A detectability criterion and data assimilation for nonlinear differential equations*Submitted to the editors. (18th October 2018)

• Record Type: Journal Article
• Authors: Frank, Jason; Zhuk, Sergiy
• Is Part Of: Nonlinearity. Volume 31:Number 11(2018:Nov.)
• Page Start: 5235
• Page End: 5257
• Publication Date: 2018-10-18
• Subjects: data assimilation -- synchronization -- filtering -- detectability -- Lyapunov exponents; 62M20 -- 37C50 -- 34D06 -- 37M25; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/aaddcb
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: In this paper we propose a new sequential data assimilation method for nonlinear 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. In particular this implies that 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 Lorenz '96 and Burgers equations with incomplete and noisy observations.