A parameter identification problem involving a nonlinear parabolic differential equation

Abstract

This work contributes to the analysis of a parameter identification task for a certain type of parabolic PDE. The first part of this work proposes an analytical approach for the parameter identification problem. The PDE is a nonlinear reaction-diffusion equation modeling the gene expression of the embryogenesis of Drosophila melanogaster. The problem is formulated in an evolution equation setting, suitable solution and parameter spaces are introduced for that purpose. We utilize novel results from parabolic regularity theory. The continuous differentiability of the parameter-to-state operator is proved by a modified implicit function theorem. Continuity properties of the derivative of the parameter-to-state operator are proved. The adjoint of this derivative operator is formulated. The second part concerns the applicability of Tikhonov regularization for the setting. A generalization of classical Tikhonov regularization is elaborated which suits our setup. As a means to compute minimizers of the Tikhonov functional, we employ iterated soft shrinkage. The theoretical underpinning for this algorithm builds on the particular choice of spaces in our setup and mentioned regularity theory. A generated numerical example with different noise levels is thoroughly inspected using an adaptive-wavelet based PDE solver. The results exhibit the desired convergence of mentioned regularization method.