國立高雄大學圖資館 |

Language: English

Back

Hilbert space splittings and iterative methods

Record Type:	Electronic resources : Monograph/item
Title/Author:	Hilbert space splittings and iterative methodsby Michael Griebel, Peter Oswald.
Author:	Griebel, Michael.
other author:	Oswald, Peter.
Published:	Cham :Springer Nature Switzerland :2024.
Description:	ix, 440 p. :ill., digital ;24 cm.
Contained By:	Springer Nature eBook
Subject:	Hilbert space.
Online resource:	https://doi.org/10.1007/978-3-031-74370-2
ISBN:	9783031743702$q(electronic bk.)

Hilbert space splittings and iterative methods
Griebel, Michael.

Hilbert space splittings and iterative methods[electronic resource] /by Michael Griebel, Peter Oswald. - Cham :Springer Nature Switzerland :2024. - ix, 440 p. :ill., digital ;24 cm. - Springer series in computational mathematics,v. 622198-3712 ;. - Springer series in computational mathematics ;42..

1 Introduction -- 2 Hilbert space splittings: Abstract theory -- References -- 3 Hilbert space splittings: Examples and extensions -- References -- 4 Schwarz iterative methods: Finite Omega -- References -- 5 Special topics and extensions -- References -- 6 Schwarz approximation methods: Infinite Omega -- References -- 7 Applications to PDE solvers -- References -- A Hilbert space basics -- A.1 Spaces: Basic notation, definitions and properties -- A.2 Operators between Hilbert spaces -- A.3 Linear equations and variational problems -- A.4 Constructions on Hilbert spaces -- A.5 Sobolev spaces on domains -- A.6 Reproducing kernel Hilbert spaces (RKHS) -- References -- Index.

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 Vi in a certain order, combining them, and using the combined approximations in an iterative procedure. The spaces Vi 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.

ISBN: 9783031743702$q(electronic bk.)

Standard No.: 10.1007/978-3-031-74370-2doiSubjects--Topical Terms:

184635
Hilbert space.

LC Class. No.: QA322.4

Dewey Class. No.: 515.733

Hilbert space splittings and iterative methods
LDR:03495nmm a2200337 a 4500 001 672196
003 DE-He213
005 20241107115753.0
006 m d
007 cr nn 008maaau
008 250325s2024 sz s 0 eng d
020 $a 9783031743702$q(electronic bk.)
020 $a 9783031743696$q(paper)
024 7 $a 10.1007/978-3-031-74370-2 $2 doi
035 $a 978-3-031-74370-2
040 $a GP $c GP
041 0 $a eng
050 4 $a QA322.4
072 7 $a PBKS $2 bicssc
072 7 $a MAT041000 $2 bisacsh
072 7 $a PBKS $2 thema
082 0 4 $a 515.733 $2 23
090 $a QA322.4 $b .G848 2024
100 1 $a Griebel, Michael. $3 211017
245 1 0 $a Hilbert space splittings and iterative methods $h [electronic resource] / $c by Michael Griebel, Peter Oswald.
260 $a Cham : $b Springer Nature Switzerland : $b Imprint: Springer, $c 2024.
300 $a ix, 440 p. : $b ill., digital ; $c 24 cm.
490 1 $a Springer series in computational mathematics, $x 2198-3712 ; $v v. 62
505 0 $a 1 Introduction -- 2 Hilbert space splittings: Abstract theory -- References -- 3 Hilbert space splittings: Examples and extensions -- References -- 4 Schwarz iterative methods: Finite Omega -- References -- 5 Special topics and extensions -- References -- 6 Schwarz approximation methods: Infinite Omega -- References -- 7 Applications to PDE solvers -- References -- A Hilbert space basics -- A.1 Spaces: Basic notation, definitions and properties -- A.2 Operators between Hilbert spaces -- A.3 Linear equations and variational problems -- A.4 Constructions on Hilbert spaces -- A.5 Sobolev spaces on domains -- A.6 Reproducing kernel Hilbert spaces (RKHS) -- References -- Index.
520 $a 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 Vi in a certain order, combining them, and using the combined approximations in an iterative procedure. The spaces Vi 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.
650 0 $a Hilbert space. $3 184635
650 0 $a Iterative methods (Mathematics) $3 185853
650 1 4 $a Computational Mathematics and Numerical Analysis. $3 274020
700 1 $a Oswald, Peter. $3 175883
710 2 $a SpringerLink (Online service) $3 273601
773 0 $t Springer Nature eBook
830 0 $a Springer series in computational mathematics ; $v 42. $3 560664
856 4 0 $u https://doi.org/10.1007/978-3-031-74370-2
950 $a Mathematics and Statistics (SpringerNature-11649)