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逆抛物问题的应用”的延续。这个项目的目的是发展最佳收敛的自适应小波方案的复杂系统。特别是，我们关注的非平凡域上的（非线性）椭圆和抛物算子方程以及相关的逆参数识别问题。对于前一个问题，我们使用广义张量积逼近技术，实现维数无关的收敛速度。在SPP 1324的第一阶段，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参数识别问题的数值解。此外，稀疏约束Tikhonov正则化应用于此类反问题的理论前提已被证明。这些调查将在第二阶段有系统地继续进行。特别是，一个中心目标将是这样的自适应小波算法的非线性方程的泛化。此外，我们的目标是扩展的理论研究的Tikhonov正则化计划的稀疏性约束，通过取消积极的约束。作为一个模型问题，我们将研究抛物反应扩散系统的参数识别问题，该系统描述了胚胎发育早期（胚胎发生）的基因浓度。