Year: 2018
Author: Christoph Reisinger, Zhenru Wang
Journal of Computational Mathematics, Vol. 36 (2018), Iss. 2 : pp. 202–236
Abstract
In this article, we propose a space-time Multi-Index Monte Carlo (MIMC) estimator for a one-dimensional parabolic stochastic partial differential equation (SPDE) of Zakai type. We compare the complexity with the Multilevel Monte Carlo (MLMC) method of Giles and Reisinger (2012), and find, by means of Fourier analysis, that the MIMC method: (i) has suboptimal complexity of $O(ε^{−2}|logε|^3)$ for a root mean square error (RMSE) $ε$ if the same spatial discretisation as in the MLMC method is used; (ii) has a better complexity of $O(ε^{−2}|logε|)$ if a carefully adapted discretisation is used; (iii) has to be adapted for non-smooth functionals. Numerical tests confirm these findings empirically.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jcm.1612-m2016-0681
Journal of Computational Mathematics, Vol. 36 (2018), Iss. 2 : pp. 202–236
Published online: 2018-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 35
Keywords: Parabolic stochastic partial differential equations Multilevel Monte Carlo Multi-index Monte Carlo Stochastic finite differences Zakai equation.