Multi-Index Sequential Monte Carlo Methods for partially observed Stochastic Partial Differential Equations

TL;DR

This paper introduces a novel multi-index Monte Carlo approach integrated with sequential Monte Carlo methods to efficiently estimate states and parameters in partially observed stochastic PDEs, reducing computational cost.

Contribution

It develops a new method combining multi-index Monte Carlo with SMC$^2$ for SPDEs, improving efficiency over traditional SMC methods.

Findings

Reduces computational cost for a given MSE compared to standard SMC$^2$

Proves theoretical cost reduction benefits of the proposed method

Demonstrates effectiveness through numerical experiments

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 (SPDE). 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 of [11] 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 advanced method such as sequential Monte Carlo (SMC). We show how one can use the MIMC method by renormalizing the MI identity and approximating the resulting identity using the SMC$^2$ method of [5]. We prove that our approach can reduce the cost to obtain a given mean square error (MSE), relative to just using SMC$^2$ on the most precise discretization. We demonstrate this with some numerical examples.