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Hackbusch Wolfgang. Tensor Spaces and Numerical Tensor Calculus

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Hackbusch Wolfgang. Tensor Spaces and Numerical Tensor Calculus
Springer, 2012. — 500 p. — (Springer Series in Computational Mathematics 42). — ISBN 978-3-642-28026-9. — e-ISBN 978-3-642-28027-6.
Special numerical techniques are already needed to deal with nxn matrices for large n.Tensor data are of size nxnx...xn=n^d, where n^d 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 stochastic coefficients, etc. ​
Algebraic Tensors
Introduction
Matrix Tools
Algebraic Foundations of Tensor Spaces
Functional Analysis of Tensor Spaces
Banach Tensor Spaces
General Techniques
Minimal Subspaces
Numerical Treatment
r-Term Representation
Tensor Subspace Representation
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
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