Temporal parallelisation of continuous-time maximum-a-posteriori trajectory estimation

TL;DR

This paper introduces a parallel-in-time method for continuous-time MAP trajectory estimation of SDEs, significantly accelerating computations on parallel architectures while maintaining accuracy.

Contribution

It reformulates the MAP estimation as a continuous-time optimal control problem and adapts parallel algorithms, including parallel Kalman-Bucy filter and smoother, for faster computation.

Findings

Achieves significant speedup on GPU implementations.

Maintains accuracy comparable to sequential algorithms.

Extends linear methods to nonlinear models via Taylor expansions.

Abstract

This paper proposes a parallel-in-time method for computing continuous-time maximum-a-posteriori (MAP) trajectory estimates of the states of partially observed stochastic differential equations (SDEs), with the goal of improving computational speed on parallel architectures. The MAP estimation problem is reformulated as a continuous-time optimal control problem based on the Onsager-Machlup functional. This reformulation enables the use of a previously proposed parallel-in-time solution for optimal control problems, which we adapt to the current problem. The structure of the resulting optimal control problem admits a parallel solution based on parallel associative scan algorithms. In the linear Gaussian special case, it yields a parallel Kalman-Bucy filter and a parallel continuous-time Rauch-Tung-Striebel smoother. These linear computational methods are further extended to nonlinear continuous-time state-space models through Taylor expansions. We also present the corresponding parallel two-filter smoother. The graphics processing unit (GPU) experiments on linear and nonlinear models demonstrate that the proposed framework achieves a significant speedup in computations while maintaining the accuracy of sequential algorithms.