Parallel Iterated Extended and Sigma-Point Kalman Smoothers

Abstract

The problem of Bayesian filtering and smoothing in nonlinear models with additive noise is an active area of research. Classical Taylor series as well as more recent sigma-point based methods are two well-known strategies to deal with this problem. However, these methods are inherently sequential and do not in their standard formulation allow for parallelization in the time domain. In this paper, we present a set of parallel formulas that replace the existing sequential ones in order to achieve lower time (span) complexity. Our experimental results done with a graphics processing unit (GPU) illustrate the efficiency of the proposed methods over their sequential counterparts.

De-Sequentialized Monte Carlo: a parallel-in-time particle smoother

“…However, this problem can be mitigated by using proposal distributions that are adapted to the model at hand. In Section 4.1 we describe how recently developed parallel-in-time Gaussian approximation based smoothing algorithms (Särkkä and García-Fernández, 2021;Yaghoobi et al, 2021) can be used to form such proposal. As these methods are also parallel in time, they do not relinquish the O(log(T )) span complexity of the dSMC algorithm.…”

“…Temporal parallelization of general Bayesian filters and smoothers have recently been discussed in Särkkä and García-Fernández (2021), Hassan et al (2021), andYaghoobi et al (2021), but only in the contexts of Gaussian approximations and finite-state models. Parallelization methods for Kalman type of (ensemble) filters via parallel matrix computations over the state dimension are presented in Lyster et al (1997) and Evensen (2003).…”

“…This leads to logarithmic span-time complexity log(T ) instead of the conventional O(T ) of sequential methods. Yaghoobi et al (2021) then extended this framework to non-linear models by developing parallelized versions of the iterated extended Kalman smoothers as well as the more general iterated posterior linearization smoothers. This framework allows for computing the marginal approximations p(x t | y 1:T ) ≈ N (x t ; m l t , P l t ) in the O(log T ) time complexity.…”

“…However, it was recently shown in Särkkä and García-Fernández (2021) that Bayesian filtering and smoothing recursions (including, e.g., the Kalman filter and smoother) can be reformulated in terms of associative operators that can be time-parallelized to O(log T ) span-complexity by using a parallel scan algorithm. In Hassan et al (2021), similar methods were developed for finite-state models, and Yaghoobi et al (2021) developed Gaussian approximation based parallel methods for non-linear SSMs. These methods reduce the computational cost from linear to logarithmic in the number of observations on highly parallel hardware such as graphics processing units (GPUs).…”

Linear-Time Probabilistic Solutions of Boundary Value Problems

“…The mesh refinement is designed to make I Mesh as small as possible. Linear complexity in N stems from the state space implementation of the IEKS and could potentially be reduced to log N by temporal parallelisation [39]. The cubic complexity in ν and in d stems from the matrix-matrix operations that are required in a Kalman filter step [22].…”