Space-time multilevel Monte Carlo methods and their application to cardiac electrophysiology
| Autor: | Pietro Benedusi, Patrick Zulian, S. Ben Bader, Rolf Krause, Alessio Quaglino |
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| Jazyk: | angličtina |
| Rok vydání: | 2019 |
| Předmět: |
FOS: Computer and information sciences
Physics and Astronomy (miscellaneous) Discretization Computer science Monte Carlo method 010103 numerical & computational mathematics 01 natural sciences Computational Engineering Finance and Science (cs.CE) Mathematics - Analysis of PDEs Convergence (routing) FOS: Mathematics Mathematics - Numerical Analysis 0101 mathematics Uncertainty quantification Computer Science - Computational Engineering Finance and Science Numerical Analysis Sequence Partial differential equation Applied Mathematics Space time Numerical Analysis (math.NA) Computer Science Applications 010101 applied mathematics Computational Mathematics Nonlinear system Modeling and Simulation Algorithm Analysis of PDEs (math.AP) |
| Popis: | 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-Loeve 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. |
| Databáze: | OpenAIRE |
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