Ill-posedness of time-dependent inverse problems in Lebesgue-Bochner spaces

IF 4.6 Q2 MATERIALS SCIENCE, BIOMATERIALS
Martin Burger, Thomas Schuster, Anne Wald
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引用次数: 0

Abstract

We consider time-dependent inverse problems in a mathematical setting using Lebesgue-Bochner spaces. Such problems arise when one aims to recover parameters from given observations where the parameters or the data depend on time. There are various important applications being subject of current research that belong to this class of problems. Typically inverse problems are ill-posed in the sense that already small noise in the data causes tremendous errors in the solution. In this article we present two different concepts of ill-posedness: temporally (pointwise) ill-posedness and uniform ill-posedness with respect to the Lebesgue-Bochner setting. We investigate the two concepts by means of a typical setting consisting of a time-depending observation operator composed by a compact operator. Furthermore we develop regularization methods that are adapted to the respective class of ill-posedness.
Lebesgue-Bochner 空间中与时间相关的逆问题的失当性
我们利用 Lebesgue-Bochner 空间考虑数学环境中与时间相关的逆问题。当人们想从给定的观测数据中恢复参数时,就会出现这类问题,而参数或数据都取决于时间。目前研究的各种重要应用都属于这类问题。通常情况下,反求问题是一种 "摆不平 "的问题,即数据中的微小噪声就会导致求解中的巨大误差。在本文中,我们提出了两种不同的条件不良概念:相对于 Lebesgue-Bochner 设定的时间(点)条件不良和均匀条件不良。我们通过一个由紧凑算子组成的随时间变化的观测算子的典型设置来研究这两个概念。此外,我们还开发了正则化方法,这些方法适用于相应类别的问题。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
ACS Applied Bio Materials
ACS Applied Bio Materials Chemistry-Chemistry (all)
CiteScore
9.40
自引率
2.10%
发文量
464
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