Optimal Transport-based Nonlinear Filtering in High-dimensional Settings

This paper addresses the problem of nonlinear filtering, i.e., computing the conditional distribution of the state of a stochastic dynamical system given a history of noisy partial observations. The primary focus is on scenarios involving degenerate likelihoods or high-dimensional states, where traditional sequential importance resampling (SIR) particle filters face the weight degeneracy issue. Our proposed method builds on an optimal transport interpretation of nonlinear filtering, leading to a simulation-based and likelihood-free algorithm that estimates the Brenier optimal transport map from the current distribution of the state to the distribution at the next time step. Our formulation allows us to harness the approximation power of neural networks to model complex and multi-modal distributions and employ stochastic optimization algorithms to enhance scalability. Extensive numerical experiments are presented that compare our method to the SIR particle filter and the ensemble Kalman filter, demonstrating the superior performance of our method in terms of sample efficiency, high-dimensional scalability, and the ability to capture complex and multi-modal distributions.Comment: 24 pages, 15 figure

Transform-based particle filtering for elliptic Bayesian inverse problems

We introduce optimal transport based resampling in adaptive SMC. We consider elliptic inverse problems of inferring hydraulic conductivity from pressure measurements. We consider two parametrizations of hydraulic conductivity: by Gaussian random field, and by a set of scalar (non-)Gaussian distributed parameters and Gaussian random fields. We show that for scalar parameters optimal transport based SMC performs comparably to monomial based SMC but for Gaussian high-dimensional random fields optimal transport based SMC outperforms monomial based SMC. When comparing to ensemble Kalman inversion with mutation (EKI), we observe that for Gaussian random fields, optimal transport based SMC gives comparable or worse performance than EKI depending on the complexity of the parametrization. For non-Gaussian distributed parameters optimal transport based SMC outperforms EKI