Adaptive wavelet methods for inverse problems and inverse parabolic equations

反问题和反抛物线方程的自适应小波方法

• 批准号: 22812949
• 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: Research Grants
• 财政年份: 2006
• 资助国家: 德国
• 起止时间: 2005-12-31 至 2009-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/22812949?language=en
• 关键词: Adaptive; wavelet; methods; inverse; problems

In this project, two fields of applied mathematics, which were developed almost independently from each other, will be consolidated. On one hand, adaptive wavelet methods for the numerical treatment of operator equations were examined for many years. However the achieved convergence results refer exclusively to continuously invertible operators.On the other hand, the general convergence theory, at least for linear inverse problems, has been worked out for a long time. However, adaptive wavelet methods and the results of non-linear wavelet approximation schemes have hardly been used so far in this context, with the exception of the recent results in [14, 42].In this project we would like to analyze different approaches to adaptive wavelet methods for inverse problems: two-step regularization schemes, where the data are pre-smoothed in a first step; regularization methods based on adaptive procedures for the forward operator combined with classical regularization procedures; regularization by wavelet discretization. These methods will be extended to some non-linear operator equations.As a prototypical application an inverse heat conduction problem will be examined.

在这个项目中，应用数学的两个领域，这两个领域几乎是相互独立的发展，将得到巩固。一方面，自适应小波方法用于算子方程的数值处理已经进行了多年的研究。然而，所得到的收敛结果只适用于连续可逆算子。另一方面，广义收敛理论，至少对于线性逆问题，已经研究了很长时间。然而，自适应小波方法和非线性小波近似方案的结果迄今为止几乎没有在这种情况下使用，除了最近的结果[14,42]。在这个项目中，我们想分析反问题的自适应小波方法的不同方法：两步正则化方案，其中数据在第一步预平滑；基于自适应过程的正演算子正则化方法与经典正则化方法相结合小波离散化正则化。这些方法将推广到一些非线性算子方程。作为一个典型的应用，反热传导问题将被检查。

项目成果

Iterated soft shrinkage with adaptive operator evaluations

• DOI: 10.1515/jiip.2009.023
• 发表时间: 2009
• 期刊: Annu. Rev. Control.
• 作者: Thomas Bonesky;Peter Maass

Minimization of Tikhonov functionals in Banach spaces

• DOI: 10.1155/2008/192679
• 发表时间: 2008-01-01
• 期刊: ABSTRACT AND APPLIED ANALYSIS
• 作者: Bonesky, Thomas;Kazimierski, Kamil S.;Schuster, Thomas
• 通讯作者: Schuster, Thomas