Hackbusch W. Tensor Spaces and Numerical Tensor Calculus

Springer, 2012. — 525 p. — (Springer Series in Computational Mathematics, 42). — e-ISBN: 978-3-642-28027-6.Large-scale problems have always been a challenge for numerical computations. An example is the treatment of fully populated n × n matrices, when n2 is close to or beyond the computer’s memory capacity. Here, the technique of hierarchical matrices can reduce the storage and the cost of numerical operations from O(n2 ) to almost O(n).
Tensors of order (or spatial dimension) d can be understood as d-dimensional generalisations of matrices, i.e., arrays with d discrete or continuous arguments. For large d > 3, the data size n d is far beyond any computer capacity. This book concerns the development of compression techniques for such high-dimensional data via suitable data sparse representations. Just as the hierarchical matrix technique was based on a successful application of the low-rank strategy, in recent years, related approaches have been used to solve high-dimensional tensor-based problems numerically
One of the aims of this monograph is to introduce a more mathematically-based treatment of this topic. Through this more abstract approach, the methods can be better understood, independently of the physical or technical details of the application.Algebraic TensorsMatrix 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
References
Index