Reconstruction of the Radiation Condition and Solution for the Helmholtz Equation in a Semi-infinite Strip from Cauchy Data on an Interior Segment

IF 1 4区数学 Q3 MATHEMATICS, APPLIED

Computational Methods in Applied Mathematics Pub Date : 2023-07-25 DOI:10.1515/cmam-2022-0244

Pauline Achieng, F. Berntsson, V. Kozlov

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Abstract

Abstract We consider an inverse problem for the Helmholtz equation of reconstructing a solution from measurements taken on a segment inside a semi-infinite strip. Homogeneous Neumann conditions are prescribed on both side boundaries of the strip and an unknown Dirichlet condition on the remaining part of the boundary. Additional complexity is that the radiation condition at infinity is unknown. Our aim is to find the unknown function in the Dirichlet boundary condition and the radiation condition. Such problems appear in acoustics to determine acoustical sources and surface vibrations from acoustic field measurements. The problem is split into two sub-problems, a well-posed and an ill-posed problem. We analyse the theoretical properties of both problems; in particular, we show that the radiation condition is described by a stable non-linear problem. The second problem is ill-posed, and we use the Landweber iteration method together with the discrepancy principle to regularize it. Numerical tests show that the approach works well.

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由内线段上的Cauchy数据重建半无限带中Helmholtz方程的辐射条件和求解

摘要我们考虑亥姆霍兹方程的一个反问题，该反问题是根据对半无限带内的一段的测量来重建解。在条的两侧边界上规定了齐次Neumann条件，在边界的剩余部分上规定了未知的Dirichlet条件。额外的复杂性是无穷远处的辐射条件是未知的。我们的目的是找到Dirichlet边界条件和辐射条件下的未知函数。这样的问题出现在声学中，以根据声场测量来确定声源和表面振动。该问题分为两个子问题，一个是适定问题，另一个是不适定问题。我们分析了这两个问题的理论性质；特别地，我们证明了辐射条件是由一个稳定的非线性问题描述的。第二个问题是不适定的，我们使用Landweber迭代方法和差分原理对其进行正则化。数值测试表明，该方法效果良好。

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

Computational Methods in Applied Mathematics MATHEMATICS, APPLIED-

CiteScore

2.40

自引率

7.70%

发文量

期刊介绍： The highly selective international mathematical journal Computational Methods in Applied Mathematics　(CMAM) considers original mathematical contributions to computational methods and numerical analysis with applications mainly related to PDEs. CMAM seeks to be interdisciplinary while retaining the common thread of numerical analysis, it is intended to be readily readable and meant for a wide circle of researchers in applied mathematics. The journal is published by De Gruyter on behalf of the Institute of Mathematics of the National Academy of Science of Belarus.