Volume 36, Issue 2
Analysis of Multi-Index Monte Carlo Estimators for a Zakai SPDE

J. Comp. Math., 36 (2018), pp. 202-236.

Published online: 2018-04

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• 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.

• Keywords

Parabolic stochastic partial differential equations, Multilevel Monte Carlo, Multi-index Monte Carlo, Stochastic finite differences, Zakai equation.

65C05, 65T50, 60H15, 65N06, 65N12.

christoph.reisinger@maths.ox.ac.uk (Christoph Reisinger)

zhenru.wang@maths.ox.ac.uk (Zhenru Wang)

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@Article{JCM-36-202, author = {Reisinger , Christoph and Wang , Zhenru }, title = {Analysis of Multi-Index Monte Carlo Estimators for a Zakai SPDE}, journal = {Journal of Computational Mathematics}, year = {2018}, volume = {36}, number = {2}, pages = {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.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1612-m2016-0681}, url = {http://global-sci.org/intro/article_detail/jcm/12256.html} }
TY - JOUR T1 - Analysis of Multi-Index Monte Carlo Estimators for a Zakai SPDE AU - Reisinger , Christoph AU - Wang , Zhenru JO - Journal of Computational Mathematics VL - 2 SP - 202 EP - 236 PY - 2018 DA - 2018/04 SN - 36 DO - http://doi.org/10.4208/jcm.1612-m2016-0681 UR - https://global-sci.org/intro/article_detail/jcm/12256.html KW - Parabolic stochastic partial differential equations, Multilevel Monte Carlo, Multi-index Monte Carlo, Stochastic finite differences, Zakai equation. AB -

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.

Christoph Reisinger & Zhenru Wang. (2020). Analysis of Multi-Index Monte Carlo Estimators for a Zakai SPDE. Journal of Computational Mathematics. 36 (2). 202-236. doi:10.4208/jcm.1612-m2016-0681
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