# SPP 1324 - Adaptive Wavelet Frame Methods for Operator Equations: Sparse Grids, Vector-Valued Spaces, and Applications to Nonlinear Inverse Parabolic Problems

**Researcher: **Rudolf Ressel, Peter Maaß**Project funding:** Deutsche Forschungsgemeinschaft (DFG)**Project sponsor:** Deutsche Forschungsgemeinschaft (DFG)**Partner: **Rob Stevenson, University of Amsterdam, Korteweg-de Vries Institute for Mathematics, Niederlande; Stephan Dahlke, Philipps-Universität Marburg**Duration:** 01.04.2009 - 01.12.2013

The realisation that the embryogenesis of organisms is controlled by genes represents a milestone in modern biological research. The award of the Nobel Prize in 1995 to Nüsslein-Volhard, Wieshaus and Lewis for their research on the genetic control of development in the Drosophila Melanogaster (fruit fly) underlines the importance of these research results.

From this fact grew the interest in understanding the present processes quantitatively as well and, if possible, modelling them. A coupled system of non-linear parabolic differential equations was proposed as a mathematical model for this purpose. In this model, the concentrations of the individual genes appear as state variables that completely describe the system (at least theoretically). The parameters appearing in the differential equation system are now of particular biological interest, because they reflect the - promoting as well as inhibiting - interaction of the individual gene levels as well as the formation of structures in the cell. However, obtaining these parameters has proven to be quite difficult in practice.

This challenge is being met by the cooperative project of the Centre for Industrial Mathematics with the University of Marburg within the framework of the DFG Priority Programme SPP 1324. Specifically, the project is concerned with the solution of an ill-posed, nonlinear operator equation Ax=y, which maps differentiably between Banach spaces. The data y are generally not known exactly, but only a noisy version of the data. Consequently, regularisation methods must be applied to solve the problem.

The objective of the project is to choose a variant of the Tikhonov regularisation, in which the parameters to be identified are reconstructed "sparsely". Sparsity here means that the representation of the parameter is sparse with respect to a frame or a Hilbert space basis. To numerically obtain the minimiser of the Tikhonov functional used, the iterated soft shrinkage procedure is employed. In each iteration step, the nonlinear parabolic differential equation of the model must be solved, i.e. the nonlinear operator A must be evaluated. On the other hand, a linearised, adjoint parabolic differential equation must then be solved in each case, i.e. (A')* must be evaluated. The solution of the inverse problem thus first requires an in-depth functional-analytical investigation of the forward problem. Then it must be examined to what extent existing regularisation methods can be applied to the problem at hand. The processing of these questions is essentially the responsibility of the Bremen side of the cooperation.

Due to the volume of data to be processed, the coupling of the systems and the non-linearity, a correspondingly efficient solver is needed for the numerical implementation of the regularisation method. The adaptive wavelet solver developed in Marburg will be used for this aspect of the joint project. In the past, a similar problem from the field of parameter identification has already been successfully dealt with using this numerical apparatus. In this sense, the present project on nonlinear equations from biology also represents a continuation of past projects. The developed theoretical and numerical tool will thus be further developed for more complicated problems.