Tensor Spaces and Numerical Tensor Calculus

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Sprache: Englisch

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Beschreibung

Special numerical techniques are already needed to deal with n × n matrices for large n. Tensor data are of size n × n ×...× n=nd, where nd exceeds the computer memory by far. They appear for problems of high spatial dimensions. Since standard methods fail, a particular tensor calculus is needed to treat such problems. This monograph describes the methods by which tensors can be practically treated and shows how numerical operations can be performed. Applications include problems from quantum chemistry, approximation of multivariate functions, solution of partial differential equations, for example with stochastic coefficients, and more. In addition to containing corrections of the unavoidable misprints, this revised second edition includes new parts ranging from single additional statements to new subchapters. The book is mainly addressed to numerical mathematicians and researchers working with high-dimensional data. It also touches problems related to Geometric Algebra.

Über den Autor

Wolfgang Hackbusch is working in the field of numerical mathematics for partial differential equations and integral equations. He has published monographs, e.g., about the multi-grid method, about the numerical analysis of elliptic pdes, about iterative solution of large systems of equation, and about the technique of hierarchical matrices.

Zusammenfassung

First monograph on this subject in the field of numerical mathematics

Covers algebraic and functional analysis aspects of tensor spaces

Focuses on the numerical treatment

Inhaltsverzeichnis

Part I: Algebraic Tensors.- 1 Introduction.- 2 Matrix Tools.- 3 Algebraic Foundations of Tensor Spaces.- Part II: Functional Analysis of Tensor Spaces.- 4 Banach Tensor Spaces.- 5 General Techniques.- 6 Minimal Subspaces.- Part III: Numerical Treatment.- 7 r-Term Representation.- 8 Tensor Subspace Represenation.- 9 r-Term Approximation.- 10 Tensor Subspace Approximation.- 11 Hierarchical Tensor Representation.- 12 Matrix Product Systems.- 13 Tensor Operations.- 14 Tensorisation.- 15 Multivariate Cross Approximation.- 16 Applications to Elliptic Partial Differential Equations.- 17 Miscellaneous Topics.

Erscheinungsjahr:	2020
Fachbereich:	Wahrscheinlichkeitstheorie
Genre:	Mathematik
Rubrik:	Naturwissenschaften & Technik
Medium:	Buch
Reihe:	Springer Series in Computational Mathematics
Inhalt:	xxviii 605 S. 5 s/w Illustr. 3 farbige Illustr. 605 p. 8 illus. 3 illus. in color.
ISBN-13:	9783030355531
ISBN-10:	3030355535
Sprache:	Englisch
Ausstattung / Beilage:	HC runder Rücken kaschiert
Einband:	Gebunden
Autor:	Hackbusch, Wolfgang
Auflage:	2nd ed. 2019
Hersteller:	Springer International Publishing Springer International Publishing AG Springer Series in Computational Mathematics
Maße:	241 x 160 x 40 mm
Von/Mit:	Wolfgang Hackbusch
Erscheinungsdatum:	24.01.2020
Gewicht:	1,109 kg

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