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 - <p style="text-align: justify;">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 <span style="text-align: justify;">$O(ε^{−2}|logε|^3)$</span> 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&nbsp;<span style="text-align: justify;"></span><span style="text-align: justify;">$O(ε^{−2}|logε|)$</span> if a carefully adapted discretisation is used; (iii) has to be adapted for
non-smooth functionals. Numerical tests confirm these findings empirically.</p>