Hilbert Space Splittings and Iterative Methods

Griebel, Michael, Oswald, Peter

Hilbert Space Splittings and Iterative Methods
  • 出版商: Springer
  • 出版日期: 2024-11-07
  • 售價: $6,270
  • 貴賓價: 9.5 折 $5,957
  • 語言: 英文
  • 頁數: 440
  • 裝訂: Hardcover - also called cloth, retail trade, or trade
  • ISBN: 3031743695
  • ISBN-13: 9783031743696

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商品描述

This book is about the theory of so-called Schwarz methods for solving variational problems in a Hilbert space V arising from linear equations and their associated quadratic minimization problems. Schwarz methods are based on the construction of a sequence of approximate solutions by solving auxiliary variational problems on a set of (smaller, finite-dimensional) Hilbert spaces $V_i$ in a certain order, combining them, and using the combined approximations in an iterative procedure. The spaces $V_i$ form a so-called space splitting for V, they need not necessarily be subspaces of V, and their number can be finite or infinite.

The convergence behavior of Schwarz methods is influenced by certain properties of the space splittings they are based on. These properties are identified, and a detailed treatment of traditional deterministic and more recent greedy and stochastic orderings in the subproblem solution process is given, together with an investigation of accelerated methods. To illustrate the abstract theory, the numerical linear algebra analogs of the iterative methods covered in the book are discussed. Its standard application to the convergence theory of multilevel and domain decomposition methods for solving PDE problems is explained, and links to optimization theory and online learning algorithms are given.

Providing an introduction and overview of iterative methods which are based on problem decompositions and suitable for parallel and distributed computing, the book could serve as the basis for a one- or two-semester course for M.S. and Ph.D. students specializing in numerical analysis and scientific computing. It will also appeal to a wide range of researchers interested in scientific computing in the broadest sense.

商品描述(中文翻譯)

本書探討所謂的Schwarz方法理論，該方法用於解決來自線性方程及其相關的二次最小化問題的變分問題，這些問題位於希爾伯特空間V中。Schwarz方法基於通過在一組（較小、有限維度的）希爾伯特空間$V_i$中以特定順序解決輔助變分問題來構造一系列近似解，然後將它們結合，並在迭代過程中使用這些結合的近似解。空間$V_i$形成了V的所謂空間分割，它們不必一定是V的子空間，且其數量可以是有限或無限的。

Schwarz方法的收斂行為受到其所基於的空間分割的某些特性的影響。這些特性被識別出來，並詳細處理了傳統的確定性方法以及較新的貪婪和隨機排序在子問題解決過程中的應用，並對加速方法進行了研究。為了說明這一抽象理論，書中討論了與迭代方法相關的數值線性代數類比。標準應用於解決偏微分方程問題的多層次和區域分解方法的收斂理論也得到了說明，並提供了與優化理論和在線學習算法的聯繫。

本書提供了基於問題分解的迭代方法的介紹和概述，適合平行和分散計算，並可作為數值分析和科學計算專攻的碩士及博士生的一個或兩個學期課程的基礎。它也將吸引對科學計算有廣泛興趣的研究人員。

作者簡介

Michael Griebel received his education at the Technical University of Munich, Germany. He is a professor at the Institute for Numerical Simulation at the University of Bonn, Germany, where he holds the Chair of Scientific Computing and Numerical Simulation. Additionally, he is the director of Fraunhofer SCAI (Institute for Algorithms and Scientific Computing), Sankt Augustin, Germany. His research interests include numerical simulation, scientific computing, machine learning, and high-dimensional approximation. Since 2002, he has served as the Editor-in-Chief of the Springer journal Numerische Mathematik.

Peter Oswald received his education at Odessa State University and Moscow State University. He has held research, teaching, and professorship positions at various institutions, including TU Dresden, FSU Jena, Kuwait University, Texas A&M University, Bell Laboratories, Jacobs University Bremen, and the University of Bonn. His research interests include approximation theory, function spaces, and numerical analysis.

作者簡介(中文翻譯)

Michael Griebel 於德國慕尼黑工業大學接受教育。他是德國波恩大學數值模擬研究所的教授，擔任科學計算與數值模擬的講座教授。此外，他還是德國聖奧古斯丁的弗勞恩霍夫SCAI（算法與科學計算研究所）的主任。他的研究興趣包括數值模擬、科學計算、機器學習和高維近似。自2002年以來，他擔任施普林格期刊《Numerische Mathematik》的主編。

彼得·奧斯瓦爾德於敖德薩國立大學和莫斯科國立大學接受教育。他曾在多個機構擔任研究、教學和教授職位，包括德國德累斯頓工業大學、耶拿大學、科威特大學、德州農工大學、貝爾實驗室、布雷門雅各布大學和波恩大學。他的研究興趣包括近似理論、函數空間和數值分析。

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