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

本项目是dfg项目“算子方程的自适应小波框架方法：稀疏网格、向量值空间及其在非线性逆抛物问题中的应用”的延续。本课题的目的是为复杂系统开发最优收敛的自适应小波格式。特别地，我们关注非平凡域上的（非线性）椭圆和抛物算子方程以及相关的逆参数辨识问题。对于正向问题，我们采用广义张量积逼近技术，实现了与维数无关的收敛速率。在spp1324的第一期中，已经提供了这些张量小波基，并设计、实现和测试了相关的自适应小波算法。在第一阶段进行的测试包括一个典型的反抛物型参数识别问题的数值解。此外，还证明了将稀疏约束Tikhonov正则化应用于此类逆问题的理论前提。这些调查将在第二阶段有系统地继续进行。特别是，一个中心目标是将这种自适应小波算法推广到非线性方程。在此基础上，通过引入正约束，扩展了具有稀疏约束的tikhonov正则化格式的理论研究。作为一个模型问题，我们将研究描述胚胎早期发育状态（胚胎发生）基因浓度的抛物反应-扩散系统的参数识别问题。

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).