数值分析

The problem of optimal precision switching for the conjugate gradient (CG) method applied to sparse linear systems is considered. A sparse matrix is defined as an $n\!\times\!n$ matrix with $m\!=\!O(n)$ nonzero entries. The algorithm first…

The computation of a matrix function $f(A)$ is an important task in scientific computing appearing in machine learning, network analysis and the solution of partial differential equations. In this work, we use only matrix-vector products…

Shock waves in gas dynamics feature jump discontinuities that hinder numerical simulations. Viscous regularizations are prone to excessive dissipation of fine-scale structures. In this work, we propose the first inviscid regularization of…

We present a neural operator framework for solving inverse scattering problems. A neural operator produces a preliminary indicator function for the scatterer, which, after appropriate rescaling, is used as a regularization parameter within…

We consider potential-based interactions between beams (or fibers) and shells (or membranes) using a coarse-grained approach with focus on van der Waals attraction and steric repulsion. The involved 6D integral over volumes of a beam and a…

In the present paper, we are concerned with the study of matrix-sequences arising from the discretization of PDEs and FDEs on domains $\Omega \subset \mathbb{R}^d$ with finite measure. When $\Omega$ is either a hypercube or a bounded…

We develop an iterative refinement method that improves the accuracy of a user-chosen subset of $k$ eigenvectors ($k\ll n$) of an $n\times n$ real symmetric matrix. Using an orthogonal matrix represented in compact WY form, the method…

We propose a sparse interpolation construction and a practical coarsening algorithm for the algebraic multigrid (AMG) method, tailored towards H(curl). Building on the generalized AMG framework, we introduce an interior/exterior splitting…

This review article was first published in 2008 as chapter 11 in the book "Fast Fourier Transforms," edited by C. S. Burrus, for the Connexions project at Rice University, which is sadly no longer online. It gives a high-level overview of…

Can Machine Learning (ML) autonomously recover interpretable mathematical structure from raw numerical data? We aim to answer this question using the classification of real-root configurations of polynomials up to degree five as a…

We propose an extension of the classical susceptible infectious recovered (SIR) model that incorporates the effects of spatial propagation of an epidemic through a small number of additional compartments. The model is designed to capture…

Phase-field simulations are a practical but also expensive tool to calculate microstructural evolution. This work aims to compare explicit time integrators for a broad class of phase-field models involving coupling between the phase-field…

This work is concerned with relaxation models arising from numerical schemes for hyperbolic-parabolic systems. Such models are a hyperbolic system with both the hyperbolic part and the stiff source term involving a small positive parameter,…

Neural operators (NOs) struggle with high-contrast multiscale partial differential equations (PDEs), where fine-scale heterogeneities cause large errors. To address this, we use the Generalized Multiscale Finite Element Method (GMsFEM) that…

The Eberlein diagonalization method is an iterative Jacobi-type method for solving the eigenvalue problem of a general complex matrix. In this paper we develop the block version of the Eberlein method. We prove the global convergence of our…

Two different Sinc-collocation methods for Volterra integral equations of the second kind have been independently proposed by Stenger and Rashidinia--Zarebnia. However, their relation remains unexplored. This study theoretically examines…

Many time-dependent problems in the field of computational fluid dynamics can be solved using space-time methods. However, such methods can encounter issues with computational cost and robustness. In order to address these issues,…

A new fixed (non-adaptive) recursive scheme for multigrid algorithms is introduced. Governed by a positive parameter $\kappa$ called the cycle counter, this scheme generates a family of multigrid cycles dubbed $\kappa$-cycles. The…

Entropy correction artificial viscosity (ECAV) is an approach for enforcing a semi-discrete entropy inequality through an entropy dissipative correction term. The resulting method can be implemented as an artificial viscosity with an…

We introduce a modeling framework for an urban tram network based on a hyperbolic partial differential equation describing the transport of passengers along the network, coupled with a family of stochastic processes representing passenger…