A Wasserstein coupled particle filter for multilevel estimation

Stochastic Analysis and Applications, vol. 41, pp. 820–859

Abstract

In this article, we consider the filtering problem for partially observed diffusions, which are regularly observed at discrete times. We are concerned with the case when one must resort to time-discretization of the diffusion process if the transition density is not available in an appropriate form. In such cases, one must resort to advanced numerical algorithms such as particle filters to consistently estimate the filter. It is also well known that the particle filter can be enhanced by considering hierarchies of discretizations and the multilevel Monte Carlo (MLMC) method, in the sense of reducing the computational effort to achieve a given mean square error (MSE). A variety of multilevel particle filters (MLPF) have been suggested in the literature, e.g., in Jasra et al., SIAM J, Numer. Anal., 55, 3068–3096. Here we introduce a new alternative that involves a resampling step based on the optimal Wasserstein coupling. We prove a central limit theorem (CLT) for the new method. On considering the asymptotic variance, we establish that in some scenarios, there is a reduction, relative to the approach in the aforementioned paper by Jasra et al., in computational effort to achieve a given MSE. These findings are confirmed in numerical examples. We also consider filtering diffusions with unstable dynamics; we empirically show that in such cases a change of measure technique seems to be required to maintain our findings.

Authors 4

1. Marco Ballesio corresponding

   King Abdullah University of Science and Technology

   Affiliation as printed

   Computer, Electrical and Mathematical Sciences and Engineering, King Abdullah University of Science and Technology (KAUST), Thuwal, Kingdom of Saudi Arabia

   SRI Center for Uncertainty Quantification in Computational Science and Engineering, King Abdullah University of Science and Technology (KAUST), Thuwal, Kingdom of Saudi Arabia

2. Ajay Jasra

   King Abdullah University of Science and Technology

   Affiliation as printed

   Computer, Electrical and Mathematical Sciences and Engineering, King Abdullah University of Science and Technology (KAUST), Thuwal, Kingdom of Saudi Arabia

3. Erik von Schwerin

   King Abdullah University of Science and Technology

   Affiliation as printed

   Computer, Electrical and Mathematical Sciences and Engineering, King Abdullah University of Science and Technology (KAUST), Thuwal, Kingdom of Saudi Arabia

   SRI Center for Uncertainty Quantification in Computational Science and Engineering, King Abdullah University of Science and Technology (KAUST), Thuwal, Kingdom of Saudi Arabia

4. Raul F. Tempone Aachen

   RWTH Aachen University · King Abdullah University of Science and Technology

   Affiliation as printed

   Alexander von Humboldt Professor in Mathematics for Uncertainty Quantification, RWTH Aachen University, Aachen, Germany

   Computer, Electrical and Mathematical Sciences and Engineering, King Abdullah University of Science and Technology (KAUST), Thuwal, Kingdom of Saudi Arabia

   SRI Center for Uncertainty Quantification in Computational Science and Engineering, King Abdullah University of Science and Technology (KAUST), Thuwal, Kingdom of Saudi Arabia