Computation of the Response Surface in the Tensor Train data format
Abstract
We apply the Tensor Train (TT) approximation to construct the Polynomial Chaos Expansion (PCE) of a random field, and solve the stochastic elliptic diffusion PDE with the stochastic Galerkin discretization. We compare two strategies of the polynomial chaos expansion: sparse and full polynomial (multiindex) sets. In the full set, the polynomial orders are chosen independently in each variable, which provides higher flexibility and accuracy. However, the total amount of degrees of freedom grows exponentially with the number of stochastic coordinates. To cope with this curse of dimensionality, the data is kept compressed in the TT decomposition, a recurrent lowrank factorization. PCE computations on sparse grids sets are extensively studied, but the TT representation for PCE is a novel approach that is investigated in this paper. We outline how to deduce the PCE from the covariance matrix, assemble the Galerkin operator, and evaluate some postprocessing (mean, variance, Sobol indices), staying within the lowrank framework. The most demanding are two stages. First, we interpolate PCE coefficients in the TT format using a few number of samples, which is performed via the block cross approximation method. Second, we solve the discretized equation (large linear system) via the alternating minimal energy algorithm. In the numerical experiments we demonstrate that the full expansion set encapsulated in the TT format is indeed preferable in cases when high accuracy and high polynomial orders are required.
 Publication:

arXiv eprints
 Pub Date:
 June 2014
 DOI:
 10.48550/arXiv.1406.2816
 arXiv:
 arXiv:1406.2816
 Bibcode:
 2014arXiv1406.2816D
 Keywords:

 Mathematics  Numerical Analysis;
 Mathematics  Probability;
 15A69 65F10 60H15 60H35 65C30
 EPrint:
 28 pages. Submitted to SIAM J. of Uncertainty Quantification