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Low-rank cross approximation algorithms for the solution of stochastic PDEs

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UNQW03 - Reducing dimensions and cost for UQ in complex systems

Co-authors: Robert Scheichl (University of Bath)

We consider the approximate solution of parametric PDEs using the low-rank Tensor Train (TT) decomposition. Such parametric PDEs arise for example in uncertainty quantification problems in engineering applications. We propose an algorithm that is a hybrid of the alternating least squares and the TT cross methods. It computes a TT approximation of the whole solution, which is particularly beneficial when multiple quantities of interest are sought. The new algorithm exploits and preserves the block diagonal structure of the discretized operator in stochastic collocation schemes. This disentangles computations of the spatial and parametric degrees of freedom in the TT representation. In particular, it only requires solving independent PDEs at a few parameter values, thus allowing the use of existing high performance PDE solvers. We benchmark the new algorithm on the stochastic diffusion equation against quasi-Monte Carlo and dimension-adaptive sparse grids methods. For sufficiently smooth random fields the new approach is orders of magnitude faster.

This talk is part of the Isaac Newton Institute Seminar Series series.

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