arXiv subjects

Larisa Beilina

Publications and source records attributed to Larisa Beilina.

At least 19 recordsLinked to original sources

arXiv2026

Variational optimization approach for reconstruction of dielectric permittivity and conductivity functions using partial boundary measurements

We present a variational optimization approach for the solution of a coefficient inverse problem of simultaneous reconstruction of the dielectric permittivity and conductivity functions in time-dependent Maxwell's system using limited boundary observations of the electric field. The variational optimization approach is based on constructing a weak form of a Lagrangian which allows to use finite element based reconstruction algorithms. The optimality conditions for the Lagrangian and stability estimate for the adjoint problem are derived, as well as Frechét differentiability of it and of the regularized Tikhonov functional are also presented. Two- and three-dimensional numerical studies confirm our theoretical investigations.

math.NA↗

arXiv2025

Reconstructing the dielectric properties of melanoma in 3D using real-life melanoma model

The paper presents performance of the adaptive domain decomposition finite element/finite difference method for reconstruction of the dielectric permittivity and conductivity functions for 3D real-life melanoma model using measurements of the backscattered electric field at the boundary of the investigated domain. We present several gradient-based reconstruction algorithms which use optimization approach to find stationary point of the Lagrangian. Our computational tests show qualitative and quantitative reconstruction of dielectric permittivity and conductivity functions using realistic model of malign melanoma at 6 GHz in 3D.

math.NA↗

arXiv2025

Finite element 3D models of melanoma growth and time-dependent backscattered data for dielectric properties of melanoma at 6 GHz

Finite element meshes for 3D models simulating realistic malignant melanoma (MM) growth, incorporating accurate dielectric properties of the skin, have been developed. Numerical simulations illustrate how 3D finite element meshes can be utilized to generate backscattered data, enabling the evaluation of reconstruction algorithms designed to determine the dielectric properties of the proposed 3D model.

math.NA↗

arXiv2025

A hybrid finite element/finite difference method for reconstruction of dielectric properties of conductive objects

The aim of this article is to present a hybrid finite element/finite difference method which is used for reconstructions of electromagnetic properties within a realistic breast phantom. This is done by studying the mentioned properties' (electric permittivity and conductivity in this case) representing coefficients in a constellation of Maxwell's equations. This information is valuable since these coefficient can reveal types of tissues within the breast, and in applications could be used to detect shapes and locations of tumours. Because of the ill-posed nature of this coefficient inverse problem, we approach it as an optimization problem by introducing the corresponding Tikhonov functional and in turn Lagrangian. These are then minimized by utilizing an interplay between finite element and finite difference methods for solutions of direct and adjoint problems, and thereafter by applying a conjugate gradient method to an adaptively refined mesh.

math.NA↗

arXiv2024

Frequency inversion method and device for malignant melanoma detection using RF/microwaves

The paper considers an inverse problem of reconstructing the spatially distributed complex dielectric permittivity function using backscattered data of the electric field in the frequency domain. We develop iterative algorithm for reconstruction of this function. Numerical example show the performance of the proposed algorithm in 2D. Additionally, we present at the first time the new prototype of the device for the malignant melanoma detection which is modelled in HFSS/Ansys. Results of measurements of scattered parameters in HFSS/Ansys show significant phase distortion in the case when melanoma is present compared to the case when melanoma is absent.

physics.med-ph↗

arXiv2023

Energy norm error estimates and convergence analysis for a stabilized Maxwell's equations in conductive media

The aim of this article is to investigate the well-posedness, stability and convergence of solutions to the time-dependent Maxwell's equations for electric field in conductive media in continuous and discrete settings. The situation we consider would represent a physical problem where a subdomain is emerged in a homogeneous medium, characterized by constant dielectric permittivity and conductivity functions. It is well known that in these homogeneous regions the solution to the Maxwell's equations also solves the wave equation which makes calculations very efficient. In this way our problem can be considered as a coupling problem for which we derive stability and convergence analysis. A number of numerical examples validate theoretical convergence rates of the proposed stabilized explicit finite element scheme.

math.NA↗

arXiv2022

An adaptive finite element/finite difference domain decomposition method for applications in microwave imaging

A new domain decomposition method for Maxwell's equations in conductive media is presented. Using this method reconstruction algorithms are developed for determination of dielectric permittivity function using time-dependent scattered data of electric field. All reconstruction algorithms are based on optimization approach to find stationary point of the Lagrangian. Adaptive reconstruction algorithms and space mesh refinement indicators are also presented. Our computational tests show qualitative reconstruction of dielectric permittivity function using anatomically realistic breast phantom.

math.NA↗

arXiv2021

Stability and convergence analysis of a domain decomposition FE/FD method for the Maxwell's equations in time domain

Stability and convergence analysis for the domain decomposition finite element/finite difference (FE/FD) method is presented. The analysis is designed for semi-discrete finite element scheme for the time-dependent Maxwell's equations. The explicit finite element schemes in different settings of the spatial domain are constructed and domain decomposition algorithm is formulated. Several numerical examples validate convergence rates obtained in the theoretical studies.

math.NA↗

arXiv2021

Interactive Change Point Detection using optimisation approach and Bayesian statistics applied to real world applications

Change point detection becomes more and more important as datasets increase in size, where unsupervised detection algorithms can help users process data. To detect change points, a number of unsupervised algorithms have been developed which are based on different principles. One approach is to define an optimisation problem and minimise a cost function along with a penalty function. In the optimisation approach, the choice of the cost function affects the predictions made by the algorithm. In extension to the existing studies, a new type of cost function using Tikhonov regularisation is introduced. Another approach uses Bayesian statistics to calculate the posterior probability distribution of a specific point being a change point. It uses a priori knowledge on the distance between consecutive change points and a likelihood function with information about the segments. The optimisation and Bayesian approaches for offline change point detection are studied and applied to simulated datasets as well as a real world multi-phase dataset. The approaches have previously been studied separately and a novelty lies in comparing the predictions made by the two approaches in a specific setting, consisting of simulated datasets and a real world example. The study has found that the performance of the change point detection algorithms are affected by the features in the data.

math.NA↗

arXiv2020

An explicit P1 finite element scheme for Maxwell's equations with constant permittivity in a boundary neighborhood

This paper is devoted to the complete convergence study of the finite-element approximation of Maxwell's equations in the case where the magnetic permeability is constant. Standard linear finite elements for the space discretization are combined with a well-known explicit finite-difference scheme for the time discretization. The analysis applies to the particular case where the dielectric permittivity has a constant value outside a sub-domain, whose closure does not intersect the boundary of the problem-definition domain. Optimal convergence results are established in natural norms under reasonable assumptions, provided a classical CFL condition holds. A numerical validation of the theoretical results is provided.

math.NA↗

arXiv2017

Numerical studies of the Lagrangian approach for reconstruction of the conductivity in a waveguide

We consider an inverse problem of reconstructing the conductivity function in a hyperbolic equation using single space-time domain noisy observations of the solution on the backscattering boundary of the computational domain. We formulate our inverse problem as an optimization problem and use Lagrangian approach to minimize the corresponding Tikhonov functional. We present a theorem of a local strong convexity of our functional and derive error estimates between computed and regularized as well as exact solutions of this functional, correspondingly. In numerical simulations we apply domain decomposition finite element-finite difference method for minimization of the Lagrangian. Our computational study shows efficiency of the proposed method in the reconstruction of the conductivity function in three dimensions.

math.NA↗

arXiv2017

Computational design of acoustic materials using an adaptive optimization algorithm

We consider the problem of the construction of the acoustic structure of arbitrary geometry with prescribed desired properties. We use optimization approach for the solution of this problem and minimize the Tikhonov functional on adaptively refined meshes. These meshes are refined locally only in places where the acoustic structure should be designed. Our special symmetric mesh refinement strategy together with interpolation procedure allows the construction of the symmetric acoustic material with prescribed properties. Efficiency of the presented adaptive optimization algorithm is illustrated on the construction of the symmetric acoustic material in two dimensions.

math.NA↗

arXiv2016

Reconstruction of annular bi-layered media in cylindrical waveguide section

A radial transverse resonance model for two cylindrical concentric layers with different complex dielectric constants is presented. An inverse problem with four unknowns - 3 physical material parameters and one dimensional dielectric layer thickness parameter- is solved by employing TE110 and TE210 modes with different radial field distribution. First a Newton-Raphson algorithm is used to solve a least square problem with a Lorentzian function (as resonance model and "measured" data generator). Then found resonance frequencies and quality factors are used in a second inverse Newton-Raphson algorithm that solves four transverse resonance equations in order to get four unknown parameters. The use of TE110 and TE210 models offers one dimensional radial tomographic capability. An open ended coax quarter-wave resonator is added to the sensor topology, and the effect on the convergence is investigated.

physics.optics↗

arXiv2016

An Adaptive Finite Element Method in Quantitative Reconstruction of Small Inclusions from Limited Observations

We consider a coefficient inverse problem for the dielectric permittivity in Maxwell's equations, with data consisting of boundary measurements of one or two backscattered or transmitted waves. The problem is treated using a Lagrangian approach to the minimization of a Tikhonov functional, where an adaptive finite element method forms the basis of the computations. A new a posteriori error estimate for the coefficient is derived. The method is tested successfully in numerical experiments for the reconstruction of two, three, and four small inclusions with low contrast, as well as the reconstruction of a superposition of two Gaussian functions.

math.NA↗

arXiv2016

An adaptive finite element method in reconstruction of coefficients in Maxwell's equations from limited observations

We propose an adaptive finite element method for the solution of a coefficient inverse problem of simultaneous reconstruction of the dielectric permittivity and magnetic permeability functions in the Maxwell's system using limited boundary observations of the electric field in 3D. We derive a posteriori error estimates in the Tikhonov functional to be minimized and in the regularized solution of this functional, as well as formulate corresponding adaptive algorithm. Our numerical experiments justify the efficiency of our a posteriori estimates and show significant improvement of the reconstructions obtained on locally adaptively refined meshes.

math.NA↗

arXiv2015

Optimization approach for the simultaneous reconstruction of the dielectric permittivity and magnetic permeability functions from limited observations

We consider the inverse problem of the simultaneous reconstruction of the dielectric permittivity and magnetic permeability functions of the Maxwell's system in 3D with limited boundary observations of the electric field. The theoretical stability for the problem is provided by the Carleman estimates. For the numerical computations the problem is formulated as an optimization problem and hybrid finite element/difference method is used to solve the parameter identification problem.

math.NA↗

arXiv2015

Numerical studies of an adaptive finite element method applied to the reconstruction of shapes of buried objects from experimental data

We perform extended studies of an adaptive finite element method applied to the reconstruction of shapes of buried objects from experimental backscattering data. We use experimental data which are collected by a microwave scattering facility located at the University of North Carolina at Charlotte, USA. Our numerical tests show accurate imaging of three components of interest of targets: shapes, locations and refractive indices.

math.NA↗

arXiv2014

Globally convergent and adaptive finite element methods in imaging of buried objects from experimental backscattering radar measurements

We consider a two-stage numerical procedure for imaging of objects buried in dry sand using time-dependent backscattering experimental radar measurements. These measurements are generated by a single point source of electric pulses and are collected using a microwave scattering facility which was built at the University of North Carolina at Charlotte. Our imaging problem is formulated as the inverse problem of the reconstruction of the spatially distributed dielectric permittivity $\varepsilon_\mathrm{r}\left(\mathbf{x}\right), \ \mathbf{x}\in \mathbb{R}^{3}$, which is an unknown coefficient in Maxwell's equations. On the first stage an approximately globally convergent method is applied to get a good first approximation for the exact solution. On the second stage a local adaptive finite element method is applied to refine the solution obtained on the first stage. The two-stage numerical procedure results in accurate imaging of all three components of interest of targets: shapes, locations and refractive indices. In this paper we briefly describe methods and present new reconstruction results for both stages.

math.NA↗