{"title":"An iterative method for the solution of Laplace-like equations in high and very high space dimensions","authors":"","doi":"10.1007/s00211-024-01401-2","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>This paper deals with the equation <span> <span>\\(-\\varDelta u+\\mu u=f\\)</span> </span> on high-dimensional spaces <span> <span>\\({\\mathbb {R}}^m\\)</span> </span>, where the right-hand side <span> <span>\\(f(x)=F(Tx)\\)</span> </span> is composed of a separable function <em>F</em> with an integrable Fourier transform on a space of a dimension <span> <span>\\(n>m\\)</span> </span> and a linear mapping given by a matrix <em>T</em> of full rank and <span> <span>\\(\\mu \\ge 0\\)</span> </span> is a constant. For example, the right-hand side can explicitly depend on differences <span> <span>\\(x_i-x_j\\)</span> </span> of components of <em>x</em>. Following our publication (Yserentant in Numer Math 146:219–238, 2020), we show that the solution of this equation can be expanded into sums of functions of the same structure and develop in this framework an equally simple and fast iterative method for its computation. The method is based on the observation that in almost all cases and for large problem classes the expression <span> <span>\\(\\Vert T^ty\\Vert ^2\\)</span> </span> deviates on the unit sphere <span> <span>\\(\\Vert y\\Vert =1\\)</span> </span> the less from its mean value the higher the dimension <em>m</em> is, a concentration of measure effect. The higher the dimension <em>m</em>, the faster the iteration converges.</p>","PeriodicalId":49733,"journal":{"name":"Numerische Mathematik","volume":null,"pages":null},"PeriodicalIF":2.1000,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Numerische Mathematik","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00211-024-01401-2","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
This paper deals with the equation \(-\varDelta u+\mu u=f\) on high-dimensional spaces \({\mathbb {R}}^m\), where the right-hand side \(f(x)=F(Tx)\) is composed of a separable function F with an integrable Fourier transform on a space of a dimension \(n>m\) and a linear mapping given by a matrix T of full rank and \(\mu \ge 0\) is a constant. For example, the right-hand side can explicitly depend on differences \(x_i-x_j\) of components of x. Following our publication (Yserentant in Numer Math 146:219–238, 2020), we show that the solution of this equation can be expanded into sums of functions of the same structure and develop in this framework an equally simple and fast iterative method for its computation. The method is based on the observation that in almost all cases and for large problem classes the expression \(\Vert T^ty\Vert ^2\) deviates on the unit sphere \(\Vert y\Vert =1\) the less from its mean value the higher the dimension m is, a concentration of measure effect. The higher the dimension m, the faster the iteration converges.
Abstract This paper deals with the equation \(-\varDelta u+\mu u=f\) on high-dimensional spaces \({\mathbb {R}}^m\) , where the right-hand side \(f(x)=F(Tx)\) is composed of a separable function F with an integrable Fourier transform on a space of dimension \(n>. m\) and linear mapping given by a matrix T full rank and\(mu\ge 0\) is a constant；m) 和一个全秩矩阵 T 给出的线性映射，并且 \(\mu \ge 0\) 是一个常数。继我们的出版物（Yserentant in Numer Math 146:219-238，2020）之后，我们展示了该方程的解可以扩展为相同结构的函数之和，并在此框架下开发了一种同样简单而快速的迭代计算方法。该方法基于以下观察：在几乎所有情况下，对于大的问题类别，表达式 \(\Vert T^ty\Vert ^2\)在单位球面上的偏差 \(\Vert y\Vert =1\)，维度 m 越高，偏离其平均值的程度越小，这是一种度量集中效应。维数 m 越高，迭代收敛越快。
期刊介绍：
Numerische Mathematik publishes papers of the very highest quality presenting significantly new and important developments in all areas of Numerical Analysis. "Numerical Analysis" is here understood in its most general sense, as that part of Mathematics that covers:
1. The conception and mathematical analysis of efficient numerical schemes actually used on computers (the "core" of Numerical Analysis)
2. Optimization and Control Theory
3. Mathematical Modeling
4. The mathematical aspects of Scientific Computing
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