Tensor Spaces and Numerical Tensor Calculus

Tensor Spaces and Numerical Tensor Calculus

by Wolfgang Hackbusch


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Special numerical techniques are already needed to deal with nxn matrices for large n.Tensor data are of size nxnx...xn=nsubd, where nsubd 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. The monograph describes the methods how tensors can be practically treated and how numerical operations can be performed. Applications are problems from quantum chemistry, approximation of multivariate functions, solution of pde, e.g., with shastic coefficients, etc.

Product Details

ISBN-13: 9783642427091
Publisher: Springer Berlin Heidelberg
Publication date: 04/12/2014
Series: Springer Series in Computational Mathematics , #42
Edition description: 2012
Pages: 500
Product dimensions: 6.10(w) x 9.25(h) x 0.04(d)

About the Author

The author 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.

Table of Contents

Part I: Algebraic Tensors.- Introduction.- Matrix Tools.- Algebraic Foundations of Tensor Spaces.- Part II: Functional Analysis of Tensor Spaces.- Banach Tensor Spaces.- General Techniques.- Minimal Subspaces.-Part III: Numerical Treatment.- r-Term Representation.- Tensor Subspace Represenation.- r-Term Approximation.- Tensor Subspace Approximation.-Hierarchical Tensor Representation.- Matrix Product Systems.- Tensor Operations.- Tensorisation.- Generalised Cross Approximation.- Applications to Elliptic Partial Differential Equations.- Miscellaneous Topics.- References.- Index.

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