Modeling a physical flow experiment on nonwovens leads to a
Cauchy-Dirichlet problem with a free boundary for the degenerate
parabolic Richards equation in the water content formulation. We study
the inverse problem of reconstructing the nonlinear diffusivity
coefficient of the differential equation from the knowledge of the free
boundary. This problem is treated as a minimization problem for an
output least squares functional and we employ iterative regularization
methods like the Levenberg-Marquardt method, parametrizing the
coefficient space by quadratic B-Splines. The derivative of the free
boundary with respect to the coefficient, which is needed for the
numerical reconstruction, is linked to a linear degenerate parabolic
boundary-value problem. We present the numerical implementation of our
reconstruction method together with numerical reconstructions
for synthetic data.
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