Regularization of nonlinear ill-posed problems in Banach spaces and conditional stability

Banach空间中非线性不适定问题的正则化和条件稳定性

• 批准号: 190672901
• 负责人: Professor Dr. Bernd Hofmann
• 金额: --
• 依托单位: Weierstraß-Institut für Angewandte Analysis und Stochastik (WIAS)
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2011
• 资助国家: 德国
• 起止时间: 2010-12-31 至 2015-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/190672901?language=en
• 关键词: Regularization; nonlinear; ill; posed; problems

The main purpose of this research project consists in finding essential progress of knowledge in the field of regularization of nonlinear ill-posed problems in Banach spaces and their crossrelations to the concept of conditional stability. Compared with regularization in Hilbert spaces, considering the regularization of ill-posed operator equations in Banach spaces gives much more freedom that can be used for enforcing a priori information about desired solution properties.A fundamental goal of the project is to get new insight into the impact of smoothness on the quality of regularized solutions for operator equations in Banach spaces. Varying fitting functionals and error measures and their consequences with respect to properties and precision of approximate solutions are components for obtaining progress in this field. We aim to explore the distinguished role of variational inequalities and approximate source conditions intensively studied by the German partners and their strong link to conditional stability for inverse PDE problems well-studied by the Chinese partners as well as consequences for obtaining convergence rates of regularized solutions. This ensures good chances for a significant synergy of such a joint project.

本研究项目的主要目的是在Banach空间中的非线性不适定问题的正则化领域及其与条件稳定性概念的相互关系中找到知识的实质性进展。与Hilbert空间正则化相比，Banach空间中不适定算子方程的正则化具有更大的自由度，可以用来加强关于期望解性质的先验信息，其基本目标是对光滑性对Banach空间中算子方程正则化解质量的影响有新的认识.不同的拟合泛函和误差测量及其后果的性质和精度的近似解是在这一领域取得进展的组成部分。我们的目标是探索德国合作伙伴深入研究的变分不等式和近似源条件的杰出作用，以及它们与中国合作伙伴深入研究的逆偏微分方程问题的条件稳定性的密切联系，以及获得正则化解的收敛速度的后果。这确保了这种联合项目产生重大协同作用的良好机会。

项目成果

OPTIMAL RATES FOR LAVRENTIEV REGULARIZATION WITH ADJOINT SOURCE CONDITIONS

• DOI: 10.1090/mcom/3237
• 发表时间: 2018-03-01
• 期刊: MATHEMATICS OF COMPUTATION
• 影响因子: 2
• 作者: Plato, Robert;Mathe, Peter;Hofmann, Bernd
• 通讯作者: Hofmann, Bernd

The index function and Tikhonov regularization for ill-posed problems

不适定问题的指数函数和吉洪诺夫正则化

• DOI: 10.1016/j.cam.2013.09.035
• 发表时间: 2014-08
• 期刊: Journal of Computational and Applied Mathematics
• 影响因子: 2.4
• 作者: Jin Cheng;Bernd Hofmann;Shuai Lu
• 通讯作者: Shuai Lu

Regularization properties of the sequential discrepancy principle for Tikhonov regularization in Banach spaces

Banach 空间中 Tikhonov 正则化的序贯差异原理的正则化性质

• DOI: 10.1080/00036811.2013.833326
• 发表时间: 2014
• 期刊: Applicable Analysis
• 影响因子: 1.1
• 作者: S. W. Anzengruber;B. Hofmann;P. Mathé
• 通讯作者: P. Mathé

A unified approach to convergence rates for ℓ1-regularization and lacking sparsity

用于 â1 正则化且缺乏稀疏性的收敛率的统一方法

• DOI: 10.1515/jiip-2015-0058
• 发表时间: 2016
• 期刊: Journal of Inverse and Ill-posed Problems
• 影响因子: 1.1
• 作者: J. Flemming;B. Hofmann;I. Veselić
• 通讯作者: I. Veselić

Elastic-net regularization versus ℓ1-regularization for linear inverse problems with quasi-sparse solutions

• DOI: 10.1088/1361-6420/33/1/015004
• 发表时间: 2016-04
• 期刊: Inverse Problems
• 影响因子: 2.1
• 作者: De-Han Chen;B. Hofmann;J. Zou