# Fundamentals of Numerical Analysis and Symbolic Computation: Numerical Tensor Calculus

Lecturer: Prof. Wolfgang Hackbusch

ECTS: 3.00, hours: 2.0

Time and place: blocked lecture between 23 October and 3 November 2017 (details see below)

Abstract:

Problems involving many variables are often related to tensors. A tensor might be regarded as a (hyper)matrix with $d$ instead of two indices. Already large-scale matrices of size $n\times n$ cause problems. In the tensor case the data size is $n^d$ which is usually beyond the computer capacity. The Numerical Tensor Calculus describes methods, how to work with tensors. A fundamental step is the efficient data-sparse representation of tensors. Next the implementation of tensor operations is of interest. In a first part we introduce the algebraic structure of tensors (tensor product of general vector spaces, mappings, functionals, symmetric and antisymmetric tensors). It turns out that most of the helpful properties of matrices do not hold for the general tensor case. In a second part we discuss the functional analysis of (infinite-dimensional) tensor spaces. This is of interest since we want to approximate functions by tensors. The third and main part introduces the numerical methods, in particular the data-sparse representation of tensors. Particular applications examples are the solution of high-dimensional linear systems or the treatment of partial differential equations with stochastic coefficients.

Literature: W.H.: Tensor spaces and numerical tensor calculus. Springer, 2012

**Starting on Monday, Oct. 23, 12:45, at S2 416-1**

Schedule:

Lecture 01: Monday, Oct 23, 12:45 - 14:15, S2 416-1

Lecture 02: Tuesday, Oct 24, 13:45 - 15:15, S2 416-1

Lecture 03: Tuesday, Oct 24, 15:30 - 17:00, S2 416-1

Lecture 04: Wednesday, Oct 25, 10:15 - 11:45, S2 416-1

Lecture 05: Wednesday, Oct 25, 13:45 - 15:15, S2 416-1

Lecture 06: Friday, Oct 27, 13:45 - 15:15, S2 416-1

Lecture 07: Monday, Oct 30, 13:45 - 15:15, S2 416-1

Lecture 08: Monday, Oct 30, 15:30 - 17:00, S2 416-1

Lecture 09: Tuesday, Oct 31, 13:45 - 15:15, S2 416-1

Lecture 10: Tuesday, Oct 31, 15:30 - 17:00, S2 416-1

Consultation / Discussion: Thursday, Nov 2, tba

Lecture 11: Friday, Nov 3, 13:45 - 15:15, S2 416-1

Lecture 12: Friday, Nov 3, 15:30 - 17:00, S2 416-1