What is it about?

We apply the tensor train technique to solve elliptic PDE with uncertain coefficients. After discretisation stochastic variables the problem dimension becomes very high (10-300). How to work in such high dimension? We applied low-rank tensor train methods to the stochastic Galerkin approach. As a results we received a multivariate polynomial approxiamation of the high-dimensional solution. We analyzed the tensor ranks, approximation errors, explained ho to work in the compressed data format, how to do post-processing in this format.

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Why is it important?

Many data sets are high-dimensional. Usual algorithms does not work. For example, how to find the mean and the variance if the sample size is 10^20? To write 20 nested loops and wait 100 years? With our low-rank tensor technique the computing time and the storage requirement drops from exponential to linear.


Stochastic PDE was just an example. We believe that the algorithms are more universal and could be extended to more general multi-dimensional data sets.

Dr. Alexander Litvinenko
Rheinisch Westfalische Technische Hochschule Aachen

Read the Original

This page is a summary of: Polynomial Chaos Expansion of Random Coefficients and the Solution of Stochastic Partial Differential Equations in the Tensor Train Format, SIAM/ASA Journal on Uncertainty Quantification, January 2015, Society for Industrial & Applied Mathematics (SIAM),
DOI: 10.1137/140972536.
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