arXiv subjects

Shubin Fu

Publications and source records attributed to Shubin Fu.

At least 19 recordsLinked to original sources

arXiv2026

A Fully Matrix-Free Three-Grid Preconditioner for the Time-Harmonic Maxwell Equations at Extreme Scale

Three-dimensional time-harmonic Maxwell simulations generate massive complex indefinite systems whose mesh coarsening is strictly limited by phase accuracy. Although matrix-free finite element kernels utilize GPU throughput efficiently, standard multilevel solvers are ultimately bottlenecked by the memory and communication costs of exact coarse-grid factorizations. We present a fully matrix-free, factorization-free three-grid preconditioner for curl-conforming N{\'e}delec discretizations with perfectly matched layers (PML) and optimally blended quadrature. The method employs an outer FGMRES to solve the unshifted fine-grid equation, while an intermediate-grid correction is computed by a fixed-work FGMRES preconditioned with a complex-shifted $2h$--$4h$ cycle. This strategically confines the complex shift to an auxiliary preconditioner, preserving the physical Maxwell operator. A local Fourier analysis derives the blended Maxwell branches and compatible edge transfers, identifying robust shift and Jacobi damping parameters. Validated against the analytical Maxwell Green tensor, our approach demonstrates extreme scalability: using a single solver configuration, both homogeneous and highly heterogeneous systems with approximately 10.89 billion complex edge unknowns are solved in 42.0--72.0 seconds on just 64 NVIDIA A100 GPUs.

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arXiv2026

A Massively Parallel Three-Grid Preconditioner for the High-Frequency Helmholtz Equation

Accurate simulation of three-dimensional time-harmonic wave propagation over many wavelengths requires control of phase error and efficient solution of large indefinite systems. We develop a three-grid solver based on the compact 27-point interpolated optimized finite-difference (IOFD) discretization. Its wavenumber-dependent stencil supports a fine-grid resolution of six points per shortest wavelength and an unshifted physical correction on the $2h$ grid at only three points per shortest wavelength. The method retains unshifted IOFD operators on the $h$ and $2h$ grids, while a complex-shifted $2h$--$4h$ auxiliary cycle preconditions a factorization-free iterative approximation of the coarse inverse. Restricting the shift to this auxiliary cycle preserves the propagative character of the coarse correction. Comparison with the outgoing Green function confirms phase and relative-amplitude accuracy on a sequence of meshes up to $6144^3$, with the largest problem spanning approximately 1024 wavelengths per coordinate. The same fixed solver configuration retains robust convergence across smooth, discontinuous, high-contrast, and geophysical velocity models and exhibits scalable parallel performance. In particular, a problem spanning approximately 340 wavelengths in each coordinate direction is solved in 18.1 seconds on just 64 NVIDIA A100 GPUs.

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arXiv2026

Edge Multiscale Finite Element Methods

The objective of this paper is to review recent developments in Edge Multiscale Finite Element Methods (EMsFEM) for partial differential equations with heterogeneous coefficients or highly oscillatory solutions. Using elliptic equations with heterogeneous coefficients as an illustrative example, we present the key ideas of the method. We also analyze the approach while accounting for the discrete error in the multiscale basis functions. Extensive numerical tests are provided to validate the performance of the method.

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arXiv2026

An algebraic multiscale preconditioner for large sparse SPD matrices

We present a two-grid algebraic multiscale preconditioner for large sparse symmetric positive definite systems arising from elliptic problems with highly heterogeneous coefficients. The coarse space is constructed directly from the system matrix by graph partitioning and local generalized eigenvalue solvers, yielding basis functions that capture the low-energy modes responsible for slow convergence. The method requires no geometric information, making it suitable for unstructured and matrix-only settings, and its construction is naturally parallelizable. Numerical results for heterogeneous Darcy flow problems show robustness with respect to coefficient contrast and problem size, better performance than standard algebraic multigrid on challenging large-scale cases, and good parallel scalability.

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arXiv2026

Applying Two-Grid Preconditioner for Subsurface Flow Simulation using Attention-enhanced Hybrid Network to Accelerate Multiscale Discretization in High-contrast Media

In this paper, we study the efficient numerical solution of Darcy equations in strongly heterogeneous media with high-contrast permeability and propose a hybrid framework that combines learning with multiscale numerical methods. The learning component is used for the prediction of multiscale basis functions in the mixed generalized multiscale finite element method (mixed GMsFEM), with the goal of reducing the repeated local computations required in the offline stage. Once these basis functions are predicted, the global system is assembled and the pressure field is computed by a two-grid preconditioned solver. The resulting method accelerates the costly local basis-construction stage while retaining the multiscale discretization and preconditioned iterative structure of the underlying solver. Numerical experiments on two-dimensional heterogeneous Darcy problems show that the proposed framework yields more accurate final pressure reconstruction than several representative learning-based methods and remains stable under strong heterogeneity and high-contrast coefficients. In comparison with the traditional mixed GMsFEM, its main advantage lies in the efficiency of the basis-generation stage, while the quality of the global solve is still ensured by the two-grid preconditioner. These results indicate that accelerating multiscale basis construction through learning, while preserving a mature numerical solver for the global problem, provides a viable approach for high-resolution Darcy-type simulations.

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arXiv2025

A residual driven multiscale method for Darcy's flow in perforated domains

In this paper, we present a residual-driven multiscale method for simulating Darcy flow in perforated domains, where complex geometries and highly heterogeneous permeability make direct simulations computationally expensive. To address this, we introduce a velocity elimination technique that reformulates the mixed velocity-pressure system into a pressure-only formulation, significantly reducing complexity by focusing on the dominant pressure variable. Our method is developed within the Generalized Multiscale Finite Element Method (GMsFEM) framework. For each coarse block, we construct offline basis functions from local spectral problems that capture key geometric and physical features. Online basis functions are then adaptively enriched using residuals, allowing the method to incorporate global effects such as source terms and boundary conditions, thereby improving accuracy. We provide detailed error analysis demonstrating how the offline and online spaces contribute to the accuracy and efficiency of the solution. Numerical experiments confirm the method's effectiveness, showing substantial reductions in computational cost while maintaining high accuracy, particularly through adaptive online enrichment. These results highlight the method's potential for efficient and accurate simulation of Darcy flow in complex, heterogeneous perforated domains.

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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 Schr\"{o}dinger-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.

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arXiv2025

A novel multipoint stress control volume method for linear elasticity on quadrilateral grids

In this paper, we develop a novel control volume method that is locally conservative and locking-free for linear elasticity problem on quadrilateral grids. The symmetry of stress is weakly imposed through the introduction of a Lagrange multiplier. As such, the method involves three unknowns: stress, displacement and rotation. To ensure the well-posedness of the scheme, a pair of carefully defined finite element spaces is used for the stress, displacement and rotation such that the inf-sup condition holds. An appealing feature of the method is that piecewise constant functions are used for the approximations of stress, displacement and rotation, which greatly simplifies the implementation. In particular, the stress space is defined delicately such that the stress bilinear form is localized around each vertex, which allows for the local elimination of the stress, resulting in a cell-centered system. By choosing different definitions of the space for rotation, we develop two variants of the method. In particular, the first method uses a constant function for rotation over the interaction region, which allows for further elimination and results in a cell-centered system involving displacement only. A rigorous error analysis is performed for the proposed scheme. We show the optimal convergence for $L^2$-error of the stress and rotation. Moreover, we can also prove the superconvergence for $L^2$-error of displacement. Extensive numerical simulations indicate that our method is efficient and accurate, and can handle problems with discontinuous coefficients.

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arXiv2024

An adaptive two-grid preconditioner and linearly implicit scheme for shale gas transport in fractured porous media

We consider a nonlinear mixed-dimensional model for simulating gas transport in shale formation. The mathematical model consists of a coupled system of nonlinear equations, where flow within fractures is represented using a lower-dimensional representation. For the numerical solution of the coupled transport problem, we construct an unstructured mesh that resolves lower dimensional fractures on the grid level and use the finite element approximation to build a discrete system. To construct an efficient scheme for the resulting nonlinear problem, we use an explicit-implicit method for time integration, where we carefully choose an additive partition of the nonlinear operators to separate the stiff linear component and integrate it implicitly to ensure the stability of the time integration. Next, we invert the linear partition of the operator by constructing an efficient two-grid preconditioner for shale gas transport in fractured porous media. We use a local pointwise smoother on the fine grid and carefully design an adaptive multiscale space for coarse grid approximation based on local generalized eigenvalue problems. We utilize an adaptive thresholding to automatically identify local dominant modes which correspond to the very small eigenvalues in local domains. We remark that such spatial features are automatically captured through our local spectral problems, and connect these to fracture information in the global formulation of the problem. Approximation properties of the local spectral space with convergence of the proposed two-grid algorithm are given. Numerical results are presented for two fracture distributions with 30 and 160 fractures, demonstrating iterative convergence independent of the contrast of fracture and porous matrix permeability.

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arXiv2024

Edge multiscale finite element methods for semilinear parabolic problems with heterogeneous coefficients

We develop a new spatial semidiscrete multiscale method based upon the edge multiscale methods to solve semilinear parabolic problems with heterogeneous coefficients and smooth initial data. This method allows for a cheap spatial discretization, which fails to resolve the spatial heterogeneity but maintains satisfactory accuracy independent of the heterogeneity. This is achieved by simultaneously constructing a steady-state multiscale ansatz space with certain approximation properties for the evolving solution and the initial data. The approximation properties of the multiscale ansatz space are derived using local-global splitting. A fully discrete scheme is analyzed using a first-order explicit exponential Euler scheme. We derive the error estimates in the $L^{2}$-norm and energy norm under the regularity assumptions for the semilinear term. The convergence rates depend on the coarse grid size and the level parameter. Finally, extensive numerical experiments are carried out to validate the efficiency of the proposed method.

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arXiv2024

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-Lo\`eve 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.

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arXiv2024

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.

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arXiv2024

On Edge Multiscale Space based Hybrid Schwarz Preconditioner for Helmholtz Problems with Large Wavenumbers

In this work, we develop a novel hybrid Schwarz method, termed as edge multiscale space based hybrid Schwarz (EMs-HS), for solving the Helmholtz problem with large wavenumbers. The problem is discretized using $H^1$-conforming nodal finite element methods on meshes of size $h$ decreasing faster than $k^{-1}$ such that the discretization error remains bounded as the wavenumber $k$ increases. EMs-HS consists of a one-level Schwarz preconditioner (RAS-imp) and a coarse solver in a multiplicative way. The RAS-imp preconditioner solves local problems on overlapping subdomains with impedance boundary conditions in parallel, and combines the local solutions using partition of unity. The coarse space is an edge multiscale space proposed in [13]. The key idea is to first establish a local splitting of the solution over each subdomain by a local bubble part and local Helmholtz harmonic extension part, and then to derive a global splitting by means of the partition of unity. This facilitates representing the solution as the sum of a global bubble part and a global Helmholtz harmonic extension part. We prove that the EMs-HS preconditioner leads to a convergent fixed-point iteration uniformly for large wavenumbers, by rigorously analyzing the approximation properties of the coarse space to the global Helmholtz harmonic extension part and to the solution of the adjoint problem. Distinctly, the theoretical convergence analysis are valid in two extreme cases: using minimal overlapping size among subdomains (of order $h$), or using coarse spaces of optimal dimension (of magnitude $k^d$, where $d$ is the spatial dimension). We provide extensive numerical results on the sharpness of the theoretical findings and also demonstrate the method on challenging heterogeneous models.

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arXiv2024

A fast cosine transformation accelerated method for predicting effective thermal conductivity

Predicting effective thermal conductivity by solving a Partial Differential Equation (PDE) defined on a high-resolution Representative Volume Element (RVE) is a computationally intensive task. In this paper, we tackle the task by proposing an efficient and implementation-friendly computational method that can fully leverage the computing power offered by hardware accelerators, namely, graphical processing units (GPUs). We first employ the Two-Point Flux-Approximation scheme to discretize the PDE and then utilize the preconditioned conjugate gradient method to solve the resulting algebraic linear system. The construction of the preconditioner originates from FFT-based homogenization methods, and an engineered linear programming technique is utilized to determine the homogeneous reference parameters. The fundamental observation presented in this paper is that the preconditioner system can be effectively solved using multiple Fast Cosine Transformations (FCT) and parallel tridiagonal matrix solvers. Regarding the fact that default multiple FCTs are unavailable on the CUDA platform, we detail how to derive FCTs from FFTs with nearly optimal memory usage. Numerical experiments including the stability comparison with standard preconditioners are conducted for 3D RVEs. Our performance reports indicate that the proposed method can achieve a $5$-fold acceleration on the GPU platform over the pure CPU platform and solve the problems with $512^3$ degrees of freedom and reasonable contrast ratios in less than $30$ seconds.

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arXiv2024

An efficient multiscale multigrid preconditioner for Darcy flow in high-contrast media

In this paper, we develop a multigrid preconditioner to solve Darcy flow in highly heterogeneous porous media. The key component of the preconditioner is to construct a sequence of nested subspaces $W_{\mathcal{L}}\subset W_{\mathcal{L}-1}\subset\cdots\subset W_1=W_h$. An appropriate spectral problem is defined in the space of $W_{i-1}$, then the eigenfunctions of the spectral problems are utilized to form $W_i$. The preconditioner is applied to solve a positive semidefinite linear system which results from discretizing the Darcy flow equation with the lowest order Raviart-Thomas spaces and adopting a trapezoidal quadrature rule. Theoretical analysis and numerical investigations of this preconditioner will be presented. In particular, we will consider several typical highly heterogeneous permeability fields whose resolutions are up to $1024^3$ and examine the computational performance of the preconditioner in several aspects, such as strong scalability, weak scalability, and robustness against the contrast of the media. We also demonstrate an application of this preconditioner for solving a two-phase flow benchmark problem.

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arXiv2024

A robust two-level overlapping preconditioner for Darcy flow in high-contrast media

In this article, a two-level overlapping domain decomposition preconditioner is developed for solving linear algebraic systems obtained from simulating Darcy flow in high-contrast media. Our preconditioner starts at a mixed finite element method for discretizing the partial differential equation by Darcy's law with the no-flux boundary condition and is then followed by a velocity elimination technique to yield a linear algebraic system with only unknowns of pressure. Then, our main objective is to design a robust and efficient domain decomposition preconditioner for this system, which is accomplished by engineering a multiscale coarse space that is capable of characterizing high-contrast features of the permeability field. A generalized eigenvalue problem is solved in each non-overlapping coarse element in a communication-free manner to form the global solver, which is accompanied by local solvers originated from additive Schwarz methods but with a non-Galerkin discretization to derive the two-level preconditioner. We provide a rigorous analysis that indicates that the condition number of the preconditioned system could be bounded above with several assumptions. Extensive numerical experiments with various types of three-dimensional high-contrast models are exhibited. In particular, we study the robustness against the contrast of the media as well as the influences of numbers of eigenfunctions, oversampling sizes, and subdomain partitions on the efficiency of the proposed preconditioner. Besides, strong and weak scalability performances are also examined.

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arXiv2023

Wavelet-based Edge Multiscale Finite Element Methods for Singularly Perturbed Convection-Diffusion Equations

We propose a novel efficient and robust Wavelet-based Edge Multiscale Finite Element Method (WEMsFEM) motivated by \cite{MR3980476,GL18} to solve the singularly perturbed convection-diffusion equations. The main idea is to first establish a local splitting of the solution over a local region by a local bubble part and local Harmonic extension part, and then derive a global splitting by means of Partition of Unity. This facilitates a representation of the solution as a summation of a global bubble part and a global Harmonic extension part, where the first part can be computed locally in parallel. To approximate the second part, we construct an edge multiscale ansatz space locally with hierarchical bases as the local boundary data that has a guaranteed approximation rate \noteLg{both inside and outside of the layers}. The key innovation of this proposed WEMsFEM lies in a provable convergence rate with little restriction on the mesh size. Its convergence rate with respect to the computational degree of freedom is rigorously analyzed, which is verified by extensive 2-d and 3-d numerical tests.

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arXiv2023

Convergence of the CEM-GMsFEM for compressible flow in highly heterogeneous media

This paper presents and analyses a Constraint Energy Minimization Generalized Multiscale Finite Element Method (CEM-GMsFEM) for solving single-phase non-linear compressible flows in highly heterogeneous media. The construction of CEM-GMsFEM hinges on two crucial steps: First, the auxiliary space is constructed by solving local spectral problems, where the basis functions corresponding to small eigenvalues are captured. Then the basis functions are obtained by solving local energy minimization problems over the oversampling domains using the auxiliary space. The basis functions have exponential decay outside the corresponding local oversampling regions. The convergence of the proposed method is provided, and we show that this convergence only depends on the coarse grid size and is independent of the heterogeneities. An online enrichment guided by \emph{a posteriori} error estimator is developed to enhance computational efficiency. Several numerical experiments on a three-dimensional case to confirm the theoretical findings are presented, illustrating the performance of the method and giving efficient and accurate numerical.

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