Multidimensional numerical algorithms and their applications

Multidimensional problems are notoriously difficult due to the curse of dimensionality. However, high-dimensional problems are usually the most interesting ones and moreover, if the problem is of a considerable practical interest, there is a method that solves it. The most vivid example is the Schrodinger equation in quantum chemistry, where efficient solution methods have been proposed.

However, such methods are usually problem-specific, require a lot of efforts to implement and difficult to be applied in other areas. In the recent year, active development of mathematical foundations for the algorithms for the solution of high-dimensional problem has begun. Novel tensor formats (Hierarchical Tucker, Tensor Train) as well as surprinsing connections with other research areas (MPS, PEPS, tensor networks, graphical models) form a new research area with new fascinating theoretical and algorithmic problems and new applications in chemistry, biology and data-mining.

This talk will be a review of the known results and as well as recent advances in several areas, including low-rank methods for solving integro-differential equations, new computation of the convolution, application to data-mining and global optimization.