Multi-index sequential monte carlo methods for partially observed stochastic partial differential equations

Abstract

In this paper we consider sequential joint state and static parameter estimation given discrete time observations associated to a partially observed stochastic partial differential equation. It is assumed that one can only estimate the hidden state using a discretization of the model. In this context, it is known that the multi-index Monte Carlo (MIMC) method can be used to improve over direct Monte Carlo from the most precise discretizaton. However, in the context of interest, it cannot be directly applied, but rather must be used within another method such as sequential Monte Carlo (SMC). We show how one can use the MIMC method by renormalizing the standard identity and approximating the resulting identity using the SMC2 method, which is an exact method that can be used in this context. We prove that our approach can reduce the cost to obtain a given mean square error, relative to just using SMC2 on the most precise discretization. We demonstrate this with some numerical examples.