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A regularizing Levenberg-Marquardt scheme, with applications to inverse groundwater filtration problems. (English) Zbl 0873.65057
Let \(F\) be a nonlinear mapping between the Hilbert spaces \(X\), \(Y\). The inverse problem corresponding to given data \(u^{\delta} \in Y\) can be solved approximately by mimimizing \({|u^{\delta} - F(a) |}^2\) assuming that \(u^{\delta}\) approximates some \(u = F(a^{+})\). The paper develops a new Levenberg-Marquardt scheme where the regularization parameter is chosen by means of an inexact Newton strategy. It is shown that the approach provides a stable approximation of \(a^{+}\) if \(F'(a)\) is locally bounded and the Taylor remainder satisfies \(|R({\tilde a}, a) |\leq C |{\tilde a}- a ||F({\tilde a})-F(a) |\) for all \({\tilde a}, a\) in some ball in the domain of \(F\) centered at \(a^{+}\). These conditions turn out to hold for an ill-posed parameter identification problem arising in groundwater hydrology. Both transient and steady-state data are considered and numerical results are given.

65J15 Numerical solutions to equations with nonlinear operators
35Q35 PDEs in connection with fluid mechanics
35R20 Operator partial differential equations (= PDEs on finite-dimensional spaces for abstract space valued functions)
76S05 Flows in porous media; filtration; seepage
47J25 Iterative procedures involving nonlinear operators
65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization
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