LEADER 04044nam 22005055 450 001 9910906200903321 005 20241107115753.0 010 $a9783031743702 010 $a3031743709 024 7 $a10.1007/978-3-031-74370-2 035 $a(MiAaPQ)EBC31756831 035 $a(Au-PeEL)EBL31756831 035 $a(CKB)36514533900041 035 $a(DE-He213)978-3-031-74370-2 035 $a(EXLCZ)9936514533900041 100 $a20241107d2024 u| 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aHilbert Space Splittings and Iterative Methods /$fby Michael Griebel, Peter Oswald 205 $a1st ed. 2024. 210 1$aCham :$cSpringer Nature Switzerland :$cImprint: Springer,$d2024. 215 $a1 online resource (444 pages) 225 1 $aSpringer Series in Computational Mathematics,$x2198-3712 ;$v62 311 08$a9783031743696 311 08$a3031743695 327 $a1 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. 330 $aThis book is about the theory of so-called Schwarz methods for solving variational problems in a Hilbert space V arising from linear equations and their associated quadratic minimization problems. Schwarz methods are based on the construction of a sequence of approximate solutions by solving auxiliary variational problems on a set of (smaller, finite-dimensional) Hilbert spaces Vi in a certain order, combining them, and using the combined approximations in an iterative procedure. The spaces Vi form a so-called space splitting for V, they need not necessarily be subspaces of V, and their number can be finite or infinite. The convergence behavior of Schwarz methods is influenced by certain properties of the space splittings they are based on. These properties are identified, and a detailed treatment of traditional deterministic and more recent greedy and stochastic orderings in the subproblem solution process is given, together with an investigation of accelerated methods. To illustrate the abstract theory, the numerical linear algebra analogs of the iterative methods covered in the book are discussed. Its standard application to the convergence theory of multilevel and domain decomposition methods for solving PDE problems is explained, and links to optimization theory and online learning algorithms are given. Providing an introduction and overview of iterative methods which are based on problem decompositions and suitable for parallel and distributed computing, the book could serve as the basis for a one- or two-semester course for M.S. and Ph.D. students specializing in numerical analysis and scientific computing. It will also appeal to a wide range of researchers interested in scientific computing in the broadest sense. 410 0$aSpringer Series in Computational Mathematics,$x2198-3712 ;$v62 606 $aMathematics$xData processing 606 $aComputational Mathematics and Numerical Analysis 615 0$aMathematics$xData processing. 615 14$aComputational Mathematics and Numerical Analysis. 676 $a518 700 $aGriebel$b Michael$0471555 701 $aOswald$b Peter$01350277 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910906200903321 996 $aHilbert Space Splittings and Iterative Methods$94290196 997 $aUNINA