Tensor Spaces and Numerical Tensor Calculus

2012
ISBN: 978-3-642-42709-1
Verlag: Springer

Buch, Englisch, Band 42, 500 Seiten, Paperback, Format (B × H): 155 mm x 235 mm, Gewicht: 785 g

Reihe: Springer Series in Computational Mathematics

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.

Zielgruppe

Research

Autoren/Hrsg.

Hackbusch, Wolfgang

Fachgebiete

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

Inhaltsverzeichnis

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.

Über Autor(innen)

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.