Asymptotically Optimal Finite-Dimensional Approximations for Linear Filtering with Infinite-Dimensional Measurements

Abstract

This work proposes a novel approach to approximate optimal linear filters for discrete-time linear Gaussian systems with infinite-dimensional measurements and finite- dimensional states. Assuming scalar-valued states for simplicity, we formulate the problem in terms of optimally selecting N points at which to sample the infinite-dimensional measurement, in order to minimize the mean-squared filtering error. We show that for large N, this problem can be expressed using the notion of an asymptotic point density function from the field of high-resolution quantization theory. To the best of the authors' knowledge, this method has not been considered in infinite- dimensional filtering previously...