Adaptive Runge–Kutta regularization for a Cauchy problem of a modified Helmholtz equation-Reference-Cited by-同舟云学术

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

Published:2023-03-18 Issue:0 Volume:0 Page:
ISSN:0928-0219
Container-title:Journal of Inverse and Ill-posed Problems
language:en
Short-container-title:

Author:

Jday Fadhel¹,Omri Haithem²

Affiliation:

1. Umm al-Qura University , Mecca , Saudi Arabia

2. LAMSIN ENIT , Campus Universitaire de Tunis , Tunis , Tunisia

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.

Publisher

Walter de Gruyter GmbH

Subject

Applied Mathematics

Link

https://www.degruyter.com/document/doi/10.1515/jiip-2020-0014/pdf

Reference36 articles.

1. G. Alessandrini, Stable determination of a crack from boundary measurements, Proc. Roy. Soc. Edinburgh Sect. A 123 (1993), no. 3, 497–516.

2. S. Andrieux, T. N. Baranger and A. Ben Abda, Solving Cauchy problems by minimizing an energy-like functional, Inverse Problems 22 (2006), no. 1, 115–133.

3. J. Barros-Neto, Problèmes aux limites non homogènes, Les Presses de l’Université de Montréal, Montreal, 1966.

4. R. Bellman, Dynamic Programming, Princeton University, Princeton, 1957.

5. A. Ben Abda, J. Henry and F. Jday, Boundary data completion: The method of boundary value problem factorization, Inverse Problems 27 (2011), no. 5, Article ID 055014.