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The numerical treatment of high dimensional problems by means of tensor format representations

Banff International Research Station for Mathematical Innovation and Discovery

The coming century is surely the century of high dimensional data. With the rapid growth of computational chemistry and distributed parameter systems, high-dimensional data becomes very common. Thus, analyzing high dimensional data is an urgent problem of great practical importance. However, there are some unique challenges for analyzing data of high dimensions, including (1) the curse of dimensionality and (2) the meaningfulness of the similarity measure in the high dimension space. With standard techniques it is impossible to store all entries of the high-dimensional data explicitly. The reason is that the computational complexity and the storage cost are growing exponentially with the number of dimensions. Besides of the storage one should also solve this high-dimensional problems in a reasonable (e.g. linear) time and obtain a solution in some compressed (low-rank/sparse) tensor formats. The complexity of many existing algorithms is exponential with respect to the number of dimensions. With increasing dimensionality, these algorithms soon become computationally intractable and therefore inapplicable in many real applications. During the last years, tensor format representation techniques were successfully applied to high-dimensional problems. In our talk, we show how these low-rank approximations can be computed, stored and manipulated with minimal effort.

Mathematics; Quantum theory; Partial differential equations; Applied computer science

video/mp4

Author affiliation: RWTH Aachen University

2016-07-27

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/58560

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/