Space-time multilevel Monte Carlo methods and their application to cardiac electrophysiology

TL;DR

This paper introduces a space-time multilevel Monte Carlo framework for efficient uncertainty quantification in time-dependent PDEs, demonstrated on cardiac electrophysiology, achieving significant computational savings.

Contribution

The paper develops a novel integrated space-time multilevel Monte Carlo method combining adaptivity, parallelization, and iterative solvers for high-dimensional uncertainty quantification.

Findings

Achieves theoretical convergence rates of multilevel Monte Carlo.

Reduces computational work by an order of magnitude compared to standard Monte Carlo.

Successfully applied to nonlinear cardiac electrophysiology models.

Abstract

We present a novel approach aimed at high-performance uncertainty quantification for time-dependent problems governed by partial differential equations. In particular, we consider input uncertainties described by a Karhunen-Loeeve expansion and compute statistics of high-dimensional quantities-of-interest, such as the cardiac activation potential. Our methodology relies on a close integration of multilevel Monte Carlo methods, parallel iterative solvers, and a space-time discretization. This combination allows for space-time adaptivity, time-changing domains, and to take advantage of past samples to initialize the space-time solution. The resulting sequence of problems is distributed using a multilevel parallelization strategy, allocating batches of samples having different sizes to a different number of processors. We assess the performance of the proposed framework by showing in detail its application to the solution of nonlinear equations arising from cardiac electrophysiology. Specifically, we study the effect of spatially-correlated perturbations of the heart fibers conductivities on the mean and variance of the resulting activation map. As shown by the experiments, the theoretical rates of convergence of multilevel Monte Carlo are achieved. Moreover, the total computational work for a prescribed accuracy is reduced by an order of magnitude with respect to standard Monte Carlo methods.