Speaker
Description
This poster concerns the coefficient inverse problem for Maxwell's equations, related to finding the space-dependent dielectric permittivity and effective conductivity functions using backscattered time-dependent electrical field in three dimensions. The aim is to reconstruct the dielectric properties of Malignant Melanoma tissue, both shape and values.
The forward problem is solved using a Domain Decomposition method developed in \cite{BL1, BR, LB1}, where both the Finite Element Method and the Finite Difference Method is used. The inverse problem is solved using Lagrangian approach and the Conjugate Gradient Descent method, and numerical examples are performed in a homogeneous and non-homogeneous setting while adaptively refining a finite element mesh using a posteriori error estimates of \cite{BL2}.
This method was applied for breast phantoms in \cite{BL1,BL2} and for malignant melanoma detection in a homogeneous setting in \cite{KLB}.
Bibliography
@article{BL1,
title = {An {Adaptive} {Finite} {Element}/{Finite} {Difference} {Domain} {Decomposition} {Method} for {Applications} in {Microwave} {Imaging}},
volume = {11},
issn = {2079-9292},
url = {https://www.mdpi.com/2079-9292/11/9/1359},
doi = {10.3390/electronics11091359},
abstract = {A new domain decomposition method for Maxwell’s equations in conductive media is presented. Using this method, reconstruction algorithms are developed for the determination of the dielectric permittivity function using time-dependent scattered data of an electric field. All reconstruction algorithms are based on an optimization approach to find the stationary point of the Lagrangian. Adaptive reconstruction algorithms and space-mesh refinement indicators are also presented. Our computational tests show the qualitative reconstruction of the dielectric permittivity function using an anatomically realistic breast phantom.},
language = {en},
number = {9},
urldate = {2026-03-13},
journal = {Electronics},
author = {Beilina, Larisa and Lindström, Eric},
month = apr,
year = {2022},
pages = {1359},
}
@article{BR,
title = {On the {Maxwell}-wave equation coupling problem and its explicit finite-element solution},
volume = {68},
url = {http://articles.math.cas.cz/10.21136/AM.2022.0210-21},
doi = {10.21136/AM.2022.0210-21},
number = {1},
urldate = {2026-03-13},
journal = {Applications of Mathematics},
author = {Beilina, Larisa and Ruas, Vitoriano},
month = feb,
year = {2023},
pages = {75--98},
}
@article{LB1,
title = {Energy norm error estimates and convergence analysis for a stabilized {Maxwell}'s equations in conductive media},
volume = {69},
url = {https://articles.math.cas.cz/10.21136/AM.2024.0248-23},
doi = {10.21136/AM.2024.0248-23},
number = {4},
urldate = {2026-03-13},
journal = {Applications of Mathematics},
author = {Lindström, Eric and Beilina, Larisa},
month = aug,
year = {2024},
pages = {415--436},
}
@incollection{BL2,
address = {Cham},
title = {A {Posteriori} {Error} {Estimates} and {Adaptive} {Error} {Control} for {Permittivity} {Reconstruction} in {Conductive} {Media}},
volume = {429},
isbn = {9783031358708 9783031358715},
url = {https://link.springer.com/10.1007/978-3-031-35871-5_7},
language = {en},
urldate = {2026-03-13},
booktitle = {Gas {Dynamics} with {Applications} in {Industry} and {Life} {Sciences}},
publisher = {Springer International Publishing},
author = {Beilina, L. and Lindström, E.},
editor = {Asadzadeh, Mohammad and Beilina, Larisa and Takata, Shigeru},
year = {2023},
doi = {10.1007/978-3-031-35871-5_7},
pages = {117--141},
}
@inproceedings{KLB,
address = {Palermo, Italy},
title = {Reconstructing the {Dielectric} {Properties} of {Melanoma} in {3D} {Using} {Real}-{Life} {Melanoma} {Model}},
copyright = {https://doi.org/10.15223/policy-029},
isbn = {9798331544720},
url = {https://ieeexplore.ieee.org/document/11305715/},
doi = {10.1109/ICEAA65662.2025.11305715},
urldate = {2026-03-13},
booktitle = {2025 {International} {Conference} on {Electromagnetics} in {Advanced} {Applications} ({ICEAA})},
publisher = {IEEE},
author = {Kyhn, Georg and Lindström, Eric and Beilina, Larisa},
month = sep,
year = {2025},
pages = {454--459},
}
