The on-line or adaptive identification of parameters in abstract linear and nonlinear infinite-dimensional dynamical systems is considered. An estimator in the form of an infinitedimensional linear evolution system having the state and parameter estimates as its states is defined. Convergence of the state estimator is established via a Lyapunov estimate. The finite-dimensional notion of a plant being sufficiently rich or persistently excited is extended to infinite dimensions. Convergence of the parameter estimates is established under the additional assumption that the plant is persistently excited. A finite-dimensional approximation theory is developed, and convergence results are established. Numerical results for examples involving the estimation of both constant and functional parameters in one-dimensional linear and nonlinear heat or diffusion equations and the estimation of stiffness and damping parameters in a one-dimensional wave equation with Kelvin-Voigt viscoelastic damping are presented.
In this article a modified Levenberg-Marquardt method coupled with a Kaczmarz strategy for obtaining stable solutions of nonlinear systems of ill-posed operator equations is investigated. We show that the proposed method is a convergent regularization method. Numerical tests are presented for a non-linear inverse doping problem based on a bipolar model.
We investigate the iterative methods proposed by Maz'ya and Kozlov (see [6,7]) for solving ill-posed inverse problems modeled by partial differential equations. We consider linear evolutionary problems of elliptic, hyperbolic and parabolic types. Each iteration of the analyzed methods consists in the solution of a well posed problem (boundary value problem or initial value problem respectively). The iterations are described as powers of affine operators, as in [7]. We give alternative convergence proofs for the algorithms by using spectral theory and the fact that the linear parts of these affine operators are non-expansive with additional functional analytical properties (see [9,10]). Also problems with noisy data are considered and estimates for the convergence rate are obtained under a priori regularity assumptions on the problem data.
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