Linearized Calderón Problem: Reconstruction and Lipschitz Stability for Infinite-Dimensional Spaces of Unbounded Perturbations

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Mathematical Analysis Pub Date : 2024-05-20 DOI:10.1137/23m1609270

Henrik Garde, Nuutti Hyvönen

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引用次数: 0

Abstract

SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3588-3604, June 2024.
Abstract. We investigate a linearized Calderón problem in a two-dimensional bounded simply connected [math] domain [math]. After extending the linearized problem for [math] perturbations, we orthogonally decompose [math] and prove Lipschitz stability on each of the infinite-dimensional [math] subspaces. In particular, [math] is the space of square-integrable harmonic perturbations. This appears to be the first Lipschitz stability result for infinite-dimensional spaces of perturbations in the context of the (linearized) Calderón problem. Previous optimal estimates with respect to the operator norm of the data map have been of the logarithmic type in infinite-dimensional settings. The remarkable improvement is enabled by using the Hilbert–Schmidt norm for the Neumann-to-Dirichlet boundary map and its Fréchet derivative with respect to the conductivity coefficient. We also derive a direct reconstruction method that inductively yields the orthogonal projections of a general [math] perturbation onto the [math] spaces, hence reconstructing any [math] perturbation.

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线性化卡尔德龙问题：无界扰动无穷维空间的重构与 Lipschitz 稳定性

SIAM 数学分析期刊》，第 56 卷第 3 期，第 3588-3604 页，2024 年 6 月。摘要。我们研究了二维有界简单相连[math]域[math]中的线性化卡尔德龙问题。在对[math]扰动的线性化问题进行扩展后，我们对[math]进行了正交分解，并证明了每个无限维[math]子空间上的 Lipschitz 稳定性。尤其是，[math] 是平方可积分谐波扰动空间。这似乎是在（线性化的）卡尔德龙问题背景下，第一个针对无穷维扰动空间的 Lipschitz 稳定性结果。之前关于数据映射的算子规范的最优估计，在无穷维环境下都是对数类型的。通过使用 Neumann-to-Dirichlet 边界图的希尔伯特-施密特规范及其相对于传导系数的弗雷谢特导数，我们取得了显著的改进。我们还推导出一种直接重构方法，可以归纳出一般[数学]扰动在[数学]空间上的正交投影，从而重构任何[数学]扰动。

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

SIAM Journal on Mathematical Analysis 数学-应用数学

CiteScore

3.30

自引率

5.00%

发文量

175

审稿时长

12 months

期刊介绍： SIAM Journal on Mathematical Analysis (SIMA) features research articles of the highest quality employing innovative analytical techniques to treat problems in the natural sciences. Every paper has content that is primarily analytical and that employs mathematical methods in such areas as partial differential equations, the calculus of variations, functional analysis, approximation theory, harmonic or wavelet analysis, or dynamical systems. Additionally, every paper relates to a model for natural phenomena in such areas as fluid mechanics, materials science, quantum mechanics, biology, mathematical physics, or to the computational analysis of such phenomena. Submission of a manuscript to a SIAM journal is representation by the author that the manuscript has not been published or submitted simultaneously for publication elsewhere. Typical papers for SIMA do not exceed 35 journal pages. Substantial deviations from this page limit require that the referees, editor, and editor-in-chief be convinced that the increased length is both required by the subject matter and justified by the quality of the paper.