In this paper, we combine concepts from two different mathematical research topics: adaptive wavelet techniques for well-posed problems and regularization theory for nonlinear inverse problems with sparsity constraints. We are concerned with identifying certain parameters in a parabolic reaction-diffusion equation from measured data. Analytical properties of the related parameter-to-state operator are summarized, which justify the application of an iterated soft shrinkage algorithm for minimizing a Tikhonov functional with sparsity constraints. The forward problem is treated by means of a new adaptive wavelet algorithm which is based on tensor wavelets. In its general form, the underlying PDE describes gene concentrations in embryos at an early state of development. We implemented an algorithm for the related nonlinear parameter identification problem and numerical results are presented for a simplified test equation.
© 2012 by Walter de Gruyter Berlin Boston