Analysis of Multi-Index Monte Carlo Estimators for a Zakai SPDE (Journal of Computational Mathematics, 2019)
In this article, we propose a space-time Multi-Index Monte Carlo (MIMC) estimator fora one-dimensional parabolic stochastic partial differential equation (SPDE) of Zakai type.We compare the complexity with the Multilevel Monte Carlo (MLMC) method of Gilesand 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 thesame spatial discretisation as in the MLMC method is used; (ii) has a better complexityof $O(ε^{−2}|logε|)$ if a carefully adapted discretisation is used; (iii) has to be adapted fornon-smooth functionals. Numerical tests confirm these findings empirically.