Approximation error analysis in nonlinear state estimation with an application to state-space identification

Abstract

Nonstationary inverse problems are usually cast in the state-space formalism. The complete statistics of linear Gaussian problems can be computed with the Kalman filters and smoothers. Nonlinear non-Gaussian problems would necessitate the adoption of particle filters or similar computationally very heavy approaches. The so-called extended Kalman filters often provide suboptimal but feasible estimates for the nonlinear problems. Several applications which lead to nonstationary inverse problems are time critical, such as process tomography and many biomedical problems. In such applications, there is typically a need to use reduced-order models which may heavily compromise the computational accuracy that is usually required for inverse problems. One approach to overcome the model reduction problem is to use the approximation error analysis. In this paper, we derive the equations for the extended Kalman filter for nonlinear state estimation problems in which the approximation error models are taken into account. We consider the approximation errors that are due to both state reduction and time stepping. As an example, we consider the identification of the coefficients of the heat equation. Our main result is that, also in nonlinear problems, approximation error analysis allows us to obtain accurate estimates and uncertainties of the parameters in reduced-order models that are suitable for fast calculation.