A detectability criterion and data assimilation for non-linear differential equations
Provides a theoretical guarantee for exponential convergence of data assimilation in nonlinear systems, addressing a key challenge for practitioners in geosciences and engineering.
Proposed a sequential data assimilation method for nonlinear ODEs that ensures exponential error decay when the observation matrix satisfies a detectability condition based on Lyapunov exponents. Numerical tests on Lorenz96 and Burgers equations with noisy observations confirmed exponential convergence.
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.