Tensor Spaces and Numerical Tensor Calculus

2. Auflage 2019
ISBN: 978-3-030-35553-1
Verlag: Springer International Publishing

Buch, Englisch, Band 56, 605 Seiten, Format (B × H): 160 mm x 241 mm, Gewicht: 1109 g

Reihe: Springer Series in Computational Mathematics

Special numerical techniques are already needed to deal with n × n matrices for large n. Tensor data are of size n × n ×...× n=n, where n 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.

Zielgruppe

Research

Autoren/Hrsg.

Hackbusch, Wolfgang

Fachgebiete

• Mathematik | Informatik Mathematik Geometrie Differentialgeometrie
• Mathematik | Informatik Mathematik Geometrie Algebraische Geometrie
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen

Inhaltsverzeichnis

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.

Über Autor(innen)

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.