Abstract
This paper deals with the numerical approximation of the Levenberg--Marquardt SQP (LMSQP) method for parameter identification problems, which has been presented and analyzed in [M. Burger and W. Mühlhuber, Inverse Problems, 18 (2002), pp. 943--969]. It is shown that a Galerkin-type discretization leads to a convergent approximation and that the indefinite system arising from the Karush--Kuhn--Tucker (KKT) system is well-posed. In addition, we present a multilevel version of the Levenberg--Marquardt method and discuss the simultaneous solution of the discretized KKT system by preconditioned iteration methods for indefinite problems. From a discussion of the numerical effort we conclude that these approaches may lead to a considerable speed-up with respect to standard iterative regularization methods that eliminate the underlying state equation. The numerical efficiency of the LMSQP method is confirmed by numerical examples.
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