Time-parallelization of sequential data assimilation problems

TL;DR

This paper introduces a method to parallelize data assimilation over time by coupling Luenberger observers with the Parareal algorithm, maintaining accuracy and efficiency in unbounded time domains.

Contribution

It presents a novel coupling of Luenberger observers with time parallelization algorithms, including error analysis and efficiency bounds, for unbounded time domain data assimilation.

Findings

The coupled method preserves the non-parallelized observer's rate.

Efficiency bounds are derived for the Parareal-based approach.

Numerical experiments demonstrate the approach's effectiveness.

Abstract

This paper is devoted to the problem of time parallelization of assimilation methods applying on unbounded time domain. In this way, we present a general procedure to couple the Luenberger observer with time parallelization algorithm. Our approach is based on a posteriori error estimates of the latter and preserves the rate of the non-parallelized observer. We then focus on the case where the Parareal algorithm is used as time parallelization algorithm, and derive a bound of the efficiency of our procedure. A variant devoted to the case a large number of processors is also proposed. We illustrate the performance of our approach with numerical experiments.