Tensor Spaces and Numerical Tensor Calculus

• Main Author: Hackbusch, Wolfgang (Author, http://id.loc.gov/vocabulary/relators/aut)
• Corporate Author: SpringerLink (Online service)
• Format: Electronic eBook
• Language: English
• Published: Cham : Springer International Publishing : Imprint: Springer, 2019.
• Edition: 2nd ed. 2019.
• Series: Springer Series in Computational Mathematics, 56
• Subjects: Numerical analysis.; Chemistry, Physical and theoretical.; Mathematical physics.; Numerical Analysis.; Theoretical and Computational Chemistry.; Theoretical, Mathematical and Computational Physics.

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...

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.