Advances in Differential Equations and Control Processes
Volume 20, Issue 1, Pages 53 - 97
(February 2019) http://dx.doi.org/10.17654/DE020010053 |
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STABLE IDENTIFICATION OF SOURCES LOCATED ON SEPARATION INTERFACES OF TWO DIFFERENT HOMOGENEOUS MEDIA
MarÃa Monserrat MorÃn Castillo, Claudia Netzahualcoyotl Bautista, José Jacobo Oliveros Oliveros, José Julio Conde Mones and Lorenzo Héctor Juárez Valencia
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Abstract: This work presents a stable algorithm for identifying sources located on the surface interface of two different homogeneous media, one enclosed into the other one. These sources are identified from measurements on the exterior boundary of the bounded domain, and the relationship between the sources and the measurements via an elliptic model. An operational statement allows us to analyze the ill-posedness of the problem and, in particular, the numerical instability. The existence and uniqueness of the classical solution of the boundary value problem is shown as well as the uniqueness of the identification problem with respect to the measurement on the exterior boundary. The forward problem is solved analytically for the case of concentric circles and it is solved numerically by a finite element discretization for the case of complex domains. A Tikhonov regularization procedure is employed for the solution of the inverse problem, where the regularization parameter is obtained by the L-curve method. Synthetic examples are presented in order to validate the proposed stable algorithm. |
Keywords and phrases: inverse problem, regularization, ill-posed problem, source identification, layer potential, finite element method.
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