A modified Landweber iteration for solving parameter estimation problems
This paper is concerned with nonlinear operator equations \[ F(x)= y,\tag{1} \] where \(F:{\mathcal D}(F)\to Y\) with domain \({\mathcal D}(F)\subseteq X\). Here, \(X\) and \(Y\) are real Hilbert spaces with inner products \((\cdot,\cdot)\) and norms \(\|\cdot\|\), respectively; they can always be identified from the context in which they appear. Throughout the paper it is assumed that the data \(y\) in (1) is attainable, i.e., that (1) has a solution \(x_*\) (which need not be unique). In practical applications the available data will be distorted by measurement errors; therefore only approximate data \(y^\delta\) of \(y\) is available, but it is assumed that an estimate for the noise level is given, i.e., \[ \| y^\delta- y\|\leq \delta.\tag{2} \] This study is devoted to the stable solution of operator equations of the form (1) for which the solution \(x_*\) does not depend continuously on the right-hand side data, \(y\). Such problems are called ill-posed and need to be regularized to obtain reasonable approximations to \(x_*\). The author presents a convergence analysis for a modified Landweber iteration for the solution of nonlinear illposed problems. A priori and a posteriori stopping criteria for terminating the iteration are compared. Some numerical results for the solution of a parameter estimation problem are presented.
scientific article; zbMATH DE number 1172134