Adaptive wavelet frame methods for operator equations: Sparse grids, vector-valued spaces and applications to nonlinear inverse parabolic problems

• Antragsteller: Professor Dr. Stephan Dahlke; Professor Dr. Peter Maaß
• Fachliche Zuordnung: Mathematik
• Förderung: Förderung von 2008 bis 2014
• Projektkennung: Deutsche Forschungsgemeinschaft (DFG) - Projektnummer 79623579

This project is the continuation of the DFG-Rroject 'Adaptive Wavelet Frame Methods for Operator Equations: Sparse Grids, Vector-Valued Spaces and Applications to Nonlinear Inverse Parabolic Problems'. The aim of this project is the development of optimally convergent adaptive wavelet schemes for complex systems. Especially, we are concerned with (nonlinear) elliptic and parabolic operator equations on nontrivial domains as well as with the related inverse parameter identification problems. For the fonward problems, we use generalized tensor product approximation techniques that realize dimension independent convergence rates. In the first period of SPP 1324, these tensor wavelet bases have already been provided and associated adaptive wavelet algorithms have been designed, implemented, and tested. The tests performed during the first period Include the numerical solution of a prototypical Inverse parabolic parameter identification problem. In addition, the theoretical prerequisites for applying sparsity constrained Tikhonov regularization to such inverse problems have been proved. These investigations will be systematically continued in the second period. In particular, one central goal will be the generalization of such adaptive wavelet algorithms to nonlinear equations. Furthermore we aim at extending the theoretical investigation of Tikhonov-regularizatlon schemes with sparsity constraints by Incorporating positivity constraints. As a model problem we will study the parameter Identification problem for a parabolic reaction-diffusion system which describes the gene concentrations in embryos at an early state of development (embryogenesis).

• DFG-Verfahren: Schwerpunktprogramme
• Teilprojekt zu: SPP 1324: Mathematische Methoden zur Extraktion quantifizierbarer Information aus komplexen Systemen
• Internationaler Bezug: Niederlande
• Beteiligte Person: Professor Dr. Rob Stevenson