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Huiyuan Li

Huiyuan Li appears in the imported research catalog. Authorship, coauthor and topic links are available while profile ownership is still unclaimed.

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math.NAMachine Learningmath.CADistributed, Parallel, and Cluster ComputingNumerical AnalysisPerformance

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Published work

13 published item(s)

preprint2026arXiv

Bridging the Gap between Sparse Matrix Reordering and Factorization: A Deep Learning Framework for Fill-in Reduction

Sparse matrix reordering can significantly reduce the fill-in during matrix factorization, thereby decreasing the computational and storage requirements in sparse matrix computations. Finding a minimal fill-in ordering is known to be an NP-hard problem. Moreover, there is a paradox: matrix reordering is applied before matrix factorization, but fill-ins that matrix reordering methods aim at are generated from matrix factorization. To bridge the gap between reordering and factorization, we propose a deep learning framework to minimize a fill-in surrogate function based on spectral embedding. First, we employ a multi-grid-like GNN architecture to learn to approximate the smallest eigenvectors of its graph Laplacian matrix, i.e. spectral embedding, and capture the global structural information of the matrix. Then, another multi-grid-like GNN architecture is used to minimize the potential space where fill-in can occur based on the rank distribution. Experimental results indicate that our approach achieves competitive performance compared with traditional graph-theoretic algorithms and deep learning methods.

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preprint2026arXiv

Learning Fill-in Reduction Ordering via Graph Policy Optimization for Sparse Matrices

Matrix reordering in large sparse solvers seeks a permutation that minimizes factorization fill-in to reduce memory and computation. Because the minimum fill-in ordering problem is NP-complete and fill-in is implicit in the sparsity pattern, graph-theoretic heuristics are used. Existing reinforcement learning methods either ignore sparsity patterns--missing the global fill-in--or lack local exact fill-in feedback. We propose a graph policy optimization method, modeling fill-ins from global and local views: both the policy and value networks use a multi-hop graph neural backbone to embed global fill-in; the policy further interacts with symbolic factorization over graphs to extract local, step-level fill-ins, and the resulting feedback is aligned with the value network via an adaptive saturation function to improve convergence. On the SuiteSparse Matrix Collection, our method achieves mean reductions of 29.3 in fill-ins and 31.3 in peak memory usage over state-of-the-art baselines.

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preprint2026arXiv

Self-Supervised Learning for Sparse Matrix Reordering

Rearranging the rows or columns of a sparse matrix using an appropriate ordering can significantly reduce fill-ins, i.e., new nonzeros introduced during matrix factorization, decreasing memory usage and runtime. However, finding an ordering that minimizes fill-ins is NP-complete. Existing approaches, including graph-theoretic and deep learning methods, rely on surrogate objectives without theoretical guarantees. The Fill-Path Theorem reveals a direct and intrinsic relationship between fill-in generation and the sparse structure of the matrix as path triplet inequalities. Here we first employ a multigrid graph network to capture structural information for each vertex. We then derive a triplet sampling strategy based on inequalities. Finally, we introduce an end-max chain loss function to reduce the number of triplets whose predicted scores satisfy these inequalities. Experimental evaluations on the publicly available SuiteSparse matrix collection demonstrate the superiority of the proposed method in terms of both fill-in reduction and speedup in LU factorization time.

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preprint2026arXiv

Shifting the Sweet Spot: High-Performance Matrix-Free Method for High-Order Elasticity

In high-order finite element analysis for elasticity, matrix-free (PA) methods are a key technology for overcoming the memory bottleneck of traditional Full Assembly (FA). However, existing implementations fail to fully exploit the special structure of modern CPU architectures and tensor-product elements, causing their performance "sweet spot" to anomalously remain at the low order of $p \approx 2$, which severely limits the potential of high-order methods. To address this challenge, we design and implement a highly optimized PA operator within the MFEM framework, deeply integrated with a Geometric Multigrid (GMG) preconditioner. Our multi-level optimization strategy includes replacing the original $O(p^6)$ generic algorithm with an efficient $O(p^4)$ one based on tensor factorization, exploiting Voigt symmetry to reduce redundant computations for the elasticity problem, and employing macro-kernel fusion to enhance data locality and break the memory bandwidth bottleneck. Extensive experiments on mainstream x86 and ARM architectures demonstrate that our method successfully shifts the performance "sweet spot" to the higher-order region of $p \ge 6$. Compared to the MFEM baseline, the optimized core operator (kernel) achieves speedups of 7x to 83x, which translates to a 3.6x to 16.8x end-to-end performance improvement in the complete solution process. This paper provides a validated and efficient practical path for conducting large-scale, high-order elasticity simulations on mainstream CPU hardware.

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preprint2020arXiv

Vectorial ball Prolate spheroidal wave functions with the divergence free constraint

In this paper, we introduce one family of vectorial prolate spheroidal wave functions of real order $α>-1$ on the unit ball in $R^3$, which satisfy the divergence free constraint, thus are termed as divergence free vectorial ball PSWFs. They are vectorial eigenfunctions of an integral operator related to the finite Fourier transform, and solve the divergence free constrained maximum concentration problem in three dimensions, i.e., to what extent can the total energy of a band-limited divergence free vectorial function be concentrated on the unit ball? Interestingly, any optimally concentrated divergence free vectorial functions, when represented in series in vector spherical harmonics, shall be also concentrated in one of the three vectorial spherical harmonics modes. Moreover, divergence free ball PSWFs are exactly the vectorial eigenfunctions of the second order Sturm-Liouville differential operator which defines the scalar ball PSWFs. Indeed, the divergence free vectorial ball PSWFs possess a simple and close relation with the scalar ball PSWFs such that they share the same merits. Simultaneously, it turns out that the divergence free ball PSWFs solve another second order Sturm-Liouville eigen equation defined through the curl operator $\nabla\times $ instead of the gradient operator $\nabla$.

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preprint2016arXiv

A fully diagonalized spectral method using generalized Laguerre functions on the half line

A fully diagonalized spectral method using generalized Laguerre functions is proposed and analyzed for solving elliptic equations on the half line. We first define the generalized Laguerre functions which are complete and mutually orthogonal with respect to an equivalent Sobolev inner product. Then the Fourier-like Sobolev orthogonal basis functions are constructed for the diagonalized Laguerre spectral method of elliptic equations. Besides, a unified orthogonal Laguerre projection is established for various elliptic equations. On the basis of this orthogonal Laguerre projection, we obtain optimal error estimates of the fully diagonalized Laguerre spectral method for both Dirichlet and Robin boundary value problems. Finally, numerical experiments, which are in agreement with the theoretical analysis, demonstrate the effectiveness and the spectral accuracy of our diagonalized method.

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preprint2016arXiv

Efficient Spectral and Spectral Element Methods for Eigenvalue Problems of Schrödinger Equations with an Inverse Square Potential

In this article, we study numerical approximation of eigenvalue problems of the Schrödinger operator $\displaystyle -Δu + \frac{c^2}{|x|^2}u$. There are three stages in our investigation: We start from a ball of any dimension, in which case the exact solution in the radial direction can be expressed by Bessel functions of fractional degrees. This knowledge helps us to design two novel spectral methods by modifying the polynomial basis to fit the singularities of the eigenfunctions. At the second stage, we move to circular sectors in the two dimensional setting. Again the radial direction can be expressed by Bessel functions of fractional degrees. Only in the tangential direction some modifications are needed from stage one. At the final stage, we extend the idea to arbitrary polygonal domains. We propose a mortar spectral element approach: a polygonal domain is decomposed into several sub-domains with each singular corner including the origin covered by a circular sector, in which origin and corner singularities are handled similarly as in the former stages, and the remaining domains are either a standard quadrilateral/triangle or a quadrilateral/triangle with a circular edge, in which the traditional polynomial based spectral method is applied. All sub-domains are linked by mortar elements (note that we may have hanging nodes). In all three stages, exponential convergence rates are achieved. Numerical experiments indicate that our new methods are superior to standard polynomial based spectral (or spectral element) methods and $hp$-adaptive methods. Our study offers a new and effective way to handle eigenvalue problems of the Schrödinger operator including the Laplacian operator on polygonal domains with reentrant corners.

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preprint2016arXiv

Spectral-Galerkin Approximation and Optimal Error Estimate for Stokes Eigenvalue Problems in Polar Geometries

In this paper we propose and analyze spectral-Galerkin methods for the Stokes eigenvalue problem based on the stream function formulation in polar geometries. We first analyze the stream function} formulated fourth-order equation under the polar coordinates, then we derive the pole condition and reduce the problem on a circular disk to a sequence of equivalent one-dimensional eigenvalue problems that can be solved in parallel. The novelty of our approach lies in the construction} of suitably weighted Sobolev spaces according to the pole conditions, based on which, the optimal error estimate for approximated eigenvalue of each one dimensional problem can be obtained. Further, we extend our method to the non-separable Stokes eigenvalue problem in an elliptic domain and establish the optimal error bounds. Finally, we provide some numerical experiments to validate our theoretical results and algorithms.

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preprint2013arXiv

Spectral approximation on the unit ball

Spectral approximation by polynomials on the unit ball is studied in the frame of the Sobolev spaces $W^{s}_p(\ball)$, $1<p<\infty$. The main results give sharp estimates on the order of approximation by polynomials in the Sobolev spaces and explicit construction of approximating polynomials. One major effort lies in understanding the structure of orthogonal polynomials with respect to an inner product of the Sobolev space $W_2^s(\ball)$. As an application, a direct and efficient spectral-Galerkin method based on our orthogonal polynomials is proposed for the second and the fourth order elliptic equations on the unit ball, its optimal error estimates are explicitly derived for both procedures in the Sobolev spaces and, finally, numerical examples are presented to illustrate the theoretic results.

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preprint2012arXiv

A New Triangular Spectral Element Method I: Implementation and Analysis on a Triangle

This paper serves as our first effort to develop a new triangular spectral element method (TSEM) on unstructured meshes, using the rectangle-triangle mapping proposed in the conference note [21]. Here, we provide some new insights into the originality and distinctive features of the mapping, and show that this transform only induces a logarithmic singularity, which allows us to devise a fast, stable and accurate numerical algorithm for its removal. Consequently, any triangular element can be treated as efficiently as a quadrilateral element, which affords a great flexibility in handling complex computational domains. Benefited from the fact that the image of the mapping includes the polynomial space as a subset, we are able to obtain optimal $L^2$- and $H^1$-estimates of approximation by the proposed basis functions on triangle. The implementation details and some numerical examples are provided to validate the efficiency and accuracy of the proposed method. All these will pave the way for developing an unstructured TSEM based on, e.g., the hybridizable discontinuous Galerkin formulation.

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preprint2012arXiv

Discrete Fourier Analysis and Chebyshev Polynomials with $G_2$ Group

The discrete Fourier analysis on the $30^{\degree}$-$60^{\degree}$-$90^{\degree}$ triangle is deduced from the corresponding results on the regular hexagon by considering functions invariant under the group $G_2$, which leads to the definition of four families generalized Chebyshev polynomials. The study of these polynomials leads to a Sturm-Liouville eigenvalue problem that contains two parameters, whose solutions are analogues of the Jacobi polynomials. Under a concept of $m$-degree and by introducing a new ordering among monomials, these polynomials are shown to share properties of the ordinary orthogonal polynomials. In particular, their common zeros generate cubature rules of Gauss type.

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preprint2008arXiv

Discrete Fourier analysis on a dodecahedron and a tetrahedron

A discrete Fourier analysis on the dodecahedron is studied, from which results on a tetrahedron is deduced by invariance. The results include Fourier analysis in trigonometric functions, interpolation and cubature formulas on these domains. In particular, a trigonometric Lagrange interpolation on the tetrahedron is shown to satisfy an explicit compact formula and the Lebesgue constant of the interpolation is shown to be in the order of $(\log n)^3$.

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preprint2008arXiv

Discrete Fourier analysis on fundamental domain of $A_d$ lattice and on simplex in $d$-variables

A discrete Fourier analysis on the fundamental domain $Ω_d$ of the $d$-dimensional lattice of type $A_d$ is studied, where $Ω_2$ is the regular hexagon and $Ω_3$ is the rhombic dodecahedron, and analogous results on $d$-dimensional simplex are derived by considering invariant and anti-invariant elements. Our main results include Fourier analysis in trigonometric functions, interpolation and cubature formulas on these domains. In particular, a trigonometric Lagrange interpolation on the simplex is shown to satisfy an explicit compact formula and the Lebesgue constant of the interpolation is shown to be in the order of $(\log n)^d$. The basic trigonometric functions on the simplex can be identified with Chebyshev polynomials in several variables already appeared in literature. We study common zeros of these polynomials and show that they are nodes for a family of Gaussian cubature formulas, which provides only the second known example of such formulas.

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