Scalable Multilevel Monte Carlo Methods Exploiting Parallel Redistribution on Coarse Levels

TL;DR

This paper introduces a scalable element agglomeration coarsening strategy for multilevel Monte Carlo methods that enhances parallel scalability by allowing redistribution at coarse levels, demonstrated through Darcy flow simulations.

Contribution

It presents a novel element agglomeration coarsening approach within AMGe that improves MLMC scalability by enabling data redistribution on coarse levels regardless of core count.

Findings

Enhanced scalability of MLMC simulations demonstrated.

Coarsening strategy maintains approximation properties at coarse levels.

Scalability benefits shown in Darcy flow model simulations.

Abstract

We study an element agglomeration coarsening strategy that requires data redistribution at coarse levels when the number of coarse elements becomes smaller than the used computational units (cores). The overall procedure generates coarse elements (general unstructured unions of fine grid elements) within the framework of element-based algebraic multigrid methods (or AMGe) studied previously. The AMGe generated coarse spaces have the ability to exhibit approximation properties of the same order as the fine-level ones since by construction they contain the piecewise polynomials of the same order as the fine level ones. These approximation properties are key for the successful use of AMGe in multilevel solvers for nonlinear partial differential equations as well as for multilevel Monte Carlo (MLMC) simulations. The ability to coarsen without being constrained by the number of available cores, as described in the present paper, allows to improve the scalability of these solvers as well as in the overall MLMC method. The paper illustrates this latter fact with detailed scalability study of MLMC simulations applied to model Darcy equations with a stochastic log-normal permeability field.