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We present a novel approach aimed at high-performance uncertaintyquantification for time-dependent problems governed by partial differentialequations. In particular, we consider input uncertainties described by aKarhunen-Loeeve expansion and compute statistics of high-dimensionalquantities-of-interest, such as the cardiac activation potential. Ourmethodology relies on a close integration of multilevel Monte Carlo methods,parallel iterative solvers, and a space-time discretization. This combinationallows for space-time adaptivity, time-changing domains, and to take advantageof past samples to initialize the space-time solution. The resulting sequenceof problems is distributed using a multilevel parallelization strategy,allocating batches of samples having different sizes to a different number ofprocessors. We assess the performance of the proposed framework by showing indetail its application to the solution of nonlinear equations arising fromcardiac electrophysiology. Specifically, we study the effect ofspatially-correlated perturbations of the heart fibers conductivities on themean 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 byan order of magnitude with respect to standard Monte Carlo methods.

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