Journal
SIAM-ASA JOURNAL ON UNCERTAINTY QUANTIFICATION
Volume 7, Issue 1, Pages 260-291Publisher
SIAM PUBLICATIONS
DOI: 10.1137/17M1138881
Keywords
stochastic PDEs; high-dimensional problems; tensor decompositions; low-rank decompositions; cross approximation
Funding
- EPSRC fellowship [EP/M019004/1]
- EPSRC [EP/M019004/1] Funding Source: UKRI
Ask authors/readers for more resources
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 beneficial when multiple quantities of interest are sought. This might be needed, for example, for the computation of the probability density function via the maximum entropy method [M. Kavehrad and M. Joseph, IEEE Trans. Comm., 34 (1986), pp. 1183-1189]. 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. In our numerical experiments, we apply the new algorithm to the stochastic diffusion equation and compare it with preconditioned steepest descent in the TT format, as well as with (multilevel) quasi-Monte Carlo and dimension-adaptive sparse grids methods. For sufficiently smooth random fields the new approach is orders of magnitude faster.
Authors
I am an author on this paper
Click your name to claim this paper and add it to your profile.
Reviews
Recommended
No Data Available