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Tensor Spaces and Numerical Tensor Calculus Wolfgang Hackbusch

Tensor Spaces and Numerical Tensor Calculus By Wolfgang Hackbusch

Tensor Spaces and Numerical Tensor Calculus by Wolfgang Hackbusch


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Summary

Special numerical techniques are already needed to deal with n x n matrices for large n. Tensor data are of size n x n x...x n=nd, where nd exceeds the computer memory by far. Since standard methods fail, a particular tensor calculus is needed to treat such problems.

Tensor Spaces and Numerical Tensor Calculus Summary

Tensor Spaces and Numerical Tensor Calculus by Wolfgang Hackbusch

Special numerical techniques are already needed to deal with n x n matrices for large n. Tensor data are of size n x n x...x 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.


About Wolfgang Hackbusch

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.

Table of Contents

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.

Additional information

NLS9783030355562
9783030355562
303035556X
Tensor Spaces and Numerical Tensor Calculus by Wolfgang Hackbusch
New
Paperback
Springer Nature Switzerland AG
2021-01-24
605
N/A
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