Abstract: 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.

Publication: Foundations of Computational MathematicsISSN: 1615-3375

ID: CaltechAUTHORS:20220919-922490500

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Abstract: 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.

Publication: Foundations of Computational MathematicsISSN: 1615-3375

ID: CaltechAUTHORS:20220919-258327700

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Abstract: 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.

Publication: Physica D Vol.: 435ISSN: 0167-2789

ID: CaltechAUTHORS:20220607-425321000

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Abstract: 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)

Publication: Annals of PDE Vol.: 8 No.: 2 ISSN: 2524-5317

ID: CaltechAUTHORS:20221128-494241100.17

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Abstract: 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.

Publication: Multiscale Modeling and Simulation Vol.: 20 No.: 1 ISSN: 1540-3459

ID: CaltechAUTHORS:20220801-490095000

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Abstract: 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.

Publication: Proceedings of Machine Learning Research Vol.: 139ISSN: 2640-3498

ID: CaltechAUTHORS:20220622-203755587

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Abstract: 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.

Publication: Multiscale Modeling and Simulation Vol.: 19 No.: 2 ISSN: 1540-3459

ID: CaltechAUTHORS:20210922-170252834

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Abstract: 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.

Publication: Communications on Pure and Applied Mathematics Vol.: 74 No.: 6 ISSN: 0010-3640

ID: CaltechAUTHORS:20200123-101534089

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Abstract: 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.

Publication: Communications in Mathematical Physics Vol.: 383 No.: 3 ISSN: 0010-3616

ID: CaltechAUTHORS:20200122-142818756

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Abstract: 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.

Publication: Discrete and Continuous Dynamical Systems - Series A Vol.: 41 No.: 1 ISSN: 1553-5231

ID: CaltechAUTHORS:20200122-075925100

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Abstract: 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.

Publication: Information and Inference Vol.: 9 No.: 2 ISSN: 2049-8772

ID: CaltechAUTHORS:20200124-145020991

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Abstract: 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).

Publication: Journal of the Royal Society Interface Vol.: 17 No.: 166 ISSN: 1742-5689

ID: CaltechAUTHORS:20200123-103312153

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Abstract: 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.

Publication: SIAM Review Vol.: 61 No.: 4 ISSN: 0036-1445

ID: CaltechAUTHORS:20191202-133631090

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Abstract: 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.

Publication: Computational Mechanics Vol.: 64 No.: 2 ISSN: 0178-7675

ID: CaltechAUTHORS:20190620-093003600

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Abstract: 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.

Publication: Multiscale Modeling and Simulation Vol.: 17 No.: 2 ISSN: 1540-3459

ID: CaltechAUTHORS:20190711-100817562

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Abstract: 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.

Publication: Information and Inference Vol.: 8 No.: 2 ISSN: 2049-8772

ID: CaltechAUTHORS:20191114-104110525

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Abstract: 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.

Publication: Multiscale Modeling and Simulation Vol.: 17 No.: 1 ISSN: 1540-3459

ID: CaltechAUTHORS:20190416-073721627

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Abstract: 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.

Publication: Journal of Nonlinear Science Vol.: 28 No.: 6 ISSN: 0938-8974

ID: CaltechAUTHORS:20170306-105506989

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Abstract: 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.

Publication: Nonlinearity Vol.: 31 No.: 5 ISSN: 0951-7715

ID: CaltechAUTHORS:20180405-130644881

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Abstract: 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.

Publication: Multiscale Modeling and Simulation Vol.: 16 No.: 2 ISSN: 1540-3459

ID: CaltechAUTHORS:20180718-150855095

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Abstract: 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.

Publication: Research in the Mathematical Sciences Vol.: 4 No.: 1 ISSN: 2197-9847

ID: CaltechAUTHORS:20171205-123956892

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Abstract: 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.

Publication: Communications on Pure and Applied Mathematics Vol.: 70 No.: 11 ISSN: 0010-3640

ID: CaltechAUTHORS:20170509-075902405

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Abstract: 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.

Publication: Science China Mathematics Vol.: 60 No.: 10 ISSN: 1674-7283

ID: CaltechAUTHORS:20170719-100127267

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Abstract: 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.

Publication: Journal of Computational Physics Vol.: 336ISSN: 0021-9991

ID: CaltechAUTHORS:20170427-144329075

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Abstract: 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.

Publication: Multiscale Modeling & Simulation Vol.: 15 No.: 2 ISSN: 1540-3459

ID: CaltechAUTHORS:20170413-142311390

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Abstract: 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].

Publication: Multiscale Modeling and Simulation Vol.: 15 No.: 1 ISSN: 1540-3459

ID: CaltechAUTHORS:20170413-141136299

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Abstract: 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.

Publication: Structural Control and Health Monitoring Vol.: 24 No.: 3 ISSN: 1545-2255

ID: CaltechAUTHORS:20170214-074909079

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Abstract: 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.

Publication: Journal of Scientific Computing Vol.: 69 No.: 2 ISSN: 0885-7474

ID: CaltechAUTHORS:20161103-105632603

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Abstract: 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.

Publication: Journal of Computational Physics Vol.: 320ISSN: 0021-9991

ID: CaltechAUTHORS:20160624-151830832

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Abstract: 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.

Publication: Discrete and Continuous Dynamical Systems Vol.: 36 No.: 8 ISSN: 1078-0947

ID: CaltechAUTHORS:20160315-134934235

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Abstract: 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.

Publication: Philosophical Transactions A: Mathematical, Physical and Engineering Sciences Vol.: 374 No.: 2065 ISSN: 1364-503X

ID: CaltechAUTHORS:20160315-092552095

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Abstract: 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.

Publication: Philosophical Transactions A: Mathematical, Physical and Engineering Sciences Vol.: 374 No.: 2065 ISSN: 1364-503X

ID: CaltechAUTHORS:20160315-095910709

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Abstract: 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.

Publication: Philosophical Transactions A: Mathematical, Physical and Engineering Sciences Vol.: 374 No.: 2065 ISSN: 1364-503X

ID: CaltechAUTHORS:20160315-095503958

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Abstract: 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.

Publication: Bulletin of the Intitute of Mathematics Academia Sinica Vol.: 11 No.: 1 ISSN: 2304-7909

ID: CaltechAUTHORS:20161014-130127343

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Abstract: 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.

Publication: Royal Society Open Science Vol.: 2 No.: 12 ISSN: 2054-5703

ID: CaltechAUTHORS:20160404-090937429

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Abstract: 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.

Publication: Multiscale Modeling and Simulation Vol.: 13 No.: 3 ISSN: 1540-3459

ID: CaltechAUTHORS:20151023-103928257

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Abstract: 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.

Publication: Multiscale Modeling and Simulation Vol.: 13 No.: 1 ISSN: 1540-3459

ID: CaltechAUTHORS:20150501-105112539

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Abstract: 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.

Publication: Journal of Computational Physics Vol.: 281ISSN: 0021-9991

ID: CaltechAUTHORS:20150115-154548810

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Abstract: 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.

Publication: Communications in Mathematical Sciences Vol.: 13 No.: 3 ISSN: 1539-6746

ID: CaltechAUTHORS:20150519-083002339

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Abstract: 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.

Publication: Multiscale Modeling and Simulation Vol.: 12 No.: 4 ISSN: 1540-3459

ID: CaltechAUTHORS:20150202-093859868

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Abstract: 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.

Publication: Proceedings of the National Academy of Sciences of the United States of America Vol.: 111 No.: 36 ISSN: 0027-8424

ID: CaltechAUTHORS:20140825-230506377

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Abstract: 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.

Publication: Communications in Computational Physics Vol.: 16 No.: 3 ISSN: 1815-2406

ID: CaltechAUTHORS:20150112-104708436

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Abstract: 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.

Publication: Applied and Computational Harmonic Analysis Vol.: 37 No.: 2 ISSN: 1063-5203

ID: CaltechAUTHORS:20140731-103641810

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Abstract: 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.

Publication: Journal of the Royal Society Interface Vol.: 11 No.: 98 ISSN: 1742-5689

ID: CaltechAUTHORS:20140825-134136758

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Abstract: 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.

Publication: Archive for Rational Mechanics and Analysis Vol.: 212 No.: 2 ISSN: 0003-9527

ID: CaltechAUTHORS:20140422-134241808

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Abstract: 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.

Publication: Multiscale Modeling & Simulation Vol.: 12 No.: 4 ISSN: 1540-3459

ID: CaltechAUTHORS:20150202-082208889

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Abstract: 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.

Publication: ESAIM-Mathematical Modelling and Numerical Analysis Vol.: 48 No.: 2 ISSN: 0764-583X

ID: CaltechAUTHORS:20140407-110251743

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Abstract: 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 methods

Publication: Science China Mathematics Vol.: 56 No.: 12 ISSN: 1674-7283

ID: CaltechAUTHORS:20140113-074436396

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Abstract: 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.

Publication: Journal of Computational Physics Vol.: 251ISSN: 0021-9991

ID: CaltechAUTHORS:20130830-131942975

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Abstract: 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.

Publication: Climate Dynamics Vol.: 41 No.: 5-6 ISSN: 0930-7575

ID: CaltechAUTHORS:20140822-131136037

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Abstract: 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.

Publication: Applied and Computational Harmonic Analysis Vol.: 35 No.: 2 ISSN: 1063-5203

ID: CaltechAUTHORS:20130725-100732399

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Abstract: 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.

Publication: Journal of Computational Physics Vol.: 242ISSN: 0021-9991

ID: CaltechAUTHORS:20130702-134527463

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Abstract: 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.

Publication: Journal of Computational Physics Vol.: 242ISSN: 0021-9991

ID: CaltechAUTHORS:20130701-154101667

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Abstract: 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.

Publication: Journal of Computational Physics Vol.: 232 No.: 1 ISSN: 0021-9991

ID: CaltechAUTHORS:20130104-102700776

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Abstract: 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.

Publication: Advances in Water Resources Vol.: 51ISSN: 0309-1708

ID: CaltechAUTHORS:20130305-095227632

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Abstract: 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.

Publication: Advances in Mathematics Vol.: 230 No.: 2 ISSN: 0001-8708

ID: CaltechAUTHORS:20120523-112912673

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Abstract: 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].

Publication: Discrete and Continuous Dynamical Systems Vol.: 32 No.: 5 ISSN: 1078-0947

ID: CaltechAUTHORS:20120326-132813054

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Abstract: 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.

Publication: Advances in Adaptive Data Analysis Vol.: 3 No.: 1-2 ISSN: 1793-5369

ID: CaltechAUTHORS:20120314-131024684

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Abstract: 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).

Publication: Advances in Adaptive Data Analysis Vol.: 3 No.: 1-2 ISSN: 1793-5369

ID: CaltechAUTHORS:20180904-093329630

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Abstract: 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.

Publication: Archive for Rational Mechanics and Analysis Vol.: 199 No.: 1 ISSN: 0003-9527

ID: CaltechAUTHORS:20110228-113453092

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Abstract: 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).

Publication: Advances in Adaptive Data Analysis Vol.: 3 No.: 1-2 ISSN: 1793-5369

ID: CaltechAUTHORS:20120322-080850518

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Abstract: 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.

Publication: Mathematics of Computation Vol.: 79 No.: 272 ISSN: 0025-5718

ID: CaltechAUTHORS:20101012-100934632

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Abstract: 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.

Publication: Communications in Mathematical Physics Vol.: 287 No.: 2 ISSN: 0010-3616

ID: CaltechAUTHORS:20090624-105309162

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Abstract: 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.

Publication: Communications on Pure and Applied Mathematics Vol.: 62 No.: 4 ISSN: 0010-3640

ID: CaltechAUTHORS:20090528-090507806

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Abstract: 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.

Publication: Discrete and Continuous Dynamical Systems Vol.: 23 No.: 1-2 ISSN: 1078-0947

ID: CaltechAUTHORS:HOUdcds09

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Abstract: 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.

Publication: Acta Numerica Vol.: 18ISSN: 0962-4929

ID: CaltechAUTHORS:20110321-155230291

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Abstract: 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.

Publication: Journal of Computational Mathematics Vol.: 27 No.: 4 ISSN: 0254-9409

ID: CaltechAUTHORS:20090819-164116745

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Abstract: 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.

Publication: Journal of Computational Physics Vol.: 227 No.: 21 ISSN: 0021-9991

ID: CaltechAUTHORS:HOUjcp08b

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Abstract: 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.

Publication: Journal of Computational Physics Vol.: 227 No.: 20 ISSN: 0021-9991

ID: CaltechAUTHORS:HOUjcp08a

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Abstract: 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.

Publication: Communications in Partial Differential Equation Vol.: 33 No.: 9 ISSN: 0360-5302

ID: CaltechAUTHORS:20091014-100827184

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Abstract: 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.

Publication: Computational Geosciences Vol.: 12 No.: 3 ISSN: 1420-0597

ID: CaltechAUTHORS:EFEcg08

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Abstract: 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.

Publication: Computer Methods in Applied Mechanics and Engineering Vol.: 197 No.: 43-44 ISSN: 0045-7825

ID: CaltechAUTHORS:DOScmame08

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Abstract: 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.

Publication: Communications on Pure and Applied Mathematics Vol.: 61 No.: 5 ISSN: 0010-3640

ID: CaltechAUTHORS:20160322-093035560

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Abstract: 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.

Publication: Multiscale Modeling and Simulation Vol.: 6 No.: 4 ISSN: 1540-3459

ID: CaltechAUTHORS:HOUmms08

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Abstract: 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.

Publication: Journal of Computational Physics Vol.: 226 No.: 1 ISSN: 0021-9991

ID: CaltechAUTHORS:20160322-083351617

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Abstract: 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.

Publication: Journal of Computational Physics Vol.: 225 No.: 2 ISSN: 0021-9991

ID: CaltechAUTHORS:20200310-145803950

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Abstract: 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.

Publication: Discrete and Continuous Dynamical Systems Vol.: 18 No.: 4 ISSN: 1078-0947

ID: CaltechAUTHORS:20160322-091323469

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Abstract: 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.

Publication: Applied Numerical Mathematics Vol.: 57 No.: 5-7 ISSN: 0168-9274

ID: CaltechAUTHORS:20100826-114814124

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Abstract: 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).

Publication: Proceedings of the National Academy of Sciences of the United States of America Vol.: 104 No.: 16 ISSN: 0027-8424

ID: CaltechAUTHORS:HOUpnas07

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Abstract: 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.

Publication: Chinese Physics Letters Vol.: 24 No.: 1 ISSN: 0256-307X

ID: CaltechAUTHORS:TANcpl07

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Abstract: 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.

Publication: Multiscale Modeling & Simulation Vol.: 5 No.: 4 ISSN: 1540-3459

ID: CaltechAUTHORS:HOUmms06

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Abstract: 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.

Publication: Journal of Nonlinear Science Vol.: 16 No.: 6 ISSN: 0938-8974

ID: CaltechAUTHORS:20160322-090700507

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Abstract: 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.

Publication: SIAM Journal on Mathematical Analysis Vol.: 38 No.: 3 ISSN: 0036-1410

ID: CaltechAUTHORS:HOUsiamjma06

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Abstract: 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.

Publication: Communications in Computational Physics Vol.: 1 No.: 2 ISSN: 1815-2406

ID: CaltechAUTHORS:HOUccp06

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Abstract: 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.

Publication: Communications in Partial Differential Equations Vol.: 31 No.: 2 ISSN: 0360-5302

ID: CaltechAUTHORS:20170408-133541409

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Abstract: 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.

Publication: SIAM Journal on Scientific Computing Vol.: 28 No.: 2 ISSN: 1064-8275

ID: CaltechAUTHORS:EFEsiamjsc06

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Abstract: 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.

Publication: Discrete and Continuous Dynamical Systems Vol.: 13 No.: 5 ISSN: 1078-0947

ID: CaltechAUTHORS:20170408-161459048

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Abstract: 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.

Publication: International Journal for Numerical Methods in Fluids Vol.: 47 No.: 8-9 ISSN: 0271-2091

ID: CaltechAUTHORS:20180404-083725024

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Abstract: 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.

Publication: Communications in Partial Differential Equations Vol.: 30 No.: 1-2 ISSN: 0360-5302

ID: CaltechAUTHORS:20160322-075428588

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Abstract: 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.

Publication: Methods and Applications of Analysis Vol.: 11 No.: 4 ISSN: 1073-2772

ID: CaltechAUTHORS:20111013-071805594

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Abstract: 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.

Publication: Communications in Mathematical Sciences Vol.: 2 No.: 4 ISSN: 1539-6746

ID: CaltechAUTHORS:20111012-093928630

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Abstract: 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.

Publication: Communications in Mathematical Sciences Vol.: 2 No.: 2 ISSN: 1539-6746

ID: CaltechAUTHORS:20111014-070953819

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Abstract: 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.

Publication: Journal of Computational Mathematics Vol.: 22 No.: 2 ISSN: 0254-9409

ID: CaltechAUTHORS:20110819-082640450

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Abstract: 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.

Publication: Physics of Fluids Vol.: 15 No.: 1 ISSN: 1070-6631

ID: CaltechAUTHORS:HOUpof03

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Abstract: 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.

Publication: Mathematics of Computation Vol.: 72 No.: 242 ISSN: 0025-5718

ID: CaltechAUTHORS:CHEmc03

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Abstract: 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.

Publication: Acta Mathematicae Applicatae Sinica, English Series Vol.: 18 No.: 1 ISSN: 0168-9673

ID: CaltechAUTHORS:20200221-160350446

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Abstract: 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.

Publication: Journal of Fluid Mechanics Vol.: 409ISSN: 0022-1120

ID: CaltechAUTHORS:CENjfm00

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Abstract: 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.

Publication: SIAM Journal on Numerical Analysis Vol.: 37 No.: 3 ISSN: 0036-1429

ID: CaltechAUTHORS:EFEsiam00

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Abstract: 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.

Publication: Physics of Fluids Vol.: 11 No.: 9 ISSN: 1070-6631

ID: CaltechAUTHORS:CENpof99

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Abstract: 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.

Publication: Physics of Fluids Vol.: 11 No.: 5 ISSN: 1070-6631

ID: CaltechAUTHORS:CENpof99a

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Abstract: 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.

Publication: Physical Review E Vol.: 58 No.: 5 ISSN: 1063-651X

ID: CaltechAUTHORS:HOUpre98

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Abstract: 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.

Publication: Documenta Mathematica Vol.: Extra No.: Volume ISSN: 1431-0635

ID: CaltechAUTHORS:HOUdmicm98

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Abstract: 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.

Publication: Journal of Computational Physics Vol.: 134 No.: 1 ISSN: 0021-9991

ID: CaltechAUTHORS:20170707-134035089

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Abstract: 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.

Publication: SIAM Journal on Numerical Analysis Vol.: 33 No.: 5 ISSN: 0036-1429

ID: CaltechAUTHORS:BEAsiamjna96

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Abstract: 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.

Publication: SIAM Journal on Numerical Analysis Vol.: 30 No.: 3 ISSN: 0036-1429

ID: CaltechAUTHORS:HOUsiamjna93

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