Identification of a discontinuous parameter in stochastic parabolic systems (Q1381321)

Consider the stochastic partial differential equation \[ {\partial u\over \partial t}= \sum^n_{i=1} {\partial\over \partial x_i} \left(a^0(x) {\partial u\over \partial x_i} \right)+ {\partial w\over \partial t} +f(x), \] with Dirichlet boundary condition on a bounded domain \(G\) of \(\mathbb{R}^n\), initial condition \(u_0\in L^2(G)\), where \(f\) is a known deterministic function and \(w\) is an \(L^2(G)\)-valued Wiener process. This paper deals with the problem of identifying the diffusion coefficient \(a^0\) in the class \[ \Theta= \{a\in L^\infty(G): 0<\alpha\leq a(x) \leq\beta\}, \] assuming a finite-dimensional noisy observation. The maximum likelihood method combined with the Kalman filter equation are used for estimating the unknown parameter \(a^0\). In order to prove the consistency property of the maximum likelihood estimate (M. L. E.), first the class of coefficients \(\Theta\) is projected into a finite-dimensional space and next the convergence of the finite-dimensional M. L. E. to the infinite-dimensional M. L. E. is established in some admissible class \(\Theta_{\text{ad}} \subset \Theta\). An iterative algorithm for generating the M. L. E. is proposed and two numerical examples are discussed.