On a Continuation Approach in Tikhonov regularization

Vladimir Vrabel* and Valdemar Melicher

We propose a new approach to convexification in the Tikhonov regularization. We embed the original minimization problem into a one-parameter family of minimization problems. We first regularize the ill-posed inverse problem in a relaxed sense thanks to the fact that the direct problem usually requires less regularity than the a priori knowledge dictates. The "less" constrained problem will be easier to solve. The extra desired properties will be then progressively imposed on the solution via a continuation-based method. We next apply the continuation method to a geometry identification problem. The resulting method combines topology and shape sensitivities in a natural way and prevents the convergence of gradient-based minimization methods to a local minima of the corresponding Tikhonov functional. We provide illustrative results for magnetic induction tomography which is an example of PDE constrained inverse problem.

Mathematics Subject Classification: 35R30 65N20 78M30

Keywords: continuation; PDEs; variational problems; optimization; inverse problems; level set method; magnetic induction tomography

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