Parameter estimation in nonlinear distributed systems - approximation, theory, and convergence results
We describe an abstract approximation framework and convergence theory for Galerkin approximations applied to inverse problems involving nonlinear distributed parameter systems. We consider parameter estimation problems formulated as the minimization of a least-squares like performance index over a compact admissible parameter set subject to state constraints given by an inhomogeneous nonlinear distributed system. Our theory applies to systems whose dynamics can be described by either time independent or nonstationary strongly maximal monotone operators defined on a reflexive Banach space which is densely and continuously embedded in a Hilbert space. Our approach involves the use of standard Galerkin techniques to obtain a sequence of approximating identification problems, each of which involves finite dimensional state constraints. We demonstrate that if readily verifiable conditions on the system's dependence on the unknown parameters are satisfied, and the usual Galerkin approximation assumption (i.e. strong convergence of the orthogonal projections in a sufficiently strong norm topology) holds, then solutions to the approximating problems exist and in some sense approximate a solution to the original infinite dimensional identification problem. These results also guarantee a type of stability - i.e. continuous dependence of approximate estimates on the observations.
scientific article; zbMATH DE number 4072274