Importance sampling approach for the nonstationary approximation error method

The approximation error approach has previously been proposed to handle modelling, numerical and computational errors. This approach has been developed both for stationary and nonstationary inverse problems (Kalman filtering). The key idea of the approach is to compute the approximate statistics of the errors over the distribution of all unknowns and uncertainties and carry out approximative marginalization with respect to these errors. In nonstationary problems, however, information is accumulated over time, and the initial uncertainties may turn out to have been exaggerated. In this paper, we propose an algorithm with which the approximation error statistics can be updated during the accumulation of measurement information. The proposed algorithm is based on importance sampling. The recursions that are proposed here are, however, based on the (extended) Kalman filter and therefore do not employ the often exceedingly heavy computational load of particle filtering. As a computational example, we study an estimation problem that is related to a convection?diffusion problem in which the velocity field is not accurately specified.