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

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反抛物线问题中的应用

• 批准号: 79623579
• 负责人: Professor Dr. Stephan Dahlke
• 金额: --
• 依托单位: Zentrum für Technomathematik (ZeTeM)
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2008
• 资助国家: 德国
• 起止时间: 2007-12-31 至 2013-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/79623579?language=en
• 关键词: Adaptive; wavelet; frame; methods; operator

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).

该项目是 DFG-Rroject“算子方程的自适应小波框架方法：稀疏网格、向量值空间和非线性反抛物线问题的应用”的延续。该项目的目的是为复杂系统开发最优收敛的自适应小波方案。特别是，我们关注非平凡域上的（非线性）椭圆和抛物线算子方程以及相关的逆参数识别问题。对于前向问题，我们使用广义张量积近似技术来实现与维度无关的收敛速度。在SPP 1324的第一阶段，这些张量小波基已经被提供，并且相关的自适应小波算法已经被设计、实现和测试。第一阶段进行的测试包括原型逆抛物线参数识别问题的数值解。此外，还证明了将稀疏约束吉洪诺夫正则化应用于此类反问题的理论前提。这些调查将在第二阶段系统地继续进行。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。此外，我们的目标是通过结合正性约束来扩展具有稀疏性约束的吉洪诺夫正则化方案的理论研究。作为模型问题，我们将研究抛物线反应扩散系统的参数识别问题，该系统描述早期发育状态（胚胎发生）胚胎中的基因浓度。