arXiv subjects

Thomas Y. Hou

Publications and source records attributed to Thomas Y. Hou.

At least 37 records · Page 2Linked to original sources

arXiv2023

Exponentially Convergent Multiscale Finite Element Method

We provide a concise review of the exponentially convergent multiscale finite element method (ExpMsFEM) for efficient model reduction of PDEs in heterogeneous media without scale separation and in high-frequency wave propagation. ExpMsFEM is built on the non-overlapped domain decomposition in the classical MsFEM while enriching the approximation space systematically to achieve a nearly exponential convergence rate regarding the number of basis functions. Unlike most generalizations of MsFEM in the literature, ExpMsFEM does not rely on any partition of unity functions. In general, it is necessary to use function representations dependent on the right-hand side to break the algebraic Kolmogorov $n$-width barrier to achieve exponential convergence. Indeed, there are online and offline parts in the function representation provided by ExpMsFEM. The online part depends on the right-hand side locally and can be computed in parallel efficiently. The offline part contains basis functions that are used in the Galerkin method to assemble the stiffness matrix; they are all independent of the right-hand side, so the stiffness matrix can be used repeatedly in multi-query scenarios.

math.NA↗

arXiv2022

Exponentially convergent multiscale methods for high frequency heterogeneous Helmholtz equations

In this paper, we present a multiscale framework for solving the Helmholtz equation in heterogeneous media without scale separation and in the high frequency regime where the wavenumber $k$ can be large. The main innovation is that our methods achieve a nearly exponential rate of convergence with respect to the computational degrees of freedom, using a coarse grid of mesh size $O(1/k)$ without suffering from the well-known pollution effect. The key idea is a non-overlapped domain decomposition and its associated coarse-fine scale decomposition of the solution space that adapts to the media property and wavenumber; this decomposition is inspired by the multiscale finite element method (MsFEM). We show that the coarse part is of \textit{low complexity} in the sense that it can be approximated with a nearly exponential rate of convergence via local basis functions, due to the compactness of a restriction operator that maps Helmholtz-harmonic functions to their interpolation residues on edges, while the fine part is \textit{local} such that it can be computed efficiently using the local information of the right hand side. The combination of the two parts yields the overall nearly exponential rate of convergence of our multiscale method. Our method draws many connections to multiscale methods in the literature, which we will comment in detail. We demonstrate the effectiveness of our methods theoretically and numerically; an exponential rate of convergence is consistently observed and confirmed. In addition, we observe the robustness of our methods regarding the high contrast in the media numerically. We specifically focus on 2D problems in our exposition since the geometry of non-overlapped domain decomposition is simplest to explain in such cases; generalizations to 3D will be outlined at the end.

math.NA↗

arXiv2022

Asymptotic Escape of Spurious Critical Points on the Low-rank Matrix Manifold

We show that on the manifold of fixed-rank and symmetric positive semi-definite matrices, the Riemannian gradient descent algorithm almost surely escapes some spurious critical points on the boundary of the manifold. Our result is the first to partially overcome the incompleteness of the low-rank matrix manifold without changing the vanilla Riemannian gradient descent algorithm. The spurious critical points are some rank-deficient matrices that capture only part of the eigen components of the ground truth. Unlike classical strict saddle points, they exhibit very singular behavior. We show that using the dynamical low-rank approximation and a rescaled gradient flow, some of the spurious critical points can be converted to classical strict saddle points in the parameterized domain, which leads to the desired result. Numerical experiments are provided to support our theoretical findings.

math.OC↗

arXiv2022

On stability and instability of $C^{1,α}$ singular solutions to the 3D Euler and 2D Boussinesq equations

Singularity formation of the 3D incompressible Euler equations is known to be extremely challenging. In [18], Elgindi proved that the 3D axisymmetric Euler equations with no swirl and $C^{1,α}$ initial velocity develops a finite time singularity. Inspired by Elgindi's work, we proved that the 3D axisymmetric Euler and 2D Boussinesq equations with $C^{1,α}$ initial velocity and boundary develop a stable asymptotically (or approximately) self-similar finite time singularity [8]. On the other hand, the authors of [35,52] recently showed that blowup solutions to the 3D Euler equations are hydrodynamically unstable. The instability results obtained in [35,52] require some strong regularity assumption on the initial data, which is not satisfied by the $C^{1,α}$ velocity field. In this paper, we generalize the analysis of [8,18,35,52] to show that the blowup solutions of the 3D Euler and 2D Boussinesq equations with $C^{1,α}$ velocity are unstable under the notion of stability introduced in [35,52]. These two seemingly contradictory results reflect the difference of the two approaches in studying the stability of 3D Euler blowup solutions. The stability analysis of the blowup solution obtained in [8,18] is based on the stability of a dynamically rescaled blowup profile in space and time, which is nonlinear in nature. The linear stability analysis in [35,52] is performed by directly linearizing the 3D Euler equations around a blowup solution in the original variables. It does not take into account the changes in the blowup time, the dynamic changes of the rescaling rate of the perturbed blowup profile and the blowup exponent of the original 3D Euler equations using a perturbed initial condition when there is an approximate self-similar blowup profile. Such information has been used in an essential way in establishing the nonlinear stability of the blowup profile in [8,18,19].

math.AP↗

arXiv2022

Finite time blowup of 2D Boussinesq and 3D Euler equations with $C^{1,α}$ velocity and boundary

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 velocity, we prove the finite time singularity for the 2D Boussinesq and the 3D axisymmetric Euler equations with boundary and $C^{1,α}$ 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 velocity. We use a dynamic rescaling formulation and follow the general framework of analysis developed 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 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. In the previous version of this paper, we proved the blowup results for the 3D axisymmetric Euler equations with initial data $(u_0^θ)^2, u_0^r, u_0^z \in C^{1,α}$. Though the velocity $u^r, u^z$ in the axisymmetric setting is $C^{1,α}$, our interpretation that the velocity is $C^{1,α}$ is not correct since the velocity in 3D also depends on $u^θ$, which is not $C^{1,α}$. This oversight can be fixed easily with a minor change in the construction of the approximate steady state and a minor modification to localize the approximate steady state.

math.AP↗

arXiv2022

Potential singularity formation of incompressible axisymmetric Euler equations with degenerate viscosity coefficients

In this paper, we present strong numerical evidences that the incompressible axisymmetric Euler equations with degenerate viscosity coefficients and smooth initial data of finite energy develop a potential finite-time locally self-similar singularity at the origin. An important feature of this potential singularity is that the solution develops a two-scale traveling wave that travels towards the origin. The two-scale feature is characterized by the scaling property that the center of the traveling wave is located at a ring of radius $O((T-t)^{1/2})$ surrounding the symmetry axis while the thickness of the ring collapses at a rate $O(T-t)$. The driving mechanism for this potential singularity is due to an antisymmetric vortex dipole that generates a strong shearing layer in both the radial and axial velocity fields. Without the viscous regularization, the $3$D Euler equations develop a sharp front and some shearing instability in the far field. On the other hand, the Navier-Stokes equations with a constant viscosity coefficient regularize the two-scale solution structure and do not develop a finite-time singularity for the same initial data.

math.AP↗

arXiv2022

Potential singularity of the 3D Euler equations in the interior domain

Whether the 3D incompressible Euler equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this paper, we present some new numerical evidence that the 3D axisymmetric incompressible Euler equations with smooth initial data of finite energy develop a potential finite time singularity at the origin. This potential singularity is different from the blow-up scenario revealed by Luo-Hou in \cite{luo2014potentially,luo2014toward}, which occurs on the boundary. Our initial condition has a simple form and shares several attractive features of a more sophisticated initial condition constructed by Hou-Huang in \cite{Hou-Huang-2021,Hou-Huang-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 \cite{Hou-Huang-2021,Hou-Huang-2022}. More importantly, the solution seems to develop nearly self-similar scaling properties that are compatible with those of the 3D Navier-Stokes equations. We will present numerical evidence that the 3D Euler equations seem to develop a potential finite time singularity. Moreover, the nearly self-similar profile seems to be very stable to the small perturbation of the initial data.

math.AP↗

arXiv2022

The potentially singular behavior of the 3D Navier-Stokes equations

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 10^7. 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)=(\infty,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 L^3 norm of the velocity experiences rapid dynamic growth relative to the localized L^3 norm of the initial velocity. This provides further evidence for the potentially singular behavior of the NavieStokes equations.

physics.flu-dyn↗

arXiv2021

Multiscale Elliptic PDEs Upscaling and Function Approximation via Subsampled Data

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 that, 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.

math.NA↗

arXiv2021

Asymptotically self-similar blowup of the Hou-Luo model for the 3D Euler equations

Inspired by the numerical evidence of a potential 3D Euler singularity \cite{luo2014potentially,luo2013potentially-2}, we prove finite time singularity from smooth initial data for the HL model introduced by Hou-Luo in \cite{luo2014potentially,luo2013potentially-2} for the 3D Euler equations with boundary. Our finite time blowup solution for the HL model and the singular solution considered in \cite{luo2014potentially,luo2013potentially-2} 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 \cite{chen2019finite} 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 $γ\approx 1/3$ is uniformly bounded up to the singularity time.

math.AP↗

arXiv2021

Multiscale Invertible Generative Networks for High-Dimensional Bayesian Inference

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.

stat.ML↗

arXiv2021

Exponential Convergence for Multiscale Linear Elliptic PDEs via Adaptive Edge Basis Functions

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 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.

math.NA↗

arXiv2021

Fast Global Convergence for Low-rank Matrix Recovery via Riemannian Gradient Descent with Random Initialization

In this paper, we propose a new global analysis framework for a class of low-rank matrix recovery problems on the Riemannian manifold. We analyze the global behavior for the Riemannian optimization with random initialization. We use the Riemannian gradient descent algorithm to minimize a least squares loss function, and study the asymptotic behavior as well as the exact convergence rate. We reveal a previously unknown geometric property of the low-rank matrix manifold, which is the existence of spurious critical points for the simple least squares function on the manifold. We show that under some assumptions, the Riemannian gradient descent starting from a random initialization with high probability avoids these spurious critical points and only converges to the ground truth in nearly linear convergence rate, i.e. $\mathcal{O}(\text{log}(\frac{1}ε)+ \text{log}(n))$ iterations to reach an $ε$-accurate solution. We use two applications as examples for our global analysis. The first one is a rank-1 matrix recovery problem. The second one is a generalization of the Gaussian phase retrieval problem. It only satisfies the weak isometry property, but has behavior similar to that of the first one except for an extra saddle set. Our convergence guarantee is nearly optimal and almost dimension-free, which fully explains the numerical observations. The global analysis can be potentially extended to other data problems with random measurement structures and empirical least squares loss functions.

stat.ML↗

arXiv2021

On the Finite Time Blowup of the De Gregorio Model for the 3D Euler Equation

We present a novel method of analysis and prove finite time asymptotically self-similar blowup of the De Gregorio model \cite{DG90,DG96} 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 \cite{OSW08} for the entire range of parameter on $\mathbb{R}$ or $S^1$ 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.

math.AP↗

arXiv2020

Function Approximation via The Subsampled Poincar\' e Inequality

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 Poincar\' e 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 Poincar\' e inequality is proposed to address this problem; its optimality is also discussed.

math.NA↗

arXiv2020

Analysis of Asymptotic Escape of Strict Saddle Sets in Manifold Optimization

In this paper, we provide some analysis on the asymptotic escape of strict saddles in manifold optimization using the projected gradient descent (PGD) algorithm. One of our main contributions is that we extend the current analysis to include non-isolated and possibly continuous saddle sets with complicated geometry. We prove that the PGD is able to escape strict critical submanifolds under certain conditions on the geometry and the distribution of the saddle point sets. We also show that the PGD may fail to escape strict saddles under weaker assumptions even if the saddle point set has zero measure and there is a uniform escape direction. We provide a counterexample to illustrate this important point. We apply this saddle analysis to the phase retrieval problem on the low-rank matrix manifold, prove that there are only a finite number of saddles, and they are strict saddles with high probability. We also show the potential application of our analysis for a broader range of manifold optimization problems.

math.OC↗

arXiv2019

A class of robust numerical methods for solving dynamical systems with multiple time scales

In this paper, we develop a class of robust numerical methods for solving dynamical systems with multiple time scales. We first represent the solution of a multiscale dynamical system as a transformation of a slowly varying solution. Then, under the scale separation assumption, we provide a systematic way to construct the transformation map and derive the dynamic equation for the slowly varying solution. We also provide the convergence analysis of the proposed method. Finally, we present several numerical examples, including ODE system with three and four separated time scales to demonstrate the accuracy and efficiency of the proposed method. Numerical results verify that our method is robust in solving ODE systems with multiple time scale, where the time step does not depend on the multiscale parameters.

math.NA↗

arXiv2019

A Pseudo Knockoff Filter for Correlated Features

In 2015, Barber and Candes introduced a new variable selection procedure called the knockoff filter to control the false discovery rate (FDR) and prove that this method achieves exact FDR control. Inspired by the work of Barber and Candes (2015), we propose and analyze 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 rigorous 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 FDP bound and an expectation inequality.

stat.ME↗