A hybrid Alternating Least Squares -- TT Cross algorithm for parametric PDEs

TL;DR

This paper introduces a hybrid algorithm combining alternating least squares and TT cross methods to efficiently compute low-rank tensor train approximations of parametric PDE solutions, significantly accelerating uncertainty quantification tasks.

Contribution

It presents a novel hybrid algorithm for low-rank TT approximation of parametric PDEs that exploits block diagonal structure and reduces computational cost.

Findings

Orders of magnitude faster for smooth random fields

Requires solving only a few PDEs at selected parameter values

Outperforms quasi-Monte Carlo and sparse grid methods

Abstract

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 (PDF) via the maximum entropy method [Kavehrad and Joseph, IEEE Trans. Comm., 1986]. 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.