Regularized Maximum Likelihood Estimate for an Infinite-Dimensional Parameter in Stochastic Parabolic Systems

Abstract

The purpose of this paper is to study the identification problem of an infinite-dimensional parameter, more precisely a spatially varying parameter, in stochastic diffusion equations. In a previous study [S. I. Aihara and Y. Sunahara, SIAM J. Control Optim., 26 (1988), pp. 1062–1075], some explicit conditions for the consistency property of the maximum likelihood estimate (MLE) is explored. Here, an algorithm for generating the MLE is developed with the aid of the regularization technique proposed by [C. Kravaris and J. H. Seinfeld, SIAM J. Control Optim., 23 (1985), pp. 217–241]. After the consistency property of the MLE by a regularization is proved, necessary conditions for the regularized MLE (RMLE) are derived. Proposed is an iterative algorithm for computing one of the solutions of the necessary conditions derived. The convergence property of the sequence generated by the proposed algorithm is also shown. Finally, numerical examples are presented.