04386nam a22005055i 4500 978-3-031-74370-2 DE-He213 20251030113301.0 cr nn 008mamaa 241107s2024 sz | s |||| 0|eng d 9783031743702 978-3-031-74370-2 10.1007/978-3-031-74370-2 doi QA71-90 PBKS bicssc MAT041000 bisacsh PBKS thema Griebel, Michael. author. aut http://id.loc.gov/vocabulary/relators/aut 87143 Hilbert Space Splittings and Iterative Methods [electronic resource] / by Michael Griebel, Peter Oswald. 1st ed. 2024. Cham : Springer Nature Switzerland : Imprint: Springer, 2024. IX, 440 p. 1 illus. in color. online resource. text txt rdacontent computer c rdamedia online resource cr rdacarrier text file PDF rda Springer Series in Computational Mathematics, 2198-3712 ; 62 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. Mathematics Data processing. 65715 Computational Mathematics and Numerical Analysis. 66073 Oswald, Peter. author. aut http://id.loc.gov/vocabulary/relators/aut 87144 SpringerLink (Online service) Springer Nature eBook Printed edition: 9783031743696 Printed edition: 9783031743719 Printed edition: 9783031743726 Springer Series in Computational Mathematics, 2198-3712 ; 62 87145 https://doi.org/10.1007/978-3-031-74370-2 ZDB-2-SMA ZDB-2-SXMS 64773 64773