Noniterative Reconstruction Method for an Inverse Potential Problem Modeled by a Modified Helmholtz Equation

A noniterative reconstruction method for the inverse potential problem with partial boundary measurements

“…with k, i ∈ R + , where i is the contrast with respect to the material property of the background k. Therefore, in comparison to Fernandez et al, 29,30 the main novelty of the current article is the consideration of a contrast on the material properties in place of compact supported anomalies. This drives us to choose completely different adjoint and auxiliary states whose analysis led us to use the Bessel's functions of different types.…”

“…Following the original ideas presented in Fernandez et al,() in this paper, we are interested in the open problem,, pp126, problem 4.2 whose physical motivation comes out from semiconductors theory. See also the book of Isakov .…”

“…It comes out from the lack of uniqueness when both parameters are unknown. () Therefore, we assume that the contrasts γ i are given and then we focus on the reconstruction of the supports of the anomalies for i = 1, …,N from total or partial boundary measurements.…”

“…This drives us to choose completely different adjoint and auxiliary states whose analysis led us to use the Bessel's functions of different types. Another vital aspect of this paper, in com-parison to Fernandez et al, 29,30 is the collection of partial data on the boundary of the domain that makes the inverse problem much more difficult and the obtained result far more relevant from the application point of view but produced the requirement to devise a completely different computational mechanism. Now, for an initial guess k of k * , we consider the substance concentration field u to be the solution to the boundary value problem…”

Reconstruction algorithms of an inverse geometric problem for the modified Helmholtz equation

“…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 . Over the past few decades, there has been extensive studies in numerical methods for the modified Helmholtz equation (Fernandez et al 2018(Fernandez et al , 2019Hua 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.…”