Tensor Spaces and Numerical Tensor Calculus

1. Front Matter

   Pages i-xxiv

   PDF

2. Algebraic Tensors

   1. Front Matter

      Pages 1-2

      PDF

   2. Introduction

      • Wolfgang Hackbusch

      Pages 3-20

   3. Matrix Tools

      • Wolfgang Hackbusch

      Pages 21-46

   4. Algebraic Foundations of Tensor Spaces

      • Wolfgang Hackbusch

      Pages 47-84

3. Functional Analysis of Tensor Spaces

   1. Front Matter

      Pages 85-86

      PDF

   2. Banach Tensor Spaces

      • Wolfgang Hackbusch

      Pages 87-156

   3. General Techniques

      • Wolfgang Hackbusch

      Pages 157-172

   4. Minimal Subspaces

      • Wolfgang Hackbusch

      Pages 173-196

4. Numerical Treatment

   1. Front Matter

      Pages 197-198

      PDF

5. Numerical Treatment

   1. r-Term Representation

      • Wolfgang Hackbusch

      Pages 199-216

   2. Tensor Subspace Representation

      • Wolfgang Hackbusch

      Pages 217-248

   3. r-Term Approximation

      • Wolfgang Hackbusch

      Pages 249-280

   4. Tensor Subspace Approximation

      • Wolfgang Hackbusch

      Pages 281-314

   5. Hierarchical Tensor Representation

      • Wolfgang Hackbusch

      Pages 315-370

   6. Matrix Product Systems

      • Wolfgang Hackbusch

      Pages 371-384

   7. Tensor Operations

      • Wolfgang Hackbusch

      Pages 385-416

   8. Tensorisation

      • Wolfgang Hackbusch

      Pages 417-446

   9. Generalised Cross Approximation

      • Wolfgang Hackbusch

      Pages 447-462

   10. Applications to Elliptic Partial Differential Equations

       • Wolfgang Hackbusch

       Pages 463-472

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. ​

• high-dimensional problems
• numerical tensor calculus
• tensor formats
• tensor spaces
• tensorization

“The present book Tensor spaces and numerical tensor calculus by Wolfgang Hackbusch to a large extent fills the gap between theory on finite-dimensional tensor spaces and tensor computations. It also gives a comprehensive overview of numerical tensor computations. … the book is most useful as a (very rich) source book for researchers in the area. It might also be used as a textbook for a Ph.D. course … .” (Lars Eldén, Mathematical Reviews, October, 2015)

• for Mathematics in the Sciences, Max-Planck-Institute, Leipzig, Germany

  Wolfgang Hackbusch

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.

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