Reconstruction algorithms of an inverse geometric problem for the modified Helmholtz equation
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(2019) 38:167
Reconstruction algorithms of an inverse geometric problem for the modified Helmholtz equation Ji-Chuan Liu1 Received: 2 February 2019 / Revised: 11 August 2019 / Accepted: 24 September 2019 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019
Abstract In this paper, we consider an inverse geometric problem for the modified Helmholtz equation from measurements of the potential taken on the boundary of the geometrical domain. Our goal is to seek reconstruction algorithms to detect the number, the location, the size and the shape of unknown obstacles from Cauchy data on the external boundary. This problem is ill-posed and nonlinear, thus we should employ regularization techniques in our proposed algorithms. We give several numerical examples to demonstrate the stability of numerical algorithms. Keywords Trust-region-reflective optimization algorithm · Levenberg–Marquardt algorithm · Modified Helmholtz equation · Ill-posed problem Mathematics Subject Classification 65N20 · 65N21
1 Introduction We consider an inverse geometric problem for the modified Helmholtz equation to determine unknown obstacles from a single pair of boundary Cauchy data on an exterior surface (Marin et al. 2004). Over the past few decades, there has been extensive studies in numerical methods for the modified Helmholtz equation (Fernandez et al. 2018, 2019; Hua et al. 2017), which arises in many important fields of physics and engineering (Lesnic and Bin-Mohsin 2012; Liang and Subramaniam 1997; Callihan and Wood 2012; Bakker and Kuhlman 2011; Politis et al. 2002). In this paper, what we are interested in using reconstruction algorithms to detect the salient features of unknown obstacles within a body. An inverse geometric problem for the modified Helmholtz equation depends strongly on the input data which is ill-posed, in the sense that small measurement errors in the input data may amplify significantly the errors in the solution for the modified Helmholtz equation. Therefore, in order to obtain the more
Communicated by Antonio José Silva Neto.
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Ji-Chuan Liu [email protected] School of Mathematics, China University of Mining and Technology, Xuzhou 221116, Jiangsu, People’s Republic of China
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accurate features of unknown obstacles, we should propose stable numerical algorithms to solve this inverse geometric problems along with regularization techniques. To our knowledge, many researchers take more attention to the research of inverse geometric problem for the modified Helmholtz equation. Isakov (2006) gave no uniqueness result for the inverse geometric problem governed by the modified Helmholtz equation for a finite number of boundary measurements. Ammari and Uhlmann (2004) proved that the knowledge of the partial Cauchy data for the Schrödinger equation on any open subset of the boundary determines uniquely the potential q provided that q is known in a neighborhood of the bound. Fernandez et al. (2018, 2019) employed noniterative reconstruction method to deal w
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