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＜E-Book＞
Tensor Spaces and Numerical Tensor Calculus / by Wolfgang Hackbusch
(Springer Series in Computational Mathematics. ISSN:21983712 ; 56)

Edition	2nd ed. 2019.
Publisher	(Cham : Springer International Publishing : Imprint: Springer)
Year	2019
Language	English
Size	XXVIII, 605 p. 8 illus., 3 illus. in color : online resource
Authors	*Hackbusch, Wolfgang author SpringerLink (Online service)
Subjects	LCSH:Numerical analysis LCSH:Chemistry, Physical and theoretical LCSH:Mathematical physics FREE:Numerical Analysis FREE:Theoretical Chemistry FREE:Theoretical, Mathematical and Computational Physics
Notes	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 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 HTTP:URL=https://doi.org/10.1007/978-3-030-35554-8

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