{"title":"ℋ²矩阵的自适应快速乘法","authors":"Steffen Börm","doi":"10.1090/mcom/3978","DOIUrl":null,"url":null,"abstract":"<p>Hierarchical matrices approximate a given matrix by a decomposition into low-rank submatrices that can be handled efficiently in factorized form. <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper H squared\"> <mml:semantics> <mml:msup> <mml:mrow> <mml:mi mathvariant=\"script\">H</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding=\"application/x-tex\">\\mathcal {H}^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-matrices refine this representation following the ideas of fast multipole methods in order to achieve linear, i.e., optimal complexity for a variety of important algorithms.</p> <p>The matrix multiplication, a key component of many more advanced numerical algorithms, has been a challenge: the only linear-time algorithms known so far either require the very special structure of HSS-matrices or need to know a suitable basis for all submatrices in advance.</p> <p>In this article, a new and fairly general algorithm for multiplying <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper H squared\"> <mml:semantics> <mml:msup> <mml:mrow> <mml:mi mathvariant=\"script\">H</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding=\"application/x-tex\">\\mathcal {H}^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-matrices in linear complexity with adaptively constructed bases is presented. The algorithm consists of two phases: first an intermediate representation with a generalized block structure is constructed, then this representation is re-compressed in order to match the structure prescribed by the application.</p> <p>The complexity and accuracy are analyzed and numerical experiments indicate that the new algorithm can indeed be significantly faster than previous attempts.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"5 1","pages":""},"PeriodicalIF":2.2000,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Adaptive fast multiplication of ℋ²-matrices\",\"authors\":\"Steffen Börm\",\"doi\":\"10.1090/mcom/3978\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Hierarchical matrices approximate a given matrix by a decomposition into low-rank submatrices that can be handled efficiently in factorized form. <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"script upper H squared\\\"> <mml:semantics> <mml:msup> <mml:mrow> <mml:mi mathvariant=\\\"script\\\">H</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathcal {H}^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-matrices refine this representation following the ideas of fast multipole methods in order to achieve linear, i.e., optimal complexity for a variety of important algorithms.</p> <p>The matrix multiplication, a key component of many more advanced numerical algorithms, has been a challenge: the only linear-time algorithms known so far either require the very special structure of HSS-matrices or need to know a suitable basis for all submatrices in advance.</p> <p>In this article, a new and fairly general algorithm for multiplying <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"script upper H squared\\\"> <mml:semantics> <mml:msup> <mml:mrow> <mml:mi mathvariant=\\\"script\\\">H</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathcal {H}^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-matrices in linear complexity with adaptively constructed bases is presented. 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引用次数: 0
摘要
层次矩阵通过分解为低秩子矩阵来近似给定矩阵，这些子矩阵可以因式分解的形式高效处理。 H 2 \mathcal {H}^2 矩阵根据快速多极方法的思想完善了这种表示方法，从而实现了线性，即各种重要算法的最佳复杂性。矩阵乘法是许多更高级数值算法的关键组成部分，但一直是个难题：迄今已知的唯一线性时间算法要么需要 HSS 矩阵的特殊结构，要么需要事先知道所有子矩阵的合适基础。本文提出了一种新的、相当通用的算法，用于以线性复杂度与自适应构造的基相乘 H 2 \mathcal {H}^2 -matrices。该算法包括两个阶段：首先构建一个具有广义块结构的中间表示，然后对该表示进行重新压缩，以匹配应用所规定的结构。对复杂性和准确性进行了分析，数值实验表明，新算法确实比以前的尝试快得多。
Hierarchical matrices approximate a given matrix by a decomposition into low-rank submatrices that can be handled efficiently in factorized form. H2\mathcal {H}^2-matrices refine this representation following the ideas of fast multipole methods in order to achieve linear, i.e., optimal complexity for a variety of important algorithms.
The matrix multiplication, a key component of many more advanced numerical algorithms, has been a challenge: the only linear-time algorithms known so far either require the very special structure of HSS-matrices or need to know a suitable basis for all submatrices in advance.
In this article, a new and fairly general algorithm for multiplying H2\mathcal {H}^2-matrices in linear complexity with adaptively constructed bases is presented. The algorithm consists of two phases: first an intermediate representation with a generalized block structure is constructed, then this representation is re-compressed in order to match the structure prescribed by the application.
The complexity and accuracy are analyzed and numerical experiments indicate that the new algorithm can indeed be significantly faster than previous attempts.
期刊介绍：
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