(PHD, 2023)

Abstract:

Low-rank matrix recovery problems are prevalent in modern data science, machine learning, and artificial intelligence, and the low-rank property of matrices is widely exploited to extract the hidden low-complexity structure in massive datasets. Compared with Burer-Monteiro factorization in the Euclidean space, using the low-rank matrix manifold has its unique advantages, as it eliminates duplicated spurious points and reduces the polynomial order of the objective function. Yet a few fundamental questions have remained unanswered until recently. We highlight two problems here in particular, which are the global geometry of the manifold and the global convergence guarantee.

As for the global geometry, we point out that there exist some spurious critical points on the boundary of the low-rank matrix manifold Mᵣ, which have rank smaller than r but can serve as limit points of iterative sequences in the manifold Mᵣ. For the least squares loss function, the spurious critical points are rank-deficient matrices that capture part of the eigen spaces of the ground truth. Unlike classical strict saddle points, their Riemannian gradient is singular and their Riemannian Hessian is unbounded.

We show that randomly initialized Riemannian gradient descent almost surely escapes some of the spurious critical points. To prove this result, we first establish the asymptotic escape of classical strict saddle sets consisting of non-isolated strict critical submanifolds on Riemannian manifolds. We then use a dynamical low-rank approximation to parameterize the manifold Mᵣ and map the spurious critical points to strict critical submanifolds in the classical sense in the parameterized domain, which leads to the desired result. Our result is the first to partially overcome the nonclosedness of the low-rank matrix manifold without altering the vanilla gradient descent algorithm. Numerical experiments are provided to support our theoretical findings.

As for the global convergence guarantee, we point out that earlier approaches to many of the low-rank recovery problems only imply a geometric convergence rate toward a second-order stationary point. This is in contrast to the numerical evidence, which suggests a nearly linear convergence rate starting from a global random initialization. To establish the nearly linear convergence guarantee, we propose a unified framework for a class of low-rank matrix recovery problems including matrix sensing, matrix completion, and phase retrieval. All of them can be considered as random sensing problems of low-rank matrices with a linear measurement operator from some random ensembles. These problems share similar population loss functions that are either least squares or its variant.

We show that under some assumptions, for the population loss function, the Riemannian gradient descent starting from a random initialization with high probability converges to the ground truth in a nearly linear convergence rate, i.e., it takes O(log 1/ϵ + log n) iterations to reach an ϵ-accurate solution. The key to establishing a nearly optimal convergence guarantee is closely intertwined with the analysis of the spurious critical points S_# on Mᵣ. Outside the local neighborhoods of spurious critical points, we use the fundamental convergence tool by the Łojasiewicz inequality to derive a linear convergence rate. In the spurious regions in the neighborhood of spurious critical points, the Riemannian gradient becomes degenerate and the Łojasiewicz inequality could fail. By tracking the dynamics of the trajectory in three stages, we are able to show that with high probability, Riemannian gradient descent escapes the spurious regions in a small number of steps.

After addressing the two problems of global geometry and global convergence guarantee, we use two applications to demonstrate the broad applicability of our analytical tools. The first is the robust principal component analysis problem on the manifold Mᵣ with the Riemannian subgradient method. The second application is the convergence rate analysis of the Sobolev gradient descent method for the nonlinear Gross-Pitaevskii eigenvalue problem on the infinite dimensional sphere manifold. These two examples demonstrate that the analysis of manifold first-order algorithms can be extended beyond the previous framework, to nonsmooth functions and subgradient methods, and to infinite dimensional Hilbert manifolds. This exemplifies that the insights gained and tools developed for the low-rank matrix manifold Mᵣ can be extended to broader scientific and technological fields.

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(PHD, 2023)

Abstract:

This thesis focuses on numerical methods for scientific computing and scientific machine learning, specifically on solving partial differential equations and inverse problems. The design of numerical algorithms usually encompasses a spectrum that ranges from specialization to generality. Classical approaches, such as finite element methods, and contemporary scientific machine learning approaches, like neural nets, can be viewed as lying at relatively opposite ends of this spectrum. Throughout this thesis, we tackle mathematical challenges associated with both ends by advancing rigorous multiscale and statistical numerical methods.

Regarding the multiscale numerical methods, we present an exponentially convergent multiscale finite element method for solving high-frequency Helmholtz’s equation with rough coefficients. To achieve this, we first identify the local low-complexity structure of Helmholtz’s equations when the resolution is smaller than the wavelength. Then, we construct local basis functions by solving local spectral problems and couple them globally through non-overlapped domain decomposition and Galerkin’s method. This results in a numerical method that achieves nearly exponentially convergent accuracy regarding the number of local basis functions, even when the solution is highly non-smooth. We also analyze the role of a subsampled lengthscale in variational multiscale methods, characterizing the tradeoff between accuracy and efficiency in the numerical upscaling of heterogeneous PDEs and scattered data approximation.

As for the statistical numerical methods, we discuss using Gaussian processes and kernel methods to solve nonlinear PDEs and inverse problems. This framework incorporates the flavor of scientific machine learning automation and extends classical meshless solvers. It transforms general PDE problems into quadratic optimization with nonlinear constraints. We present the theoretical underpinning of the methodology. For the scalability of the method, we develop state-of-the-art algorithms to handle dense kernel matrices in both low and high-dimensional scientific problems. For adaptivity, we analyze the convergence and consistency of hierarchical learning algorithms that adaptively select kernel functions. Additionally, we note that statistical numerical methods offer natural uncertainty quantification within the Bayesian framework. In this regard, our further work contributes to some new understanding of efficient statistical sampling techniques based on gradient flows.

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(PHD, 2023)

Abstract:

High dimensionality brings both opportunities and challenges to the study of applied mathematics. This thesis consists of two parts. The first part explores the singularity formation of the axisymmetric incompressible Euler equations with no swirl in ℝⁿ, which is closely related to the Millennium Prize Problem on the global singularity of the Navier-Stokes equations. In this part, the high dimensionality contributes to the singularity formation in finite time by enhancing the strength of the vortex stretching term. The second part focuses on sampling from a high-dimensional distribution using deep generative networks, which has wide applications in the Bayesian inverse problem and the image synthesis task. The high dimensionality in this part becomes a significant challenge to the numerical algorithms, known as the curse of dimensionality.

In the first part of this thesis, we consider the singularity formation in two scenarios. In the first scenario, for the axisymmetric Euler equations with no swirl, we consider the case when the initial condition for the angular vorticity is C^{α} Hölder continuous. We provide convincing numerical examples where the solutions develop potential self-similar blow-up in finite time when the Hölder exponent α < α*, and this upper bound α* can asymptotically approach 1 - 2/n. This result supports a conjecture from Drivas and Elgindi [37], and generalizes it to the high-dimensional case. This potential blow-up is insensitive to the perturbation of initial data. Based on assumptions summarized from numerical experiments, we study a limiting case of the Euler equations, and obtain α* = 1 - 2/n which agrees with the numerical result. For the general case, we propose a relatively simple one-dimensional model and numerically verify its approximation to the Euler equations. This one-dimensional model might suggest a possible way to show this finite-time blow-up scenario analytically. Compared to the first proved blow-up result of the 3D axisymmetric Euler equations with no swirl and Hölder continuous initial data by Elgindi in [40], our potential blow-up scenario has completely different scaling behavior and regularity of the initial condition. In the second scenario, we consider using smooth initial data, but modify the Euler equations by adding a factor ε as the coefficient of the convection terms to weaken the convection effect. The new model is called the weak convection model. We provide convincing numerical examples of the weak convection model where the solutions develop potential self-similar blow-up in finite time when the convection strength ε < ε*, and this upper bound ε* should be close to 1 - 2/n. This result is closely related to the infinite-dimensional case of an open question [37] stated by Drivas and Elgindi. Our numerical observations also inspire us to approximate the weak convection model with a one-dimensional model. We give a rigorous proof that the one-dimensional model will develop finite-time blow-up if ε < 1 - 2/n, and study the approximation quality of the one-dimensional model to the weak convection model numerically, which could be beneficial to a rigorous proof of the potential finite-time blow-up.

In the second part of the thesis, we propose the Multiscale Invertible Generative Network (MsIGN) to sample from high-dimensional distributions by exploring the low-dimensional structure in the target distribution. The MsIGN models a transport map from a known reference distribution to the target distribution, and thus is very efficient in generating uncorrelated samples compared to MCMC-type methods. The MsIGN captures multiple modes in the target distribution by generating new samples hierarchically from a coarse scale to a fine scale with the help of a novel prior conditioning layer. The hierarchical structure of the MsIGN also allows training in a coarse-to-fine scale manner. The Jeffreys divergence is used as the objective function in training to avoid mode collapse. Importance sampling based on the prior conditioning layer is leveraged to estimate the Jeffreys divergence, which is intractable in previous deep generative networks. Numerically, when applied to two Bayesian inverse problems, the MsIGN clearly captures multiple modes in the high-dimensional posterior and approximates the posterior accurately, demonstrating its superior performance compared with previous methods. We also provide an ablation study to show the necessity of our proposed network architecture and training algorithm for the good numerical performance. Moreover, we also apply the MsIGN to the image synthesis task, where it achieves superior performance in terms of bits-per-dimension value over other flow-based generative models and yields very good interpretability of its neurons in intermediate layers.

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(PHD, 2022)

Abstract:

Whether the three-dimensional (3D) incompressible Euler equations can develop a finite-time singularity from smooth initial data with finite energy is a major open problem in partial differential equations. A few years ago, Tom Hou and Guo Luo obtained strong numerical evidence of a potential finite time singularity of the 3D axisymmetric Euler equations with boundary from smooth initial data. So far, there is no rigorous justification. In this thesis, we develop a framework to study the Hou-Luo blowup scenario and singularity formation in related equations and models. In addition, we analyze the obstacle to singularity formation.

In the first part, we propose a novel framework of analysis based on the dynamic rescaling formulation to study singularity formation. Our strategy is to reformulate the problem of proving finite time blowup into the problem of establishing the nonlinear stability of an approximate self-similar blowup profile using the dynamic rescaling equations. Then we prove finite time blowup of the 2D Boussinesq and the 3D Euler equations with C^{1,α} velocity and boundary. This result provides the first rigorous justification of the Hou-Luo scenario using C^{1,α} velocity.

In the second part, we further develop the framework for smooth data. The method in the first part relies crucially on the low regularity of the data, and there are several essential difficulties to generalize it to study the Hou-Luo scenario with smooth data. We demonstrate that some of the challenges can be overcome by proving the asymptotically self-similar blowup of the Hou-Luo model. Applying this framework, we establish the finite time blowup of the De Gregorio (DG) model on the real line (ℝ) with smooth data. Our result resolves the open problem on the regularity of this model on ℝ that has been open for quite a long time.

In the third part, we investigate the competition between advection and vortex stretching, an essential difficulty in studying the regularity of the 3D Euler equations. This competition can be modeled by the DG model on S^{1}. We consider odd initial data with a specific sign property and show that the regularity of the initial data in this class determines the competition between advection and vortex stretching. For any 0 < α < 1, we construct a finite time blowup solution from some C^{α} initial data. On the other hand, we prove that the solution exists globally for C^{1} initial data. Our results resolve some conjecture on the finite time blowup of this model and imply that singularities developed in the DG model and the generalized Constantin-Lax-Majda model on S^{1} can be prevented by stronger advection.

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(PHD, 2020)

Abstract:

For many decades, the study of positive-definite (PD) matrices has been one of the most popular subjects among a wide range of scientific researches. A huge mass of successful models on PD matrices has been proposed and developed in the fields of mathematics, physics, biology, etc., leading to a celebrated richness of theories and algorithms. In this thesis, we draw our attention to a general class of PD matrices that can be decomposed as the sum of a sequence of positive-semidefinite matrices. For this class of PD matrices, we will develop theories and algorithms on operator compression, multilevel decomposition, eigenpair computation, and spectrum concentration. We divide these contents into three main parts.

In the first part, we propose an adaptive fast solver for the preceding class of PD matrices which includes the well-known graph Laplacians. We achieve this by establishing an adaptive operator compression scheme and a multiresolution matrix factorization algorithm which have nearly optimal performance on both complexity and well-posedness. To develop our methods, we introduce a novel notion of energy decomposition for PD matrices and two important local measurement quantities, which provide theoretical guarantee and computational guidance for the construction of an appropriate partition and a nested adaptive basis.

In the second part, we propose a new iterative method to hierarchically compute a relatively large number of leftmost eigenpairs of a sparse PD matrix under the multiresolution matrix compression framework. We exploit the well-conditioned property of every decomposition components 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.

In the third part, we derive concentration inequalities on partial sums of eigenvalues of random PD matrices by introducing the notion of *k*-trace. For this purpose, we establish a generalized Lieb’s concavity theorem, which extends the original Lieb’s concavity theorem from the normal trace to *k*-traces. Our argument employs a variety of matrix techniques and concepts, including exterior algebra, mixed discriminant, and operator interpolation.

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(PHD, 2017)

Abstract:

Given a positive semidefinite (PSD) operator, such as a PSD matrix, an elliptic operator with rough coefficients, a covariance operator of a random field, or the Hamiltonian of a quantum system, we would like to find its best finite rank approximation with a given rank. One way to achieve this objective is to project the operator to its eigenspace that corresponds to the smallest or largest eigenvalues, depending on the setting. The eigenfunctions are typically global, i.e. nonzero almost everywhere, but our interest is to find the sparsest or most localized bases for these subspaces. The sparse/localized basis functions lead to better physical interpretation and preserve some sparsity structure in the original operator. Moreover, sparse/localized basis functions also enable us to develop more efficient numerical algorithms to solve these problems.

In this thesis, we present two methods for this purpose, namely the sparse operator compression (Sparse OC) and the intrinsic sparse mode decomposition (ISMD). The Sparse OC is a general strategy to construct finite rank approximations to PSD operators, for which the range space of the finite rank approximation is spanned by a set of sparse/localized basis functions. The basis functions are energy minimizing functions on local patches. When applied to approximate the solution operator of elliptic operators with rough coefficients and various homogeneous boundary conditions, the Sparse OC achieves the optimal convergence rate with nearly optimally localized basis functions. Our localized basis functions can be used as multiscale basis functions to solve elliptic equations with multiscale coefficients and provide the optimal convergence rate *O*(*h*^{k}) for 2*k*’th order elliptic problems in the energy norm. From the perspective of operator compression, these localized basis functions provide an efficient and optimal way to approximate the principal eigen-space of the elliptic operators. From the perspective of the Sparse PCA, we can approximate a large set of covariance functions by a rank-*n* operator with a localized basis and with the optimal accuracy. While the Sparse OC works well on the solution operator of elliptic operators, we also propose the ISMD that works well on low rank or nearly low rank PSD operators. Given a rank-*n* PSD operator, say a *N*-by-*N* PSD matrix *A* (*n* ≤ *N*), the ISMD *decomposes* it into *n* rank-one matrices Σ^{n}_{i=1}*g*_{i}*g*^{T}_{i} where the mode {*g*_{i}}^{n}_{i=1} are required to be as sparse as possible. Under the regular-sparse assumption (see Definition 1.3.2), we have proved that the ISMD gives the optimal patchwise sparse decomposition, and is stable to small perturbations in the matrix to be decomposed. We provide several applications in both the physical and data sciences to demonstrate the effectiveness of the proposed strategies.

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(PHD, 2017)

Abstract:

The partial differential equations (PDE) governing the motions of incompressible ideal fluid in three dimensional (3D) space are among the most fundamental nonlinear PDEs in nature and have found a lot of important applications. Due to the presence of super-critical non-linearity, the fundamental question of global well-posedness still remains open and is generally viewed as one of the most outstanding open questions in mathematics. In this thesis, we investigate the potential finite-time singularity formation of the 3D Euler equations and simplified models by studying the self-similar spatial profiles in the potentially singular solutions.

In the first part, we study the self-similar singularity of two 1D models, the CKY model and the HL model, which approximate the dynamics of the 3D axisymmtric Euler equations on the solid boundary of a cylindrical domain. The two models are both numerically observed to develop self-similar singularity. We prove the existence of a discrete family of self-similar profiles for the CKY model, using a combination of analysis and computer-aided verification. Then we employ a dynamic rescaling formulation to numerically study the evolution of the spatial profiles for the two 1D models, and demonstrate the stability of the self-similar singularity. We also study a singularity scenario for the HL model with multi-scale feature.

In the second part, we study the self-similar singularity for the 3D axisymmetric Euler equations. We first prove the local existence of a family of analytic self-similar profiles using a modified Cauchy-Kowalevski majorization argument. Then we use the dynamic rescaling formulation to investigate two types of initial data with different leading order properties. The first initial data correspond to the singularity scenario reported by Luo and Hou. We demonstrate that the self-similar profiles enjoy certain stability, which confirms the finite-time singularity reported by Luo and Hou. For the second initial data, we show that the solutions develop singularity in a different manner from the first case, which is unknown previously. The spatial profiles in the solutions become singular themselves, which means that the solutions to the Euler equations develop singularity at multiple spatial scales.

In the third part, we propose a family of 3D models for the 3D axisymmetric Euler and Navier-Stokes equations by modifying the amplitude of the convection terms. The family of models share several regularity results with the original Euler and Navier-Stokes equations, and we study the potential finite-time singularity of the models numerically. We show that for small convection, the solutions of the inviscid model develop self-similar singularity and the profiles behave like travelling waves. As we increase the amplitude of the velocity field, we find a critical value, after which the travelling wave self-similar singularity scenario disappears. Our numerical results reveal the potential stabilizing effect the convection terms.

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(PHD, 2014)

Abstract:

Partial differential equations (PDEs) with multiscale coefficients are very difficult to solve due to the wide range of scales in the solutions. In the thesis, we propose some efficient numerical methods for both deterministic and stochastic PDEs based on the model reduction technique.

For the deterministic PDEs, the main purpose of our method is to derive an effective equation for the multiscale problem. An essential ingredient is to decompose the harmonic coordinate into a smooth part and a highly oscillatory part of which the magnitude 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 smooth, and could be resolved on a regular coarse mesh grid. Furthermore, we provide error analysis and show that the solution to the effective equation plus a correction term is close to the original multiscale solution.

For the stochastic PDEs, we propose the model reduction based data-driven stochastic method and multilevel Monte Carlo method. In the multiquery, setting and on the assumption that the ratio of the smallest scale and largest scale is not too small, we propose the multiscale data-driven stochastic method. We construct a data-driven stochastic basis and solve the coupled deterministic PDEs to obtain the solutions. For the tougher problems, we propose the multiscale multilevel Monte Carlo method. We apply the multilevel scheme to the effective equations and assemble the stiffness matrices efficiently on each coarse mesh grid. In both methods, the $\KL$ expansion plays an important role in extracting the main parts of some stochastic quantities.

For both the deterministic and stochastic PDEs, numerical results are presented to demonstrate the accuracy and robustness of the methods. We also show the computational time cost reduction in the numerical examples.

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(PHD, 2014)

Abstract: In this work, we further extend the recently developed adaptive data analysis method, the Sparse Time-Frequency Representation (STFR) method. This method is based on the assumption that many physical signals inherently contain AM-FM representations. We propose a sparse optimization method to extract the AM-FM representations of such signals. We prove the convergence of the method for periodic signals under certain assumptions and provide practical algorithms specifically for the non-periodic STFR, which extends the method to tackle problems that former STFR methods could not handle, including stability to noise and non-periodic data analysis. This is a significant improvement since many adaptive and non-adaptive signal processing methods are not fully capable of handling non-periodic signals. Moreover, we propose a new STFR algorithm to study intrawave signals with strong frequency modulation and analyze the convergence of this new algorithm for periodic signals. Such signals have previously remained a bottleneck for all signal processing methods. Furthermore, we propose a modified version of STFR that facilitates the extraction of intrawaves that have overlaping frequency content. We show that the STFR methods can be applied to the realm of dynamical systems and cardiovascular signals. In particular, we present a simplified and modified version of the STFR algorithm that is potentially useful for the diagnosis of some cardiovascular diseases. We further explain some preliminary work on the nature of Intrinsic Mode Functions (IMFs) and how they can have different representations in different phase coordinates. This analysis shows that the uncertainty principle is fundamental to all oscillating signals.

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(PHD, 2013)

Abstract: Many physical and engineering problems involving uncertainty enjoy certain low-dimensional structures, e.g., in the sense of Karhunen-Loeve expansions (KLEs), which in turn indicate the existence of reduced-order models and better formulations for efficient numerical simulations. In this thesis, we target a class of time-dependent stochastic partial differential equations whose solutions enjoy such structures at any time and propose a new methodology (DyBO) to derive equivalent systems whose solutions closely follow KL expansions of the original stochastic solutions. KL expansions are known to be the most compact representations of stochastic processes in an L^{2} sense. Our methods explore such sparsity and offer great computational benefits compared to other popular generic methods, such as traditional Monte Carlo (MC), generalized Polynomial Chaos (gPC) method, and generalized Stochastic Collocation (gSC) method. Such benefits are demonstrated through various numerical examples ranging from spatially one-dimensional examples, such as stochastic Burgers’ equations and stochastic transport equations to spatially two-dimensional examples, such as stochastic flows in 2D unit square. Parallelization is also discussed, aiming toward future industrial-scale applications. In addition to numerical examples, theoretical aspects of DyBO are also carefully analyzed, such as preservation of bi-orthogonality, error propagation, and computational complexity. Based on theoretical analysis, strategies are proposed to overcome difficulties in numerical implementations, such as eigenvalue crossing and adaptively adding or removing mode pairs. The effectiveness of the proposed strategies is numerically verified. Generalization to a system of SPDEs is considered as well in the thesis, and its success is demonstrated by applications to stochastic Boussinesq convection problems. Other generalizations, such as generalized stochastic collocation formulation of DyBO method, are also discussed.

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(PHD, 2012)

Abstract:

In the first part, we present a mathematical derivation of a closure relating the Reynolds stress to the mean strain rate for incompressible turbulent flows. This derivation is based on a systematic multiscale analysis that 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 Smagorinsky model for homogeneous turbulence is recovered as an example to illustrate our mathematical derivation. Another example is turbulent channel flow, where we derive a simplified turbulence model based on the asymptotic flow structure near the wall. Additionally, we obtain a nonlinear model by using a second order approximation of the inverse flow map function. This nonlinear model captures the effects of the backscatter of kinetic energy and dispersion and is consistent with other models, such as a mixed model that combines the Smagorinsky and gradient models, and the generic nonlinear model of Lund and Novikov.

Numerical simulation results at two Reynolds numbers using our simplified turbulence model are in good agreement with both experiments and direct numerical simulations in turbulent channel flow. However, due to experimental and modeling errors, we do observe some noticeable differences, e.g. , root mean square velocity fluctuations at Re_{τ} = 180.

In the second part, we present a new perspective on calculating fully developed turbulent flows using a data-driven stochastic method. General polynomial chaos (gPC) bases are obtained based on the mean velocity profile of turbulent channel flow in the offline part. The velocity fields are projected onto the subspace spanned by these gPC bases and a coupled system of equations is solved to compute the velocity components in the Karhunen-Loeve expansion in the online part. Our numerical results have shown that the data-driven stochastic method for fully developed turbulence offers decent approximations of statistical quantities with a coarse grid and a relatively small number of gPC base elements.

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(PHD, 2010)

Abstract:

Multiscale problems arise in many scientific and engineering disciplines. A typical example is the modelling of flow in a porous medium containing a number of low and high permeability embedded in a matrix. Due to the high degrees of variability and the multiscale nature of formation properties, not only is a complete analysis of these problems extremely difficult, but also numerical solvers require an excessive amount of CPU time and storage. In this thesis, we study multiscale numerical methods for the elliptic equations arising in interface and two-phase flow problems. The model problems we consider are motivated by the multiscale computations of flow and transport of two-phase flow in strongly heterogeneous porous media. Although the analysis is carried out for simplified model problems, it does provide valuable insight in designing accurate multiscale methods for more realistic applications.

In the first part, 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. The method is H¹-conforming, and has an optimal convergence rate of O(h) in the energy norm and O(h²) in the L₂ norm, where h is the 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’s coefficients. 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. We conduct some numerical experiments to confirm the optimal rate of convergence of the proposed method and its independence from the aspect ratio of the coefficients.

In the second part, we propose a flow-based oversampling method where the actual two-phase flow boundary conditions are used to construct oversampling auxiliary functions. Our numerical results show that the flow-based oversampling approach is several times more accurate than the standard oversampling method. A partial theoretical explanation is provided for these numerical observations.

In the third part, we discuss “metric-based upscaling” for the pressure equation in two-phase flow problem. We show a compensation phenomenon and design a multiscale method for the pressure equation with highly oscillatory permeability.

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(PHD, 2006)

Abstract:

We derive the two-scale limit of a linear or nonlinear saturation equation with a flow-based coordinate transformation. This transformation consists of the pressure and the streamfunction. In this framework the saturation equation is decoupled to a family of one-dimensional nonconservative transport equations along streamlines. This simplifies the derivation of the two-scale limit. Moreover it allows us to obtain the convergence independent of the assumptions of periodicity and scale separation. We provide a rigorous estimate on the convergence rate. We combine the two-scale limit with Tartar’s method to complete the homogenization.

To design an efficient numerical method, we use an averaging approach across the streamlines on the two-scale limit equations. The resulting numerical method for the saturation has all the advantages in terms of adaptivity that methods have. We couple it with a moving mesh along the streamlines to resolve the shock more efficiently. We use the multiscale finite element method to upscale the pressure equation because it gives access to the fine scale velocity, which enters in the saturation equation, through the basis functions. We propose to solve the pressure equation in the coordinate frame of the initial pressure and saturation, which is similar to the modified multiscale finite element method.

We test our numerical method in realistic permeability fields, such as the Tenth SPE Comparative Solution Project permeabilities, for accuracy and computational cost.

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(PHD, 2006)

Abstract:

Stochastic partial differential equations (SPDEs) are important tools in modeling complex phenomena, and they arise in many physics and engineering applications. Developing efficient numerical methods for simulating SPDEs is a very important while challenging research topic. In this thesis, we study a numerical method based on the Wiener chaos expansion (WCE) for solving SPDEs driven by Brownian motion forcing. WCE represents a stochastic solution as a spectral expansion with respect to a set of random basis. By deriving a governing equation for the expansion coefficients, we can reduce a stochastic PDE into a system of deterministic PDEs and separate the randomness from the computation. All the statistical information of the solution can be recovered from the deterministic coefficients using very simple formulae.

We apply the WCE-based method to solve stochastic Burgers equations, Navier-Stokes equations and nonlinear reaction-diffusion equations with either additive or multiplicative random forcing. Our numerical results demonstrate convincingly that the new method is much more efficient and accurate than MC simulations for solutions in short to moderate time. For a class of model equations, we prove the convergence rate of the WCE method. The analysis also reveals precisely how the convergence constants depend on the size of the time intervals and the variability of the random forcing. Based on the error analysis, we design a sparse truncation strategy for the Wiener chaos expansion. The sparse truncation can reduce the dimension of the resulting PDE system substantially while retaining the same asymptotic convergence rates.

For long time solutions, we propose a new computational strategy where MC simulations are used to correct the unresolved small scales in the sparse Wiener chaos solutions. Numerical experiments demonstrate that the WCE-MC hybrid method can handle SPDEs in much longer time intervals than the direct WCE method can. The new method is shown to be much more efficient than the WCE method or the MC simulation alone in relatively long time intervals. However, the limitation of this method is also pointed out.

Using the sparse WCE truncation, we can resolve the probability distributions of a stochastic Burgers equation numerically and provide direct evidence for the existence of a unique stationary measure. Using the WCE-MC hybrid method, we can simulate the long time front propagation for a reaction-diffusion equation in random shear flows. Our numerical results confirm the conjecture by Jack Xin that the front propagation speed obeys a quadratic enhancing law.

Using the machinery we have developed for the Wiener chaos method, we resolve a few technical difficulties in solving stochastic elliptic equations by Karhunen-Loeve-based polynomial chaos method. We further derive an upscaling formulation for the elliptic system of the Wiener chaos coefficients. Eventually, we apply the upscaled Wiener chaos method for uncertainty quantification in subsurface modeling, combined with a two-stage Markov chain Monte Carlo sampling method we have developed recently.

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(PHD, 2005)

Abstract:

In this thesis, new results excluding finite time singularities with localized assumptions/conditions are obtained for the 3D incompressible Euler equations.

The 3D incompressible Euler equations are some of the most important nonlinear equations in mathematics. They govern the motion of ideal fluids. After hundreds of years of study, they are still far from being well-understood. In particular, a long-outstanding open problem asks whether finite time singularities would develop for smooth initial values. Much theoretical and numerical study on this problem has been carried out, but no conclusion can be drawn so far.

In recent years, several numerical experiments have been carried out by various authors, with results indicating possible breakdowns of smooth solutions in finite time. In these numerical experiments, certain properties of the velocity and vorticity field are observed in near-singular flows. These properties violate the assumptions of existing theoretical theorems which exclude finite time singularities. Thus there is a gap between current theoretical and numerical results. To narrow this gap is the main purpose of the work presented in this thesis.

In this thesis, a new framework of investigating flows carried by divergence-free velocity fields is developed. Using this new framework, new, localized sufficient conditions for the flow to remain smooth are obtained rigorously. These new results can deal with fast shrinking large vorticity regions and are applicable to recent numerical experiments. The application of the theorems in this thesis reveals new subtleties, and yields new understandings of the 3D incompressible Euler flow.

This new framework is then further applied to a two-dimensional model equation, the 2D quasi-geostrophic equation, for which global existence is still unproved. Under certain assumptions, we obtain new non-blowup results for the 2D quasi-geostrophic equation.

Finally, future plans of applying this new framework to some other PDEs as well as other possibilities of attacking the 3D Euler and 2D quasi-geostrophic singularity problems are discussed.

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(PHD, 2005)

Abstract:

In this thesis we investigate the locomotion of fish and birds by applying both new and well known mathematical techniques.

The two-dimensional model is first studied using Krasny’s vortex blob method, and then a new numerical method based on Wu’s theory is developed. To begin with, we will implement Krasny’s ideas for a couple of examples and then switch to the numerical implementation of the nonlinear analytical mathematical model presented by Wu. We will demonstrate the superiority of this latter method both by applying it to some specific cases and by comparing with the experiments. The nonlinear effects are very well observed and this will be shown by analyzing Wagner’s result for a wing abruptly undergoing an increase in incidence angle, and also by analyzing the vorticity generated by a wing in heaving, pitching and bending motion. The ultimate goal of the thesis is to accurately represent the vortex structure behind a flying wing and its influence on the bound vortex sheet.

In the second part of this work we will introduce a three-dimensional method for a flat plate advancing perpendicular to the flow. The accuracy of the method will be shown both by comparing its results with the two-dimensional ones and by validating them versus the experimental results obtained by Ringuette in the towing tank of the Aeronautics Department at Caltech.

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(PHD, 2005)

Abstract:

The understanding and modeling of flow through porous media is an important issue in several branches of engineering. In petroleum engineering, for instance, one wishes to model the “enhanced oil recovery” process, whereby water or steam is injected into an oil saturated porous media in an attempt to displace the oil so that it can be collected. In groundwater contaminant studies the transport of dissolved material, such as toxic metals or radioactive waste, and how it affects drinking water supplies, is of interest.

Numerical simulation of these flow are generally difficult. The principal reason for this is the presence of many different length scales in the physical problem, and resolving all these is computationally expensive. To circumvent these difficulties a class of methods known as upscaling methods has been developed where one attempts to solve only for large scale features of interest and model the effect of the small scale features.

In this thesis, we review some of the previous efforts in upscaling and introduce a new scheme that attempts to overcome some of the existing shortcomings of these methods. In our analysis, we consider the flow problem in two distinct stages: the first is the determination of the velocity field which gives rise to an elliptic partial differential equation (PDE) and the second is a transport problem which gives rise to a hyperbolic PDE.

For the elliptic part, we make use of existing upscaling methods for elliptic equations. In particular, we use the multi-scale finite element method of Hou et al. to solve for the velocity field on a coarse grid, and yet still be able to obtain fine scale information through a special means of interpolation.

The analysis of the hyperbolic part forms the main contribution of this thesis. We first analyze the problem by restricting ourselves to the case where the small scales have a periodic structure. With this assumption, we are able to derive a coupled set of equations for the large scale average and the small scale fluctuations about this average. This is done by means of a special averaging, which is done along the fine scale streamlines. This coupled set of equations provides better starting point for both the modeling of the largescale small-scale interactions and the numerical implementation of any scheme. We derive an upscaling scheme from this by tracking only a sub-set of the fluctuations, which are used to approximate the scale interactions. Once this model has been derived, we discuss and present a means to extend it to the case where the fluctuations are more general than periodic.

In the sections that follow we provide the details of the numerical implementation, which is a very significant part of any practical method. Finally, we present numerical results using the new scheme and compare this with both resolved computations and some existing upscaling schemes.

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(PHD, 2003)

Abstract:

In this thesis we investigate nonsmooth classical and continuum mechanics and its discretizations by means of variational numerical and geometric methods.

The theory of smooth Lagrangian mechanics is extended to a nonsmooth context appropriate for collisions and it is shown in what sense the system is symplectic and satisfies a Noether-style momentum conservation theorem.

Next, we develop the foundations of a multisymplectic treatment of nonsmooth classical and continuum mechanics. This work may be regarded as a PDE generalization of the previous formulation of a variational approach to collision problems. The multisymplectic formulation includes a wide collection of nonsmooth dynamical models such as rigid-body collisions, material interfaces, elastic collisions, fluid-solid interactions and lays the groundwork for a treatment of shocks.

Discretizations of this nonsmooth mechanics are developed by using the methodology of variational discrete mechanics. This leads to variational integrators which are symplectic-momentum preserving and are consistent with the jump conditions given in the continuous theory. Specific examples of these methods are tested numerically and the longtime stable energy behavior typical of variational methods is demonstrated.

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(PHD, 2001)

Abstract: In this thesis, we investigate both theoretically and numerically the singularity formation and long time existence of three-dimensional vortex sheets. For the theoretical work, we divide it into two parts. In the first part, we study the early time singularity formation and the local form of the vortex sheet in the neighborhood of a singularity near the singularity time. We show that under a special set of coordinates, the three-dimensional vortex sheet can be viewed as a two-dimensional vortex sheet along certain space curves. As a result, the study of singularity formation of a three-dimensional vortex sheet can be related to that of the corresponding two-dimensional vortex sheet. And the singular behavior of these two problems is very similar. Moreover, by performing a transformation in the interface variables and deriving leading order asymptotic approximations for the evolution of these transformed variables, we show that the Kelvin-Helmholtz instability is a result of the coupling of two of these three variables to the leading order. This observation simplifies significantly our singularity analysis for three-dimensional vortex sheets and allows us to reveal clearly the nature of the curvature singularity in the three-dimensional vortex sheet equation. In the second part of our theoretical work, we prove the long time existence of the three-dimensional vortex sheet problem for analytic initial conditions near equilibrium. Moreover, the existence time is almost optimal if the initial perturbation over the equilibrium is sufficiently small. We have performed careful numerical study to validate our theoretical results. Well-resolved numerical study of the three-dimensional vortex sheet equation is difficult due to the complexity in evaluating the interface velocity. To alleviate this difficulty, we introduce two model equations. An important feature of these models equations is that they can be expressed in terms of convolution operators and consequently they can be computed efficiently by Fast Fourier Transform. Moreover, we show by asymptotic analysis that these model equations preserve the singularity type of the full equations. Our analysis also suggests that the model equations generate the same local form of curvature singularity near the physical singularity time as that of the full equations. Our detailed numerical computations on the two-dimensional problem show that the model equation captures all the essential singularity behavior of the full vortex sheet equation. Our calculations based on the three-dimensional model equation provide convincing evidences that a curvature singularity develops in finite time in the three-dimensional vortex sheet. And the type of the singularity is of order -1/2 in the mean curvature.

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(PHD, 2000)

Abstract:

We develop efficient and robust numerical methods in the finite element framework for numerical solutions of the singularly perturbed convection-diffusion equation and of a degenerate elliptic equation. The standard methods for purely elliptic or hyperbolic problems perform poorly when there are sharp boundary and internal layers in the solution caused by the dominant convective effect. We offer a new approach in which we design the finite element basis functions that capture the local behavior correctly.

When the structure of the layers can be determined locally, we apply the multiscale finite element method in which we solve the corresponding homogeneous equation on each element to capture the small scale features of the differential operator. We demonstrate the effectiveness of this method by computing the enhanced diffusivity scaling for a passive scalar in the cellular flow. We carry out the asymptotic error analysis for its convergence rate and perform numerical experiments for verification. When the layer structure is nonlocal, we use a variational principle to gain additional information. For a random velocity field, this variational principle provides correct scaling results. This allows us to design asymptotic basis functions that can capture the global layers correctly.

The same approach is also extended to elliptic problems with high contrast coefficients. When an asymptotic result is available, it is incorporated naturally into the finite element setting developed earlier. When there is a strong singularity due to a discontinuous coefficient, we construct the basis functions using the infinite element method. Our methods can handle singularities efficiently and are not sensitive to the large contrast.

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(PHD, 2000)

Abstract: In the first part of this thesis, we present new formulations for computing the motion of curvature driven 3-D filament and surface. The new numerical methods have no high order time step stability constraints that are usually associated with curvature regularization. This result generalizes the previous work in [23] for 2-D fluid interfaces with surface tension. Applications to 2-D vortex sheets, the Kirchhoff rod model, nearly anti—parallel vortex filaments, motion by mean curvature in 3-D and simplified water wave model are presented to demonstrate the robustness of the methods. In the second part of this thesis, we investigate numerically the effects of surface tension on the evolution of 2-D Hele-Shaw flows and 3-D axisymmetric flows through porous media with suction. Hele-Shaw flows with suction are known to form cusp singularities in finite time with zero-surface-tension. 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. 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. Our computations reveal an asymptotic shape of the wedge as surface tension tends to zero. Moreover, 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. For 3-D axisymmetric flow, similar behavior is observed. The surface develops a narrow finger which evolves into a cone as it approaches the sink. The finger diameter is smaller than the finger width for Hele-Shaw flow and the surface moves faster. The azimuthal component of the mean curvature enhances the definition of the finger neck while smoothing the interface there.

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(PHD, 1999)

Abstract:

Multiscale problems occur in many scientific and engineering disciplines, in petroleum engineering, material science, etc. These problems are characterized by the great deal of spatial and time scales which make it difficult to analyze theoretically or solve numerically. On the other hand, the large scale features of the solutions are often of main interest. Thus, it is desirable to have a numerical method that can capture the effect of small scales on large scales without resolving the small scale details.

In the first part of this work we analyze the multiscale finite element method (MsFEM) introduced in [28] for elliptic problems with oscillatory coefficients. The idea behind MsFEM is to capture the small scale information through the base functions constructed in elements that are larger than the small scale of the problem. This is achieved by solving for the finite element base functions from the leading order of homogeneous elliptic equation. We analyze MsFEM for different situations both analytically and numerically. We also investigate the origin of the resonance errors associated with the method and discuss the ways to improve them.

In the second part we discuss flow based upscaling of absolute permeability which is an important step in the practical simulations of flow through heterogeneous formations. The central idea is to compute the upscaled, grid-block permeability from fine scale solutions of the flow equation. It is well known that the grid block permeability may be strongly influenced by the boundary conditions imposed on the flow equations and the size of grid blocks. We analyze the effects of the boundary conditions and grid block sizes on the computed grid block absolute permeabilities. Moreover, we employ the ideas developed in the analysis of MsFEM to improve the computed values of absolute permeability.

The last part of the work is the application of MsFEM as well as upscaling of absolute permeability on upscaling of two-phase flow. In this part we consider coarse models using MsFEM. We demonstrate the efficiency of these models for practical problems. Moreover, we show that these models improve the existing approaches.

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(PHD, 1997)

Abstract:

In this work, we consider the numerical calculation of water waves in three dimensions. One well accepted method for studying surface waves is the boundary integral method, which defines the fluid velocities at the interface in terms of integrals over the boundary of the domain in which the problem is posed. There exists a considerable body of work on the numerical study of surface waves in two dimensions. However, until recently the numerical study of surface waves was considered intractable because of the high computational cost of approximating the defining integrals.

We discuss the boundary integral formulation for the three-dimensional water wave problem and present the point vortex approximation to the singular integrals which define the particle velocities. We consider three aspects of the point vortex approximation: accuracy of the approximation, efficient means of computing solutions, and numerical stability of the scheme.

Concerning the accuracy of the point vortex method, we analyze the error associated with the approximation and show that it can be expressed as a series in odd powers of the discretization parameter h. We present quadrature rules which are highly accurate.

The efficient computation of the point vortex approximation is achieved through the use of the fast multipole algorithm, which combines long distance particle inter-actions into multipole expansions which can be efficiently evaluated. The underlying periodicity of the problem is reduced to a lattice sum which can be rapidly evaluated. We discuss the implementation of the numerical schemes in both serial and parallel computing environments.

The point vortex method is shown to be highly unstable for straightforward discretizations of the surface. We analyze the stability of the method about equilibrium and discuss methods for stabilizing the numerical schemes for both the linear and nonlinear regimes. We present numerical results which show that the method can be effectively stabilized.

In the final chapter, we present numerical results from several calculations of three-dimensional waves using the methods developed in the previous chapters.
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