The aim of this project is the development of optimal convergent adaptive wavelet schemes for complex systems. Especially, we shall be concerned with (nonlinear) elliptic and parabolic operator equations on nontrivial domains and manifolds. Then, one principle problem is always the construction of suitable wavelet bases on these domains. In this project, we shall use variants of adaptive frame schemes in combination with domain decomposition ideas. The goals of this project can summarized as follows:

• To generalize the concept of tensor product approximation to non-product domains via domain decomposition techniques. To construct adaptive wavelet or frame methods that realize the rate of best N-term approximation in (the union of) these tensor product bases. The benefits will be two-fold: Dimension independent convergence rates, and the optimal convergence rate for solutions of operator equations that have limited smoothness in the standard (isotropic) Besov scale.
• To combine our findings with adaptive regularization schemes for inverse problems. Especially, we shall be concerned with parameter identification problems stemming from nonlinear parabolic equations with a high-dimensional target space.

The concrete objectives can be described as follows.

1. Adaptive Tensor Product Wavelet Schemes for Elliptic Operator Equations. The goal is to develop an adaptive wavelet or frame method for solving linear operator equations that realizes the rate of best N-term approximation in (the union of) tensor product bases. Afterwards, we generalize the method to nonlinear equations. Afterwards, we generalize the method to nonlinear equations and vector valued problems.
2. Adaptive Tensor Product Wavelet Schemes for Parabolic Equations. We will develop and test two different schemes; a variant of Rothe's method, but also quite recently developed adaptive space-time refinements.
3. Parameter Identification with Sparsity Constraints for Parabolic Problems. The techniques developed in the first two steps will be applied to parameter identification problems where the forward problem is given by a nonlinear parabolic equation. As an important model problem, a semi-linear parabolic system that describes the pattern formation in the evolution of Drosophila blastoderms will be studied.
4. Numerical Realization and an Application in Embryogenesis. We intend to develop and implement effcient adaptive numerical algorithms for identifying the unknown parameters from the measurement. At least for the forward problem, we shall employ the aforementioned adaptive parabolic solvers. The inverse problem shall be solved with gradient- or Newton-type methods for sparsity constraints.