A Multi-Index Quasi-Monte Carlo Algorithm for Lognormal Diffusion Problems

TL;DR

This paper introduces a Multi-Index Quasi-Monte Carlo method combining multi-index sampling and lattice rules to efficiently estimate expectations in elliptic PDEs with lognormal coefficients, demonstrated on complex 3D flow problems.

Contribution

The paper develops a novel Multi-Index Quasi-Monte Carlo algorithm that improves computational efficiency for high-dimensional stochastic PDEs with challenging random fields.

Findings

Achieves cost inversely proportional to the RMS error tolerance.

Effectively handles small correlation length in lognormal diffusion.

Demonstrates efficiency on a 3D subsurface flow problem.

Abstract

We present a Multi-Index Quasi-Monte Carlo method for the solution of elliptic partial differential equations with random coefficients. By combining the multi-index sampling idea with randomly shifted rank-1 lattice rules, the algorithm constructs an estimator for the expected value of some functional of the solution. The efficiency of this new method is illustrated on a three-dimensional subsurface flow problem with lognormal diffusion coefficient with underlying Mat\'ern covariance function. This example is particularly challenging because of the small correlation length considered, and thus the large number of uncertainties that must be included. We show numerical evidence that it is possible to achieve a cost inversely proportional to the requested tolerance on the root-mean-square error, for problems with a smoothly varying random field