A detectability criterion and data assimilation for non-linear differential equations

TL;DR

This paper introduces a new sequential data assimilation method for non-linear differential equations that ensures exponential decay of estimation errors by satisfying detectability conditions related to Lyapunov exponents.

Contribution

The paper presents a novel data assimilation approach that guarantees exponential error decay for non-linear ODEs under specific detectability criteria.

Findings

Method ensures negative Lyapunov exponents for estimation error dynamics.

Numerical experiments confirm exponential convergence on Lorenz96 and Burgers equations.

Detectability conditions are sharp and necessary for the method's success.

Abstract

In this paper we propose a new sequential data assimilation method for non-linear ordinary differential equations with compact state space. The method is designed so that the Lyapunov exponents of the corresponding estimation error dynamics are negative, i.e. the estimation error decays exponentially fast. The latter is shown to be the case for generic regular flow maps if and only if the observation matrix H satisfies detectability conditions: the rank of H must be at least as great as the number of nonnegative Lyapunov exponents of the underlying attractor. Numerical experiments illustrate the exponential convergence of the method and the sharpness of the theory for the case of Lorenz96 and Burgers equations with incomplete and noisy observations.