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An Introduction to Structured Tensor-Product Representation of Discrete Nonlocal Operators.
Boris N. Khoromskij
These notes are based on a lecture course given by the author in the summer semester of 2005 for postgraduate students at the University of Leipzig/Max-Planck-Institute for Mathematics in the Sciences. The purpose of this course was to provide an introduction to modern methods of a data-sparse representation to integral and more general nonlocal operators based on the use of Kronecker tensor-product decomposition.
In recent years multifactor analysis has been recognised as a powerful (and really indispensable) tool to represent multi-dimensional data arising in various applications. Well-known since three decades in chemometics, physicometrics, statistics, signal processing, data mining and in complexity theory, nowadays this tool has also become attractive in numerical PDEs, many-particle calculations, and in solving integral equations.
Our goal is to introduce the main mathematical ideas and principles which allow effective representation of some classes of high-dimensional operators in the Kronecker tensor-product form, as well as rigorous analysis of the arising approximations. Low Kronecker-rank representation of operators not only relaxes the "curse of dimensionality", but also provides efficient numerical methods of sub-linear complexity to approximate 2D- and 3D-problems.