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020 _a9783031743702
_9978-3-031-74370-2
024 7 _a10.1007/978-3-031-74370-2
_2doi
050 4 _aQA71-90
072 7 _aPBKS
_2bicssc
072 7 _aMAT041000
_2bisacsh
072 7 _aPBKS
_2thema
100 1 _aGriebel, Michael.
_eauthor.
_4aut
_4http://id.loc.gov/vocabulary/relators/aut
_987143
245 1 0 _aHilbert Space Splittings and Iterative Methods
_h[electronic resource] /
_cby Michael Griebel, Peter Oswald.
250 _a1st ed. 2024.
264 1 _aCham :
_bSpringer Nature Switzerland :
_bImprint: Springer,
_c2024.
300 _aIX, 440 p. 1 illus. in color.
_bonline resource.
336 _atext
_btxt
_2rdacontent
337 _acomputer
_bc
_2rdamedia
338 _aonline resource
_bcr
_2rdacarrier
347 _atext file
_bPDF
_2rda
490 1 _aSpringer Series in Computational Mathematics,
_x2198-3712 ;
_v62
505 0 _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.
520 _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.
650 0 _aMathematics
_xData processing.
_965715
650 1 4 _aComputational Mathematics and Numerical Analysis.
_966073
700 1 _aOswald, Peter.
_eauthor.
_4aut
_4http://id.loc.gov/vocabulary/relators/aut
_987144
710 2 _aSpringerLink (Online service)
773 0 _tSpringer Nature eBook
776 0 8 _iPrinted edition:
_z9783031743696
776 0 8 _iPrinted edition:
_z9783031743719
776 0 8 _iPrinted edition:
_z9783031743726
830 0 _aSpringer Series in Computational Mathematics,
_x2198-3712 ;
_v62
_987145
856 4 0 _uhttps://doi.org/10.1007/978-3-031-74370-2
912 _aZDB-2-SMA
912 _aZDB-2-SXMS
999 _c64773
_d64773