State estimation approach to nonstationary inverse problems: discretization error and filtering problem

Abstract

We examine a certain class of linear nonstationary inverse problems. We view them as a state estimation problem. The time evolution of the state of the system is modelled by a stochastic differential or partial differential equation. The observation equation is linear with additive measurement noise. The time discretization of the continuous infinite-dimensional state estimation system is exact since the solution to the state evolution equation is given by an analytic semigroup and the observation equation depends only on the given time instant. For computational reasons, the space discretization of the time discrete infinite-dimensional state estimation system is performed. The novel contribution of the paper is the analysis of the space discretization. The distributions of the discretization errors in the discretized state evolution and observation equations are introduced. The solution to the corresponding finite-dimensional filtering problem is presented.