An inverse source problem for the stochastic wave equation
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doi: 10.3934/ipi.2021071
The domain derivative for semilinear elliptic inverse obstacle problems
Frank Hettlich ,
Institute for Applied and Numerical Mathematics, Karlsruhe Institute of Technology, Karlsruhe, Germany
*Corresponding author: Frank Hettlich
We consider the recovering of the shape of a cavity from the Cauchy datum on an accessible boundary in case of semilinear boundary value problems. Existence and a characterization of the domain derivative of solutions of semilinear elliptic equations are proven. Furthermore, the result is applied to solve an inverse obstacle problem with an iterative regularization scheme. By some numerical examples its performance in case of a Kerr type nonlinearity is illustrated.
Keywords: Inverse obstacle problem, semilinear elliptic equation, domain derivative, Kerr type nonlinearity, iterative regularization method.
Mathematics Subject Classification: Primary: 35R30; Secondary: 35J61.
Citation: Frank Hettlich. The domain derivative for semilinear elliptic inverse obstacle problems. Inverse Problems & Imaging, doi: 10.3934/ipi.2021071
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Figure 1. First and 10. iteration in case of noise free data, and the approximation errors versus iteration steps (reconstructions (red), initial guess (dotted, blue), exact object (dashed, black), residual error (blue), reconstruction error (dashed, red))
Figure 2. The worst and the best result from noisy data
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Frank Hettlich