Sparse Compression of Expected Solution Operators

TL;DR

This paper introduces a method to efficiently approximate the expected solution operator of elliptic PDEs with random coefficients using sparse matrices and wavelet-based decompositions, enabling faster computations.

Contribution

It presents a novel sparse approximation technique for the expected solution operator leveraging localized multiresolution decompositions and wavelet bases.

Findings

Efficient sparse representation of expected solution operators.

Applicable to elliptic PDEs with random coefficients.

Improves computational speed for stochastic PDE solutions.

Abstract

We show that the expected solution operator of prototypical linear elliptic partial differential operators with random coefficients is well approximated by a computable sparse matrix. This result is based on a random localized orthogonal multiresolution decomposition of the solution space that allows both the sparse approximate inversion of the random operator represented in this basis as well as its stochastic averaging. The approximate expected solution operator can be interpreted in terms of classical Haar wavelets. When combined with a suitable sampling approach for the expectation, this construction leads to an efficient method for computing a sparse representation of the expected solution operator.