arXiv subjects

Eric T. Chung

Publications and source records attributed to Eric T. Chung.

At least 19 recordsLinked to original sources

arXiv2026

Topological spectral gap, multiscale Weyl's law, and homogenization in high-contrast PDEs

This paper introduces a unified abstract variational and topological framework to characterize the spectral gap and eigenvalue distribution in high-contrast multiscale partial differential equations (PDEs). We rigorously prove that the exact location of the spectral gap is universally determined by the dimension of the local null space associated with the high-contrast inclusions. For systems with infinite-dimensional kernels, this location is strictly determined by the topological Betti numbers. Furthermore, we establish a multiscale Weyl's law via a spectral decoupling theorem, demonstrating that as the contrast approaches infinity, the multiscale spectrum bifurcates into two independent components: the Dirichlet spectrum of the background matrix and the internal Neumann spectrum of the inclusions. Using spectral homogenization theory, we also show that in the limit of vanishing periodicity, this expanding topological eigenspace asymptotically spans the entire spectral space of the macroscopic homogenized operator. These theoretical results are comprehensively verified through numerical experiments on diffusion, elasticity, fourth-order plate, Maxwell, and grad-div operators.

math.NA↗

arXiv2026

Multiscale modeling for problems with high contrast heterogeneous coefficients by the CEM-GMsFEM

This review paper provides a comprehensive overview of the Constrained Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM) for solving elliptic PDEs characterized by highly heterogeneous, high-contrast coefficients. We detail the construction of multiscale basis functions via spectral auxiliary spaces, combined with an oversampling strategy that enables localized computations and guarantees exponential error decay. Rigorous error estimates are outlined for reference to confirm the method's optimal convergence and robustness. Numerical simulations are provided to verify the exponential decay property of the multiscale basis functions. Additionally, we discuss and comment several up-to-date applications of CEM-GMsFEMs.

math.NA↗

arXiv2026

MsFEM-Inspired CNNs with Transfer Learning for Multiscale Model Reduction

Deep learning-based surrogate models have been extensively developed for efficiently approximating multiscale systems with random input fields. However, most existing approaches require retraining neural networks from scratch when source terms, boundary conditions, or differential operators change, resulting in significant computational costs and limited adaptability. To address this challenge, we integrate our previous CNN-based reduced-order model (ROM) framework with the multiscale finite element method (MsFEM) and propose an MsFEM-inspired transfer learning strategy, termed MITL. The CNN-based ROM consists of two components: Basis CNNs, which learn reduced basis functions, and Coef CNNs, which predict the corresponding linear combination coefficients. To enhance the transferability of learned multiscale representations, global MsFEM basis problems are employed as source tasks during pretraining. For new target problems, MITL requires training only lightweight adaptation networks to construct task-specific reduced bases and coefficients, thereby substantially reducing the computational burden. Numerical experiments demonstrate that MITL achieves accurate and efficient predictions across a range of target tasks, with particularly significant advantages in data-scarce scenarios.

cs.CE↗

arXiv2026

Efficient Multiscale Methods for Highly Heterogeneous Spatial Network Models

Modeling complex spatial networks with multiscale heterogeneity poses significant mathematical and computational challenges. Lacking explicit PDE discretizations and facing excessive degrees of freedom, conventional methods often become computationally prohibitive. To address these challenges, we propose an efficient multiscale modeling for highly heterogeneous spatial networks. We construct multiscale basis functions tailored to spatial network models with heterogeneous edge weights and node degrees. A key novelty is that the proposed method doesn't introduce geometric parameters (such as Dirichlet nodes, distances, or mesh sizes), thereby preserving its purely algebraic nature and ensuring broad applicability. By incorporating a subgraph-wise estimate, we define a Poincar\'e constant $C_{\mathrm{po}}$ that renders the method independent of the underlying graph geometry. Then through an appropriate choice of the number of graph oversampling layers, we establish an $O(C_{\mathrm{po}})$ convergence independent of the local heterogeneity contrast. Notably, our scheme operates entirely within an algebraic framework, eliminating the need for Dirichlet nodes and positive-definiteness on specific matrices arising in the model. This flexibility enables the simulation of a wider range of physical models and accommodates various boundary conditions. Rigorous theoretical analyses are provided under suitable assumptions, and extensive numerical experiments validate the effectiveness of the proposed approach.

math.NA↗

arXiv2026

Decoupling scales via localized subspace iteration and temporal splitting for multiscale parabolic equations

Simulating diffusion in heterogeneous media presents a significant computational challenge, as resolving microscopic physical scales traditionally demands excessively fine computational grids. To overcome this barrier, we extend the Localized Subspace Iteration (LSI) framework to multiscale parabolic equations. The proposed method constructs optimal, low-dimensional trial spaces by iteratively approximating the dominant eigenspaces of local inverse operators via Localized Standard Subspace Iteration (LSSI) or Localized Krylov Subspace Iteration (LKSI). Because these LSI basis functions are inherently tailored to capture the slow-decaying, low-frequency modes of the parabolic solution, they naturally suppress error accumulation over long-term integration. To further improve computational efficiency, we decouple the basis construction into an offline phase and implement a contrast-independent, partially explicit temporal splitting scheme for online time-stepping. By explicitly advancing the dominant macroscopic modes while implicitly treating high-frequency microscopic corrections, this scheme guarantees stability without imposing restrictive time-step constraints. We establish rigorous a priori error estimates in both the energy and $L^2$ norms. Numerical experiments illustrate the accuracy and efficiency of the LSI framework, particularly highlighting the LKSI method's advantages in handling high-contrast, complex multiscale media.

math.NA↗

arXiv2026

Multiscale Modeling for Time-harmonic Maxwell equations with impedance boundary conditions in highly heterogeneous media

Modeling time-harmonic Maxwell problems in heterogeneous media presents significant mathematical and computational challenges. Due to the inherent non-elliptic structure and non-coercive nature of Maxwell equations, conventional methods face severe numerical instabilities, particularly in high-contrast media and at high wave numbers. These challenges often lead to ill-conditioned discrete systems and prohibitively high computational costs, limiting their practical applicability. To overcome these challenges, we introduce an efficient multiscale framework for time-harmonic Maxwell equations with impedance boundary conditions in high-contrast media. A major novelty of this study lies in circumventing the need for an explicit divergence-free constraint on multiscale basis functions. To achieve this, an auxiliary space is constructed via local spectral problems incorporating a mass term and a Silver-M\"uller-type boundary penalty. This novel design guarantees the coercivity of the corresponding bilinear form and automatically excludes the kernel of the curl operator from the leading eigenspaces. Building upon the auxiliary space, we then construct the multiscale space by using a distinct bilinear form. By exploiting a resolution condition and establishing key norm relationships, we rigorously prove the coercivity of this modified bilinear form a crucial property that underpins the whole analysis. Theoretical analysis shows that, with appropriate oversampling, the method achieves $O (H)$ convergence independent of the local contrast and the approximation error increases with the wave number $k$. Extensive numerical experiments are reported to validate the effectiveness of the proposed approach.

math.NA↗

arXiv2026

Graph-Based Meshfree Multi-scale Coarse Space Approximation for Two-Level Schwarz Methods

Efficient simulation of Darcy flow in highly heterogeneous porous media requires iterative solvers that remain robust under large permeability contrasts and mixed boundary conditions. Spectral coarse spaces in two-level overlapping Schwarz methods provide such robustness, but their practical use is often limited by an expensive setup phase dominated by many local generalized eigenvalue solves. We propose a purely algebraic, coarse-space approximation that avoids these repeated local eigensolves by using a graph neural network operating on the system-matrix graph. On the analysis side, we introduce a coefficient-weighted subspace-distance measure to quantify the discrepancy between the approximated and target local multiscale coarse spaces, and we derive a condition-number bound for the resulting preconditioned operator in terms of this distance. This bound yields a principled supervised-training objective and links learning error to solver performance. Numerical experiments on 2D and 3D high-contrast Darcy systems with varying mixed boundary conditions demonstrate that the proposed approach substantially reduces setup cost and improves end-to-end time-to-solution, while preserving robust convergence across the tested contrasts and boundary configurations.

math.NA↗

arXiv2025

Iterative Contact-resolving Hybrid Methods for Multiscale Contact Mechanics

Modeling contact mechanics with high contrast coefficients presents significant mathematical and computational challenges, especially in achieving strongly symmetric stress approximations for mixed formulations. Due to the inherent nonlinearity of contact problems, conventional methods that treat the entire domain as a monolithic system often lead to high global complexity. To address this, we develop an iterative contact-resolving hybrid method by localizing nonlinear contact constraints within a smaller subdomain, while the larger subdomain is governed by a linear system. Our system employs variational inequality theory, minimization principles, and penalty methods. More importantly, we propose four discretization types within the two-subdomain framework, ranging from applying standard/mixed FEM across the entire domain to combining standard/mixed multiscale methods in the larger subdomain with standard/mixed FEM in the smaller one. % The standard finite element method and standard constraint energy minimizing generalized multiscale finite element method are simple and easy to demonstrate. By employing a multiscale reduction technique, the method avoids excessive degrees of freedom inherent in conventional methods in the larger domain, while the mixed formulation enables direct stress computation, ensures local momentum conservation, and resists locking in nearly incompressible materials. Convergence analysis and the corresponding algorithms are provided for all cases. Extensive numerical experiments are presented to validate the effectiveness of the approaches.

math.NA↗

arXiv2025

Multiscale Methods for wave propagation in materials with sign-changing coefficients

From a mathematical perspective, the extraordinary properties of metamaterials are often reflected in the coefficients of the governing partial differential equations (PDEs). These coefficients may fall outside the assumptions of classical theory, particularly when the effective dielectric permittivity and/or magnetic permeability are negative. This situation can transform a coercive operator into a non-coercive one, potentially leading to ill-posedness. In this paper, we utilize the Constraint Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM), specifically designed for time-harmonic electromagnetic wave problems, where the construction of auxiliary spaces in the original CEM-GMsFEM is tailored to accommodate the sign-changing setting. Based on the framework of \texttt{T}-coercivity theory and resolution conditions, we establish the inf-sup stability and provide an a priori error estimate for the proposed method. The numerical results demonstrate the effectiveness and robustness of our approach in handling such sophisticated coefficient profiles.

math.NA↗

arXiv2025

Multiscale modeling for contact problem with high-contrast heterogeneous coefficients with primary-dual formulation

In this paper, we propose a novel iterative multiscale framework for solving high-contrast contact problems of Signorini type. The method integrates the constrained energy minimizing generalized multiscale finite element method (CEM-GMsFEM) with a primal-dual active set strategy derived from semismooth Newton methods. First, local spectral problems are employed to construct an auxiliary multiscale space, from which energy minimizing multiscale basis functions are derived on oversampled domains, yielding a contrast-robust reduced-order approximation of the underlying partial differential equation. The multiscale bases are updated iteratively, but only at contact boundary, during the active set evolution process. Rigorous analysis is provided to establish error estimates and finite step convergence of the iterative scheme. Numerical experiments on heterogeneous media with high-contrast coefficients demonstrate that the proposed approach is both robust and efficient in capturing fine-scale features near contact boundaries.

math.NA↗

arXiv2025

Deep Global Model Reduction Learning in Porous Media Flow Simulation

In this paper, we combine deep learning concepts and some proper orthogonal decomposition (POD) model reduction methods for predicting flow in heterogeneous porous media. Nonlinear flow dynamics is studied, where the dynamics is regarded as a multi-layer network. The solution at the current time step is regarded as a multi-layer network of the solution at the initial time and input parameters. As for input, we consider various sources, which include source terms (well rates), permeability fields, and initial conditions. We consider the flow dynamics, where the solution is known at some locations and the data is integrated to the flow dynamics by modifying the reduced-order model. This approach allows modifying the reduced-order formulation of the problem. Because of the small problem size, limited observed data can be handled. We consider enriching the observed data using the computational data in deep learning networks. The basis functions of the global reduced order model are selected such that the degrees of freedom represent the solution at observation points. This way, we can avoid learning basis functions, which can also be done using neural networks. We present numerical results, where we consider channelized permeability fields, where the network is constructed for various channel configurations. Our numerical results show that one can achieve a good approximation using forward feed maps based on multi-layer networks.

math.NA↗

arXiv2025

A locking free multiscale method for linear elasticity in stress-displacement formulation with high contrast coefficients

Achieving strongly symmetric stress approximations for linear elasticity problems in high-contrast media poses a significant computational challenge. Conventional methods often struggle with prohibitively high computational costs due to excessive degrees of freedom, limiting their practical applicability. To overcome this challenge, we introduce an efficient multiscale model reduction method and a computationally inexpensive coarse-grid simulation technique for linear elasticity equations in highly heterogeneous, high-contrast media. We first utilize a stable stress-displacement mixed finite element method to discretize the linear elasticity problem and then present the construction of multiscale basis functions for the displacement and the stress. The mixed formulation offers several advantages such as direct stress computation without post-processing, local momentum conservation (ensuring physical consistency), and robustness against locking effects, even for nearly incompressible materials. Theoretical analysis confirms that our method is inf-sup stable and locking-free, with first-order convergence relative to the coarse mesh size. Notably, the convergence remains independent of contrast ratios as enlarging oversampling regions. Numerical experiments validate the method's effectiveness, demonstrating its superior performance even under extreme contrast conditions.

math.NA↗

arXiv2025

Homogenization rates of beam lattices to micropolar continua

As the size of a mechanical lattice with beam-modeled edges approaches zero, it undergoes homogenization into a continuum model, which exhibits unusual mechanical properties that deviate from classical Cauchy elasticity, named micropolar elasticity. Typically, the homogenization process is qualitative in the engineering community, lacking quantitative homogenization error estimates. In this paper, we rigorously analyze the homogenization process of a beam lattice to a continuum. Our approach is initiated from an engineered mechanical problem defined on a triangular lattice with periodic boundary conditions. By applying Fourier transformations, we reduce the problem to a series of equations in the frequency domain. As the lattice size approaches zero, this yields a homogenized model in the form of a partial differential equation with periodic boundary conditions. This process can be easily justified if the external conditions in the frequency domain are nonzero only at low-frequency modes. However, through numerical experiments, we discover that beyond the low-frequency regime, the homogenization of the beam lattice differs from classical periodic homogenization theory due to the additional rotational degrees of freedom in the beams. A crucial technique in our analysis is the decoupling of displacement and rotation fields, achieved through a linear algebraic manipulation known as the Schur complement. Through dedicated analysis, we establish the coercivity of the Schur complements in both lattice and continuum models, which enables us to derive convergence rate estimates for homogenization errors. Numerical experiments validate the optimality of the homogenization rate estimates.

math.NA↗

arXiv2025

Efficient numerical method for the Schrödinger equation with high-contrast potentials

In this paper, we study the Schrödinger equation in the semiclassical regime and with multiscale potential function. We develop the so-called constraint energy minimization generalized multiscale finite element method (CEM-GMsFEM), in the framework of Crank-Nicolson (CN) discretization in time. The localized multiscale basis functions are constructed by addressing the spectral problem and a constrained energy minimization problem related to the Hamiltonian norm. A first-order convergence in the energy norm and second-order convergence in the $L^2$ norm for our numerical scheme are shown, with a relation between oversampling number in the CEM-GMsFEM method, spatial mesh size and the semiclassical parameter provided. Furthermore, we demonstrate the convergence of the proposed Crank-Nicolson CEM-GMsFEM scheme. The convergence requires $H/\sqrtΛ=O(\varepsilon^{\frac{5}{4}})$, $Δt=O(\varepsilon^{\frac{5}{4}})$ if $\varepsilon\leq δ$; while if $δ<\varepsilon$, the convergence requires $H/\sqrtΛ=O(\varepsilon^{\frac{1}{4}}δ)$, $Δt=O(\frac{δ^2}{\varepsilon^{3/4}})$ (where $H$ represents the maximum diameter of coarse elements, $Λ$ is the minimal eigenvalue associated with the eigenvector not included in the auxiliary space, $Δt$ is the time step, $0 < \varepsilon\ll 1$ is the Planck constant and $δ$ describes the multiscale structure of the potential).Several numerical examples including 1D and 2D in space, with high-contrast potential are conducted to demonstrate the efficiency and accuracy of our proposed scheme.

math.NA↗

arXiv2025

MatExPre: A matrix exponential preconditioner for the high-frequency Helmholtz equation

In this article, we present a new preconditioner, MatExPre, for the high-frequency Helmholtz equation by leveraging the properties of matrix exponentials. Our approach begins by reformulating the Helmholtz equation into a Schrödinger-like equation and constructing a time-domain solver based on a fixed-point iteration. We then establish a rigorous connection between the time-domain solver and matrix exponential integrators, which enables us to derive algebraic preconditioners that rely solely on sparse matrix-vector products. Spectral analysis and a detailed numerical implementation strategy, including performance improvements achieved through complex shifting, are discussed. Finally, numerical experiments on 2D and large-scale 3D homogeneous and inhomogeneous models, including benchmark seismic examples, substantiate the effectiveness and scalability of the proposed methods.

math.NA↗

arXiv2025

Learning a generalized multiscale prolongation operator

In this research, we address Darcy flow problems with random permeability using iterative solvers, enhanced by a two-grid preconditioner based on a generalized multiscale prolongation operator, which has been demonstrated to be stable for high contrast profiles. To circumvent the need for repeatedly solving spectral problems with varying coefficients, we harness deep learning techniques to expedite the construction of the generalized multiscale prolongation operator. Considering linear transformations on multiscale basis have no impact on the performance of the preconditioner, we devise a loss function by the coefficient-based distance between subspaces instead of the plain $l^2$-norm of the difference of the corresponding multiscale bases. We discover that leveraging the inherent symmetry in the local spectral problem can effectively accelerate the neural network training process. In scenarios where training data are limited, we utilize the Karhunen-Loève expansion to augment the dataset. Extensive numerical experiments with various types of random coefficient models are exhibited, showing that the proposed method can significantly reduce the time required to generate the prolongation operator while maintaining the original efficiency of the two-grid preconditioner. Notably, the neural network demonstrates strong generalization capabilities, as evidenced by its satisfactory performance on unseen random permeability fields.

math.NA↗

arXiv2025

A robust solver for large-scale heat transfer topology optimization

This paper presents a large-scale parallel solver, specifically designed to tackle the challenges of solving high-dimensional and high-contrast linear systems in heat transfer topology optimization. The solver incorporates an interpolation technique to accelerate convergence in high-resolution domains, along with a multiscale multigrid preconditioner to handle complex coefficient fields with significant contrast. All modules of the optimization solver are implemented on a high performance computing cluster by the PETSc numerical library. Through a series of numerical investigations, we demonstrate the effectiveness of our approach in enhancing convergence and robustness during the optimization process, particularly in high-contrast scenarios with resolutions up to $1024^3$. Our performance results indicate that the proposed preconditioner achieves over $2\times$ speedup against the default algebraic multigrid in PETSc for high-contrast cases.

math.NA↗

arXiv2024

Constraint Energy Minimizing Generalized Multiscale Finite Element Method for Convection Diffusion Equations with Inhomogeneous Boundary Conditions

In this paper, we develop the constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) for convection-diffusion equations with inhomogeneous Dirichlet, Neumann and Robin boundary conditions, along with high-contrast coefficients. For time independent problems, boundary correctors $\mathcal{D}^m$ and $\mathcal{N}^{m}$ for Dirichlet, Neumann, and Robin conditions are designed. For time dependent problems, a scheme to update the boundary correctors is formulated. Error analysis in both cases is given to show the first-order convergence in energy norm with respect to the coarse mesh size $H$ and second-order convergence in $L^2-$norm, as verified by numerical examples, with which different finite difference schemes are compared for temporal discretization. Nonlinear problems are also demonstrated in combination with Strang splitting.

math.NA↗