Scalable hierarchical PDE sampler for generating spatially correlated random fields using non-matching meshes

TL;DR

This paper introduces a scalable hierarchical PDE-based sampling method for generating spatially correlated random fields on unstructured meshes, enabling efficient large-scale uncertainty quantification in PDEs with random inputs.

Contribution

It proposes a novel, scalable approach using non-matching mesh embedding and reaction-diffusion PDEs for efficient Gaussian random field generation on unstructured meshes.

Findings

Method achieves high parallel scalability in 3D.

Supports large-scale MLMC simulations with up to 1.9 billion unknowns.

Demonstrates effective uncertainty quantification for subsurface flow.

Abstract

This work describes a domain embedding technique between two non-matching meshes used for generating realizations of spatially correlated random fields with applications to large-scale sampling-based uncertainty quantification. The goal is to apply the multilevel Monte Carlo (MLMC) method for the quantification of output uncertainties of PDEs with random input coefficients on general, unstructured computational domains. We propose a highly scalable, hierarchical sampling method to generate realizations of a Gaussian random field on a given unstructured mesh by solving a reaction-diffusion PDE with a stochastic right-hand side. The stochastic PDE is discretized using the mixed finite element method on an embedded domain with a structured mesh, and then the solution is projected onto the unstructured mesh. This work describes implementation details on how to efficiently transfer data from the structured and unstructured meshes at coarse levels, assuming this can be done efficiently on the finest level. We investigate the efficiency and parallel scalability of the technique for the scalable generation of Gaussian random fields in three dimensions. An application of the MLMC method is presented for quantifying uncertainties of subsurface flow problems. We demonstrate the scalability of the sampling method with non-matching mesh embedding, coupled with a parallel forward model problem solver, for large-scale 3D MLMC simulations with up to $1.9\cdot 10^9$ unknowns.