Lectures on Tensor Approximation Methods for Integral-Differential Equations in $R^d$

Dr.Sci. Boris N. Khoromskij PhD

Dec. 2, 2008, 9:15 a.m. HF 136

The purpose of this course is to provide an introduction to modern methods of data-sparse representation of multi-variate nonlocal operators and functions based on tensor product approximation. Based on tensor formats, we consider the rank structured iterative methods for solving integral-differential equations in $R^d$, which scale linearly in $d$.

In the recent years multifactor analysis has been recognisedas a powerful (and really indispensable) tool to represent multi-dimensional data arising in various applications. Well-known since three decades in chemometrics, physicometrics, statistics, signal processing and data mining, nowadays this tool has become attractive in numerical PDEs, many-particle calculations, stochastic PDEs, financial mathematics.

We will discuss the main mathematical ideas which allow effective representation of operators and functions, numerical multilinear algebra, iterative methods with rank truncation for solving boundary-value/eigenvalue problems in $R^d$, and present MATLAB illustarations of basic numerical algorithms.

Main topics:

  1. Polynomial approximation of multivariate functions.

  2. Introduction to wavelet techniques, look on the Fourier kingdom.

  3. Sinc interpolation and quadratures.

  4. Separable approximation of the classical Green’s kernels in $R^d$.

  5. Introduction to multilinear algebra, low rank approximation of tensors. Rank structured tensor formats.

  6. Low tensor rank approximation of operators (analytic methods).

For more details, please see the personal homepage of PD Dr. Boris Khoromskij and the course overview.