Stable Numerical Identification of Sources in Non-Homogeneous Media

José Julio Conde Mones, Carlos Arturo Hernández Gracidas, María Monserrat Morín Castillo, José Jacobo Oliveros Oliveros and Lorenzo Héctor Juárez Valencia
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José Julio Conde Mones: Facultad de Ciencias Físico Matemáticas, Benemérita Universidad Autónoma de Puebla, Avenida San Claudio y 18 Sur, Colonia San Manuel, Ciudad Universitaria, Puebla 72570, Mexico
Carlos Arturo Hernández Gracidas: CONACYT-BUAP, Facultad de Ciencias Físico Matemáticas, Benemérita Universidad Autónoma de Puebla, Avenida San Claudio y 18 Sur, Colonia San Manuel, Ciudad Universitaria, Puebla 72570, Mexico
María Monserrat Morín Castillo: Facultad de Ciencias de la Electrónica, Benemérita Universidad Autónoma de Puebla, Avenida San Claudio y 18 Sur, Colonia San Manuel, Ciudad Universitaria, Puebla 72570, Mexico
José Jacobo Oliveros Oliveros: Facultad de Ciencias Físico Matemáticas, Benemérita Universidad Autónoma de Puebla, Avenida San Claudio y 18 Sur, Colonia San Manuel, Ciudad Universitaria, Puebla 72570, Mexico
Lorenzo Héctor Juárez Valencia: Departamento de Matemáticas, División de Ciencias Básicas e Ingeniería, UAM-Izt., Edificio AT-314, Unidad Iztapalapa, San Rafael Atlixco No. 186, Col. Vicentina, Iztapalapa 09340, Mexico

Mathematics, 2022, vol. 10, issue 15, 1-31

Abstract: In this work, we present a numerical algorithm to solve the inverse problem of volumetric sources from measurements on the boundary of a non-homogeneous conductive medium, which is made of conductive layers with constant conductivity in each layer. This inverse problem is ill-posed since there is more than one source that can generate the same measurement. Furthermore, the ill-posedness is due to the fact that small variations (or errors) in the measurement (input data) can produce substantial variations in the identified source location. We propose two steps to solve this inverse problem in some classes of sources: we first recover the harmonic part of the volumetric source, and, in a second step, we compute the non-harmonic part of the source. For the reconstruction of the harmonic part of the source, we follow a variational approach based on the reformulation of the inverse problem as a distributed control problem, for which the cost function incorporates a penalized term with the input data on the boundary. This cost function is minimized by a conjugate gradient algorithm in combination with a finite element discretization. We recover the non-harmonic component of the source using a priori information and an iterative algorithm for some particular classes of sources. To validate the numerical methodology, we develop synthetic examples both in circular (simple) and irregular (complex) regions. The numerical results show that the proposed methodology allows to recover the complete source and produce stable and accurate numerical solutions.

Keywords: inverse problem; harmonic sources; control theory; conjugate gradient method; finite element method (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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