Asymptotic spectral properties and preconditioning of an approximated nonlocal Helmholtz equation with fractional Laplacian and variable coefficient wave number μ

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications Pub Date : 2024-12-27 DOI:10.1016/j.laa.2024.12.015

Andrea Adriani , Rosita L. Sormani , Cristina Tablino-Possio , Rolf Krause , Stefano Serra-Capizzano

{"title":"Asymptotic spectral properties and preconditioning of an approximated nonlocal Helmholtz equation with fractional Laplacian and variable coefficient wave number μ","authors":"Andrea Adriani , Rosita L. Sormani , Cristina Tablino-Possio , Rolf Krause , Stefano Serra-Capizzano","doi":"10.1016/j.laa.2024.12.015","DOIUrl":null,"url":null,"abstract":"<div><div>The current study investigates the asymptotic spectral properties of a finite difference approximation of nonlocal Helmholtz equations with a fractional Laplacian and a variable coefficient wave number <em>μ</em>, as it occurs when considering a wave propagation in complex media, characterized by nonlocal interactions and spatially varying wave speeds. More specifically, by using tools from Toeplitz and generalized locally Toeplitz theory, the present research delves into the spectral analysis of nonpreconditioned and preconditioned matrix sequences, with the main novelty regarding a complete picture of the case where <span><math><mi>μ</mi><mo>=</mo><mi>μ</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span> is nonconstant. We report numerical evidence supporting the theoretical findings. Finally, open problems and potential extensions in various directions are presented and briefly discussed.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"708 ","pages":"Pages 551-584"},"PeriodicalIF":1.0000,"publicationDate":"2024-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Linear Algebra and its Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0024379524004877","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

The current study investigates the asymptotic spectral properties of a finite difference approximation of nonlocal Helmholtz equations with a fractional Laplacian and a variable coefficient wave number μ, as it occurs when considering a wave propagation in complex media, characterized by nonlocal interactions and spatially varying wave speeds. More specifically, by using tools from Toeplitz and generalized locally Toeplitz theory, the present research delves into the spectral analysis of nonpreconditioned and preconditioned matrix sequences, with the main novelty regarding a complete picture of the case where

μ = μ (x, y)

is nonconstant. We report numerical evidence supporting the theoretical findings. Finally, open problems and potential extensions in various directions are presented and briefly discussed.

查看原文本刊更多论文

一类带分数阶拉普拉斯变系数波数μ的非局部近似Helmholtz方程的渐近谱性质和预处理

本文研究了具有分数阶拉普拉斯和变系数波数μ的非局部Helmholtz方程的有限差分近似的渐近谱性质，因为它发生在考虑具有非局部相互作用和空间变化波速的复杂介质中的波传播时。更具体地说，通过使用Toeplitz和广义局部Toeplitz理论的工具，本研究深入研究了非预条件和预条件矩阵序列的谱分析，主要新颖之处是关于μ=μ(x,y)是非常数情况的全貌。我们报告了支持理论发现的数值证据。最后，提出并简要讨论了有待解决的问题和在各个方向上的潜在扩展。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Linear Algebra and its Applications 数学-应用数学

CiteScore

2.20

自引率

9.10%

发文量

333

审稿时长

13.8 months

期刊介绍： Linear Algebra and its Applications publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra in their algebraic, arithmetic, combinatorial, geometric, or numerical aspects. It also publishes articles that give significant applications of matrix theory or linear algebra to other branches of mathematics and to other sciences. Articles that provide new information or perspectives on the historical development of matrix theory and linear algebra are also welcome. Expository articles which can serve as an introduction to a subject for workers in related areas and which bring one to the frontiers of research are encouraged. Reviews of books are published occasionally as are conference reports that provide an historical record of major meetings on matrix theory and linear algebra.