Adaptive wavelet methods for inverse problems and inverse parabolic equations

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In this project, two fields of applied mathematics, which were developed almost independently from each other, will be consolidated. On one hand, adaptive wavelet methods for the numerical treatment of operator equations were examined for many years. However the achieved convergence results refer exclusively to continuously invertible operators.On the other hand, the general convergence theory, at least for linear inverse problems, has been worked out for a long time. However, adaptive wavelet methods and the results of non-linear wavelet approximation schemes have hardly been used so far in this context, with the exception of the recent results in [14, 42].In this project we would like to analyze different approaches to adaptive wavelet methods for inverse problems: two-step regularization schemes, where the data are pre-smoothed in a first step; regularization methods based on adaptive procedures for the forward operator combined with classical regularization procedures; regularization by wavelet discretization. These methods will be extended to some non-linear operator equations.As a prototypical application an inverse heat conduction problem will be examined.