Space-time multilevel quadrature methods and their application for cardiac electrophysiology

TL;DR

This paper introduces a high-performance, parallelized multilevel quadrature framework for uncertainty quantification in cardiac electrophysiology simulations, effectively handling complex geometries and stochastic perturbations.

Contribution

The work develops an integrated multilevel quadrature approach combined with space-time finite element discretizations for efficient uncertainty quantification in cardiac models.

Findings

Multilevel methods outperform classical Monte Carlo in convergence speed.

Parallel implementation enables large-scale, efficient simulations.

The approach effectively handles realistic heart geometries.

Abstract

We present a novel approach which aims at high-performance uncertainty quantification for cardiac electrophysiology simulations. Employing the monodomain equation to model the transmembrane potential inside the cardiac cells, we evaluate the effect of spatially correlated perturbations of the heart fibers on the statistics of the resulting quantities of interest. Our methodology relies on a close integration of multilevel quadrature methods, parallel iterative solvers and space-time finite element discretizations, allowing for a fully parallelized framework in space, time and stochastics. Extensive numerical studies are presented to evaluate convergence rates and to compare the performance of classical Monte Carlo methods such as standard Monte Carlo (MC) and quasi-Monte Carlo (QMC), as well as multilevel strategies, i.e. multilevel Monte Carlo (MLMC) and multilevel quasi-Monte Carlo (MLQMC) on hierarchies of nested meshes. Finally, we employ a recently suggested variant of the multilevel approach for non-nested meshes to deal with a realistic heart geometry.