Adaptive Runge–Kutta regularization for a Cauchy problem of a modified Helmholtz equation

IF 0.9 4区数学 Q2 MATHEMATICS

Journal of Inverse and Ill-Posed Problems Pub Date : 2023-03-18 DOI:10.1515/jiip-2020-0014

Fadhel Jday, H. Omri

引用次数: 0

Abstract

Abstract In this paper, we investigate the Cauchy problem for the modified Helmholtz equation. We consider the data completion problem in a bounded cylindrical domain on which the Neumann and the Dirichlet conditions are given in a part of the boundary. Since this problem is ill-posed, we reformulate it as an optimal control problem with an appropriate cost function. The method of factorization of boundary value problems is used to immediately obtain an approximation of the missing boundary data. In order to regularize this problem, we firstly scrutinize two classical regularizations for the cost function. Then we propose a new numerical regularization named “adaptive Runge–Kutta regularization”, which does not require any penalization term. Finally, we compare them numerically.

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一类修正Helmholtz方程Cauchy问题的自适应龙格-库塔正则化

摘要本文研究了修正Helmholtz方程的Cauchy问题。我们考虑有界圆柱域中的数据完备问题，其中Neumann和Dirichlet条件在部分边界中给出。由于这个问题是不适定的，我们将其重新表述为具有适当成本函数的最优控制问题。边值问题的因子分解方法用于立即获得缺失边界数据的近似值。为了正则化这个问题，我们首先考察了代价函数的两个经典正则化。然后，我们提出了一种新的数值正则化方法，称为“自适应Runge–Kutta正则化”，它不需要任何惩罚项。最后，我们对它们进行了数值比较。

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

Journal of Inverse and Ill-Posed Problems MATHEMATICS, APPLIED-MATHEMATICS

CiteScore

2.60

自引率

9.10%

发文量

审稿时长

>12 weeks

期刊介绍： This journal aims to present original articles on the theory, numerics and applications of inverse and ill-posed problems. These inverse and ill-posed problems arise in mathematical physics and mathematical analysis, geophysics, acoustics, electrodynamics, tomography, medicine, ecology, financial mathematics etc. Articles on the construction and justification of new numerical algorithms of inverse problem solutions are also published. Issues of the Journal of Inverse and Ill-Posed Problems contain high quality papers which have an innovative approach and topical interest. The following topics are covered: Inverse problems existence and uniqueness theorems stability estimates optimization and identification problems numerical methods Ill-posed problems regularization theory operator equations integral geometry Applications inverse problems in geophysics, electrodynamics and acoustics inverse problems in ecology inverse and ill-posed problems in medicine mathematical problems of tomography