Nonlinearity parameter imaging in the frequency domain

IF 1.2 4区数学 Q2 MATHEMATICS, APPLIED

Inverse Problems and Imaging Pub Date : 2023-01-01 DOI:10.3934/ipi.2023037

Barbara Kaltenbacher, William Rundell

引用次数: 0

Abstract

Nonlinearity parameter tomography leads to the problem of identifying a coefficient in a nonlinear wave equation (such as the Westervelt equation) modeling ultrasound propagation. In this paper we transfer this into frequency domain, where the Westervelt equation gets replaced by a coupled system of Helmholtz equations with quadratic nonlinearities. For the case of the to-be-determined nonlinearity coefficient being a characteristic function of an unknown, not necessarily connected domain $ D $, we devise and test a reconstruction algorithm based on weighted point source approximations combined with Newton's method. In a more abstract setting, convergence of a regularised Newton type method for this inverse problem is proven by verifying a range invariance condition of the forward operator and establishing injectivity of its linearisation.

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频域非线性参数成像

非线性参数层析成像导致在模拟超声传播的非线性波动方程(如Westervelt方程)中识别系数的问题。在本文中，我们将其转移到频域，其中Westervelt方程被二次非线性的亥姆霍兹方程的耦合系统所取代。对于待确定的非线性系数是未知且不一定连通域D $的特征函数的情况，我们设计并测试了一种基于加权点源近似和牛顿方法相结合的重构算法。在更抽象的情况下，通过验证正演算子的范围不变性条件和建立其线性化的注入性，证明了正则牛顿型方法的收敛性。

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

Inverse Problems and Imaging 数学-物理：数学物理

CiteScore

2.50

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Inverse Problems and Imaging publishes research articles of the highest quality that employ innovative mathematical and modeling techniques to study inverse and imaging problems arising in engineering and other sciences. Every published paper has a strong mathematical orientation employing methods from such areas as control theory, discrete mathematics, differential geometry, harmonic analysis, functional analysis, integral geometry, mathematical physics, numerical analysis, optimization, partial differential equations, and stochastic and statistical methods. The field of applications includes medical and other imaging, nondestructive testing, geophysical prospection and remote sensing as well as image analysis and image processing. This journal is committed to recording important new results in its field and will maintain the highest standards of innovation and quality. To be published in this journal, a paper must be correct, novel, nontrivial and of interest to a substantial number of researchers and readers.