Analysis of sparse grid multilevel estimators for multi-dimensional Zakai equations

In this article, we analyse the accuracy and computational complexity of estimators for expected functionals of the solution to multi-dimensional parabolic stochastic partial differential equations (SPDE) of Zakai-type. Here, we use the Milstein scheme for time integration and an alternating direction implicit (ADI) splitting of the spatial finite difference discretisation, coupled with the sparse grid combination technique and multilevel Monte Carlo sampling (MLMC). In the two-dimensional case, we find by detailed Fourier analysis that for a root-mean-square error (RMSE) $\varepsilon$, MLMC on sparse grids has the optimal complexity $O(\varepsilon^{-2})$, whereas MLMC on regular grids has $O(\varepsilon^{-2}(\log\varepsilon)^2)$, standard MC on sparse grids $O(\varepsilon^{-7/2}(|\log\varepsilon|)^{5/2})$, and MC on regular grids $O(\varepsilon^{-4})$. Numerical tests confirm these findings empirically. We give a discussion of the higher-dimensional setting without detailed proofs, which suggests that MLMC on sparse grids always leads to the optimal complexity, standard MC on sparse grids has a fixed complexity order independent of the dimension (up to a logarithmic term), whereas the cost of MLMC and MC on regular grids increases exponentially with the dimension.