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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.

##### MSC:
 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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