data develops a finite time blowup. To obtain sharp estimates for the nonlocal terms, we perform an exact computation for the low-frequency Fourier modes and extract damping in leading order estimates for the high-frequency modes using singularly weighted norms in the energy estimates. The analysis for the viscous case is more subtle since the viscous terms produce some instability if we just use singular weights. We establish the blowup analysis for the viscous model by carefully designing an energy norm that combines a singularly weighted energy norm and a sum of high-order Sobolev norms.</p>
</div>https://authors.library.caltech.edu/records/ca73j-gkv41