IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v10y2022i15p2726-d878291.html
   My bibliography  Save this article

Stable Numerical Identification of Sources in Non-Homogeneous Media

Author

Listed:
  • 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
    These authors contributed equally to this work.)

  • 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
    These authors contributed equally to this work.)

  • 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
    These authors contributed equally to this work.)

  • 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
    These authors contributed equally to this work.)

  • 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
    These authors contributed equally to this work.)

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.

Suggested Citation

  • José Julio Conde Mones & Carlos Arturo Hernández Gracidas & María Monserrat Morín Castillo & José Jacobo Oliveros Oliveros & Lorenzo Héctor Juárez Valencia, 2022. "Stable Numerical Identification of Sources in Non-Homogeneous Media," Mathematics, MDPI, vol. 10(15), pages 1-31, August.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:15:p:2726-:d:878291
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/10/15/2726/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/10/15/2726/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. María Monserrat Morín-Castillo & Jesús Arriaga-Hernández & Bolivia Cuevas-Otahola & José Jacobo Oliveros-Oliveros, 2022. "Analysis of Dipolar Sources in the Solution of the Electroencephalographic Inverse Problem," Mathematics, MDPI, vol. 10(11), pages 1-22, June.
    2. José Julio Conde Mones & Emmanuel Roberto Estrada Aguayo & José Jacobo Oliveros Oliveros & Carlos Arturo Hernández Gracidas & María Monserrat Morín Castillo, 2021. "Stable Identification of Sources Located on Interface of Nonhomogeneous Media," Mathematics, MDPI, vol. 9(16), pages 1-23, August.
    3. Collar, Andrés Fraguela & Castillo, Monserrat Morín & Oliveros, Jacobo Oliveros, 2008. "Inverse electroencephalography for volumetric sources," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 78(4), pages 481-492.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. María Monserrat Morín-Castillo & Jesús Arriaga-Hernández & Bolivia Cuevas-Otahola & José Jacobo Oliveros-Oliveros, 2022. "Analysis of Dipolar Sources in the Solution of the Electroencephalographic Inverse Problem," Mathematics, MDPI, vol. 10(11), pages 1-22, June.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:10:y:2022:i:15:p:2726-:d:878291. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.