  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.
