Parameter estimation in nonlinear distributed systems - approximation, theory, and convergence results
type: Item
instance of: scholarly article
title: Parameter estimation in nonlinear distributed systems - approximation, theory, and convergence results
zbMATH Open document ID: 0656.65062
DOI: 10.1016/0893-9659(88)90077-8
author: Simeon Reich
author: I. G. Rosen
author: H. Thomas Banks
published in: Applied Mathematics Letters
publication date: 1988-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 47J25
Mathematics Subject Classification ID: 47J05
zbMATH DE Number: 4072274
zbMATH Keywords: convergence
zbMATH Keywords: Galerkin approximations
zbMATH Keywords: inverse problems
zbMATH Keywords: nonlinear distributed parameter systems
zbMATH Keywords: parameter estimation
zbMATH Keywords: maximal monotone operators
zbMATH Keywords: Banach space
zbMATH Keywords: Hilbert space
zbMATH Keywords: identification problem
zbMATH Keywords: stability
MaRDI profile type: Publication
full work available at URL: 0893-9659(88)90077-8
OpenAlex ID: W2135852055
cites work: An Approximation Theory for the Identification of Nonlinear Distributed Parameter Systems
cites work: Galerkin approximation for inverse problems for nonautonomous nonlinear distributed systems
cites work: Q4094029
description: scientific article; zbMATH DE number 4072274