Ruipeng Li

Ruipeng Li, Yuanzhe Xi, Eugene Vecharynski et al.

Polynomial filtering can provide a highly effective means of computing all eigenvalues of a real symmetric (or complex Hermitian) matrix that are located in a given interval, anywhere in the spectrum. This paper describes a technique for tackling this problem by combining a Thick-Restart version of the Lanczos algorithm with deflation (`locking') and a new type of polynomial filters obtained from a least-squares technique. The resulting algorithm can be utilized in a `spectrum-slicing' approach whereby a very large number of eigenvalues and associated eigenvectors of the matrix are computed by extracting eigenpairs located in different sub-intervals independently from one another.

Ruipeng Li, Yuanzhe Xi, Lucas Erlandson et al.

This paper describes a software package called EVSL (for EigenValues Slicing Library) for solving large sparse real symmetric standard and generalized eigenvalue problems. As its name indicates, the package exploits spectrum slicing, a strategy that consists of dividing the spectrum into a number of subintervals and extracting eigenpairs from each subinterval independently. In order to enable such a strategy, the methods implemented in EVSL rely on a quick calculation of the spectral density of a given matrix, or a matrix pair. What distinguishes EVSL from other currently available packages is that EVSL relies entirely on filtering techniques. Polynomial and rational filtering are both implemented and are coupled with Krylov subspace methods and the subspace iteration algorithm. On the implementation side, the package offers interfaces for various scenarios including matrix-free modes, whereby the user can supply his/her own functions to perform matrix-vector operations or to solve sparse linear systems. The paper describes the algorithms in EVSL, provides details on their implementations, and discusses performance issues for the various methods.

Colin Ponce, Ruipeng Li, Christina Mao et al.

A common challenge in regression is that for many problems, the degrees of freedom required for a high-quality solution also allows for overfitting. Regularization is a class of strategies that seek to restrict the range of possible solutions so as to discourage overfitting while still enabling good solutions, and different regularization strategies impose different types of restrictions. In this paper, we present a multilevel regularization strategy that constructs and trains a hierarchy of neural networks, each of which has layers that are wider versions of the previous network's layers. We draw intuition and techniques from the field of Algebraic Multigrid (AMG), traditionally used for solving linear and nonlinear systems of equations, and specifically adapt the Full Approximation Scheme (FAS) for nonlinear systems of equations to the problem of deep learning. Training through V-cycles then encourage the neural networks to build a hierarchical understanding of the problem. We refer to this approach as \emph{multilevel-in-width} to distinguish from prior multilevel works which hierarchically alter the depth of neural networks. The resulting approach is a highly flexible framework that can be applied to a variety of layer types, which we demonstrate with both fully-connected and convolutional layers. We experimentally show with PDE regression problems that our multilevel training approach is an effective regularizer, improving the generalize performance of the neural networks studied.

Shao-Ting Chiu, Siu Wun Cheung, Ulisses Braga-Neto et al.

Kolmogorov-Arnold Networks (KANs) have shown strong potential for efficiently approximating complex nonlinear functions. However, the original KAN formulation relies on B-spline basis functions, which incur substantial computational overhead due to De Boor's algorithm. To address this limitation, recent work has explored alternative basis functions such as radial basis functions (RBFs) that can improve computational efficiency and flexibility. Yet, standard RBF-KANs often sacrifice accuracy relative to the original KAN design. In this work, we propose Free-RBF-KAN, a RBF-based KAN architecture that incorporates adaptive learning grids and trainable smoothness to close this performance gap. Our method employs freely learnable RBF shapes that dynamically align grid representations with activation patterns, enabling expressive and adaptive function approximation. Additionally, we treat smoothness as a kernel parameter optimized jointly with network weights, without increasing computational complexity. We provide a general universality proof for RBF-KANs, which encompasses our Free-RBF-KAN formulation. Through a broad set of experiments, including multiscale function approximation, physics-informed machine learning, and PDE solution operator learning, Free-RBF-KAN achieves accuracy comparable to the original B-spline-based KAN while delivering faster training and inference. These results highlight Free-RBF-KAN as a compelling balance between computational efficiency and adaptive resolution, particularly for high-dimensional structured modeling tasks.

Ru Huang, Kai Chang, Huan He et al.

We propose a data-driven and machine-learning-based approach to compute non-Galerkin coarse-grid operators in algebraic multigrid (AMG) methods, addressing the well-known issue of increasing operator complexity. Guided by the AMG theory on spectrally equivalent coarse-grid operators, we have developed novel ML algorithms that utilize neural networks (NNs) combined with smooth test vectors from multigrid eigenvalue problems. The proposed method demonstrates promise in reducing the complexity of coarse-grid operators while maintaining overall AMG convergence for solving parametric partial differential equation (PDE) problems. Numerical experiments on anisotropic rotated Laplacian and linear elasticity problems are provided to showcase the performance and compare with existing methods for computing non-Galerkin coarse-grid operators.

Aoming Liang, Zhaoyang Mu, Qi liu et al.

Partial Differential Equations are foundational in modeling science and natural systems such as fluid dynamics and weather forecasting. The Latent Evolution of PDEs method is designed to address the computational intensity of classical and deep learning-based PDE solvers by proposing a scalable and efficient alternative. To enhance the efficiency and accuracy of LE-PDE, we incorporate the Mamba model, an advanced machine learning model known for its predictive efficiency and robustness in handling complex dynamic systems with a progressive learning strategy. The LE-PDE was tested on several benchmark problems. The method demonstrated a marked reduction in computational time compared to traditional solvers and standalone deep learning models while maintaining high accuracy in predicting system behavior over time. Our method doubles the inference speed compared to the LE-PDE while retaining the same level of parameter efficiency, making it well-suited for scenarios requiring long-term predictions.

Ru Huang, Ruipeng Li, Yuanzhe Xi

Multigrid methods are one of the most efficient techniques for solving linear systems arising from Partial Differential Equations (PDEs) and graph Laplacians from machine learning applications. One of the key components of multigrid is smoothing, which aims at reducing high-frequency errors on each grid level. However, finding optimal smoothing algorithms is problem-dependent and can impose challenges for many problems. In this paper, we propose an efficient adaptive framework for learning optimized smoothers from operator stencils in the form of convolutional neural networks (CNNs). The CNNs are trained on small-scale problems from a given type of PDEs based on a supervised loss function derived from multigrid convergence theories, and can be applied to large-scale problems of the same class of PDEs. Numerical results on anisotropic rotated Laplacian problems demonstrate improved convergence rates and solution time compared with classical hand-crafted relaxation methods.

Yuanzhe Xi, Ruipeng Li, Yousef Saad

The distribution of the eigenvalues of a Hermitian matrix (or of a Hermitian matrix pencil) reveals important features of the underlying problem, whether a Hamiltonian system in physics, or a social network in behavioral sciences. However, computing all the eigenvalues explicitly is prohibitively expensive for real-world applications. This paper presents two types of methods to efficiently estimate the spectral density of a matrix pencil $(A, B)$ when both $A$ and $B$ are Hermitian and, in addition, $B$ is positive definite. The first one is based on the Kernel Polynomial Method (KPM) and the second on Gaussian quadrature by the Lanczos procedure. By employing Chebyshev polynomial approximation techniques, we can avoid direct factorizations in both methods, making the resulting algorithms suitable for large matrices. Under some assumptions, we prove bounds that suggest that the Lanczos method converges twice as fast as the KPM method. Numerical examples further indicate that the Lanczos method can provide more accurate spectral densities when the eigenvalue distribution is highly non-uniform. As an application, we show how to use the computed spectral density to partition the spectrum into intervals that contain roughly the same number of eigenvalues. This procedure, which makes it possible to compute the spectrum by parts, is a key ingredient in the new breed of eigensolvers that exploit "spectrum slicing".

Ruipeng Li, Yousef Saad

This paper presents a parallel preconditioning method for distributed sparse linear systems, based on an approximate inverse of the original matrix, that adopts a general framework of distributed sparse matrices and exploits the domain decomposition method and low-rank corrections. The domain decomposition approach decouples the matrix and once inverted, a low-rank approximation is applied by exploiting the Sherman-Morrison-Woodbury formula, which yields two variants of the preconditioning methods. The low-rank expansion is computed by the Lanczos procedure with reorthogonalizations. Numerical experiments indicate that, when combined with Krylov subspace accelerators, this preconditioner can be efficient and robust for solving symmetric sparse linear systems. Comparisons with other distributed-memory preconditioning methods are presented.

Ruipeng Li, Yuanzhe Xi, Yousef Saad

This paper introduces a robust preconditioner for general sparse symmetric matrices, that is based on low-rank approximations of the Schur complement in a Domain Decomposition (DD) framework. In this "Schur Low Rank" (SLR) preconditioning approach, the coefficient matrix is first decoupled by DD, and then a low-rank correction is exploited to compute an approximate inverse of the Schur complement associated with the interface points. The method avoids explicit formation of the Schur complement matrix. We show the feasibility of this strategy for a model problem, and conduct a detailed spectral analysis for the relationship between the low-rank correction and the quality of the preconditioning. Numerical experiments on general matrices illustrate the robustness and efficiency of the proposed approach.