arXiv subjects

Rui Peng Li

Publications and source records attributed to Rui Peng Li.

7 recordsLinked to original sources

arXiv2026

Graph Neural Multilevel Preconditioners for Iterative Solvers

Solving large, sparse linear systems is a core task in scientific computing, and efficient iterative solvers rely critically on effective and robust preconditioning. While classical methods such as algebraic multigrid (AMG) are highly scalable, their robustness can degrade on indefinite or nonsymmetric systems where heuristics originally developed for elliptic PDEs are less reliable. Recently, Graph Neural Networks (GNNs) have emerged as data-driven preconditioners; yet, the practical impact of imposing an AMG-style hierarchy remains underexplored for general sparse matrices. In this work, we propose a Graph Neural Multilevel Preconditioner (GMP) that adopts an AMG hierarchy as a structural prior and learns smoothing, restriction, and interpolation operators in a unified framework. Our method targets general sparse systems and is instantiated as a drop-in preconditioner for standard Krylov solvers. On a benchmark of over 800 sparse matrices, we compare against classical AMG, single-level ILUT, and state-of-the-art GNN preconditioners, and characterize the regimes where multilevel graph neural preconditioning improves convergence or, conversely, introduces overhead relative to strong single-level baselines. These results highlight both the promise and the limitations of enforcing AMG-style multilevel structure in learned preconditioners for large-scale scientific simulations.

math.NA↗

arXiv2026

Factored Sparse Approximate Inverse Preconditioning via Spectral Optimization

In this paper, we study value selection for fixed-pattern factorized sparse approximate inverse preconditioners. Given a prescribed sparsity pattern for a factor $G,$ we choose its admissible entries by optimizing spectral objectives of the congruent preconditioned operator $P(G)=GAG^T.$ This differs from classical sparse approximate inverse and FSAI constructions, which choose entries through algebraic Frobenius-residual criteria. For symmetric positive definite systems, the spectral target is a cluster near $+1.$ For symmetric indefinite systems, where congruence preserves inertia, we introduce a bimodal loss that drives positive and negative eigenvalues toward separated clusters near $+1$ and $-1,$ while penalizing eigenvalues near zero. To make these objectives practical for large sparse matrices, we derive projected Krylov support-gradients. Lanczos runs provide both a stochastic trace estimate of the spectral objective and a Ritz approximation to the exact gradient. We implement the resulting gradient through a detached Rayleigh surrogate: the Lanczos data are computed without gradient tracking and held fixed, while the backward pass differentiates only recomputed Rayleigh quotients with respect to the admissible entries of $G.$ This avoids differentiating through the Lanczos recurrence while returning a matrix-free gradient on the prescribed support. We also discuss a projected Kernel Polynomial Method rule as a finite polynomial comparison. Experiments on finite-element test problems show that spectral value selection improves fixed-support preconditioners, especially for symmetric indefinite saddle-point systems. We further demonstrate a graph neural network model for predicting admissible factor entries across related matrices.

math.NA↗

arXiv2026

Hybrid Digital-Analog Approximate Inverse Preconditioning for Krylov Methods

Analog in-memory computing enables highly parallel matrix-vector multiplications with reduced data movement, but the resulting operations are noisy, quantized, and affected by device- and circuit-level non-idealities. This paper studies approximate inverse preconditioning for Krylov subspace methods in a hybrid digital-analog setting. The digital host performs sparse products with the coefficient matrix and the precision-sensitive Krylov operations, while preconditioner applications are performed through analog crossbar matrix-vector multiplications. Since the realized preconditioner is inexact and application-dependent, the outer iteration is formulated as the flexible GMRES method. We show that analog execution changes the usual preconditioner design problem in the sense that a stronger digital preconditioner may be less effective after analog scaling, write noise, input/output perturbations, quantization, and clipping are taken into account. We compare various block Jacobi preconditioning schemes including exact block inverses, sparse approximate inverses, Monte Carlo approximate inverses (MCAI), damping, and nested block Jacobi schemes. Numerical experiments with realistic analog matrix-vector simulations show that analog-aware choices of block size, damping, MCAI construction accuracy, and nesting are important for robust convergence.

math.NA↗

arXiv2026

Free-RBF-KAN: Kolmogorov-Arnold Networks with Adaptive Radial Basis Functions for Efficient Function Learning

Kolmogorov-Arnold Networks (KANs) offer a promising framework for approximating complex nonlinear functions, yet the original B-spline formulation suffers from significant computational overhead due to De Boor algorithm. While recent RBF-based variants improve efficiency, they often sacrifice the approximation accuracy inherent in the original spline-based design. To bridge this gap, we propose Free-RBF-KAN, an architecture that integrates adaptive learning grids and trainable smoothness parameters to enable expressive, high-resolution function approximation. Our method utilizes learnable RBF shapes that dynamically align with activation patterns, and we provide the first formal universal approximation proof for the RBF-KAN family. Empirical evaluations across multiscale regression, physics-informed PDEs, and operator learning demonstrate that Free-RBF-KAN can achieve accuracy comparable to its B-spline counterparts while delivering significantly faster training and inference. These results establish Free-RBF-KAN as an efficient and adaptive alternative for high-dimensional structured modeling tasks.

cs.LG↗

arXiv2025

Multiscale Neural Networks for Approximating Green's Functions

Neural networks (NNs) have been widely used to solve partial differential equations (PDEs) in the applications of physics, biology, and engineering. One effective approach for solving PDEs with a fixed differential operator is learning Green's functions. However, Green's functions are notoriously difficult to learn due to their poor regularity, which typically requires larger NNs and longer training times. In this paper, we address these challenges by leveraging multiscale NNs to learn Green's functions. Through theoretical analysis using multiscale Barron space methods and experimental validation, we show that the multiscale approach significantly reduces the necessary NN size and accelerates training.

math.NA↗

arXiv2025

Neural Approximate Inverse Preconditioners

In this paper, we propose a data-driven framework for constructing efficient approximate inverse preconditioners for elliptic partial differential equations (PDEs) by learning the Green's function of the underlying operator with neural networks (NNs). The training process integrates four key components: an adaptive multiscale neural architecture ($α$MSNN) that captures hierarchical features across near-, middle-, and far-field regimes; the use of coarse-grid anchor data to ensure physical identifiability; a multi-$\varepsilon$ staged training protocol that progressively refines the Green's function representation across spatial scales; and an overlapping domain decomposition that enables local adaptation while maintaining global consistency. Once trained, the NN-approximated Green's function is directly compressed into either a hierarchical ($\mathcal{H}$-) matrix or a sparse matrix-using only the mesh geometry and the network output. This geometric construction achieves nearly linear complexity in both setup and application while preserving the spectral properties essential for effective preconditioning. Numerical experiments on challenging elliptic PDEs demonstrate that the resulting preconditioners consistently yield fast convergence and small iteration counts.

math.NA↗

arXiv2023

Optimizing Irregular Communication with Neighborhood Collectives and Locality-Aware Parallelism

Irregular communication often limits both the performance and scalability of parallel applications. Typically, applications individually implement irregular messages using point-to-point communications, and any optimizations are added directly into the application. As a result, these optimizations lack portability. There is no easy way to optimize point-to-point messages within MPI, as the interface for single messages provides no information on the collection of all communication to be performed. However, the persistent neighbor collective API, released in the MPI 4 standard, provides an interface for portable optimizations of irregular communication within MPI libraries. This paper presents methods for optimizing irregular communication within neighborhood collectives, analyzes the impact of replacing point-to-point communication in existing codebases such as Hypre BoomerAMG with neighborhood collectives, and finally shows an up to 1.32x speedup on sparse matrix-vector multiplication within a BoomerAMG solve through the use of our optimized neighbor collectives. The authors analyze multiple implementations of neighborhood collectives, including a standard implementation, which simply wraps standard point-to-point communication, as well as multiple implementations of locality-aware aggregation. All optimizations are available in an open-source codebase, MPI Advance, which sits on top of MPI, allowing for optimizations to be added into existing codebases regardless of the system MPI install.

cs.DC↗