arXiv subjects

Leonidas Mindrinos

Publications and source records attributed to Leonidas Mindrinos.

At least 19 recordsLinked to original sources

arXiv2026

The unified transform for Burgers' equation: Application to unsaturated flow in finite interval

In this paper, we focus on one-dimensional vertical infiltration, assuming constant diffusivity and a quadratic relationship between hydraulic conductivity and water content. Under these assumptions, Richards' equation reduces to Burgers' equation, which we then linearize via the Hopf-Cole transformation. This turns the initial boundary value problem into a diffusion equation on a finite interval with mixed boundary conditions. To solve it, we use the Unified Transform Method (also known as the Fokas method). This approach gives an explicit integral representation of the solution, and when evaluated numerically, the results match classical Fourier series solutions exactly, but with better convergence and stability. Two examples from hydrological applications are examined.

math.AP↗

arXiv2025

Estimating parameters of the diffusion model via asymptotic expansions

A broad class of inverse problems deals with determining certain parameters, from measurement data, in models which are associated to certain partial differential equations. In this work we focus on the heat equation on a finite interval and we determine the dimensionless diffusion parameter from a single measurement. Our results extend to estimating additional parameters of the initial-boundary value problem, such as the length of the interval and/or the time required for the solution to achieve a specific state. Our approach relies on the asymptotic solution of an integral equation: The formulation of this integral equation is based on the solution of the direct problem via the Fokas method; the solution of this equation is achieved through the asymptotic evaluation of the associated integrals which yield an effective approximate solution, supported by numerical verifications. We apply these approximations to well-established problems in soil science and we compare our results with existing ones, displaying clear improvement.

math.AP↗

arXiv2025

Inverse Electromagnetic Scattering for Doubly-Connected Cylinders using Convolutional Neural Networks

In this work, we consider the inverse electromagnetic scattering problem for a magneto-dielectric cylinder covering an impedance cylinder of arbitrary shape. We solve it by introducing a divide-and-conquer framework using specially designed 1D multi-channel, circular-padding Convolutional Neural Networks. The solution of the direct problem provides us with the real and imaginary components of the far-field measurements representing the input data. We first classify the shape of the impedance cylinder and then reconstruct the unknown boundary curve and the impedance function. Through extensive numerical experiments, including noisy scenarios, we demonstrate the efficiency and robustness of our approach.

math.NA↗

arXiv2025

Estimating properties of a homogeneous bounded soil using machine learning models

This work focuses on estimating soil properties from water moisture measurements. We consider simulated data generated by solving the initial-boundary value problem governing vertical infiltration in a homogeneous, bounded soil profile, with the usage of the Fokas method. To address the parameter identification problem, which is formulated as a two-output regression task, we explore various machine learning models. The performance of each model is assessed under different data conditions: full, noisy, and limited. Overall, the prediction of diffusivity $D$ tends to be more accurate than that of hydraulic conductivity $K.$ Among the models considered, Support Vector Machines (SVMs) and Neural Networks (NNs) demonstrate the highest robustness, achieving near-perfect accuracy and minimal errors.

physics.geo-ph↗

arXiv2025

On the inverse elastic problem for isotropic media using Eshelby and Lippmann-Schwinger integral formulations

We present two applications of the integro-differential volume equation for the eigenstrain, building on Eshelby's inclusion method [15,16], in the contexts of both static and dynamic linear elasticity. The primary objective is to address the inverse problem of recovering the elastic moduli of the inhomogeneity using a limited number of incident fields. In the static case, we adopt an efficient reformulation of Eshelby's equation proposed by Bonnet [7]. By employing a first-order approximation in addition with a limited number of incident loadings and measurements, we numerically determine the material coefficients of the inclusion. In elastodynamics, we focus on the inverse scattering problem, utilizing the Lippmann-Schwinger integral equation to reconstruct the elastic properties of the inclusion through a Newton-type iterative scheme. We construct the Frechet derivative and we formulate the linearized far-field equation. Additionally, the corresponding plane strain problems are analyzed in both static and dynamic elasticity.

math.AP↗

arXiv2024

Rainfall infiltration: Direct and Inverse problems on a linear evolution equation

Originating from the mathematical modelling of rainfall infiltration, we derive the solution of an initial-boundary value problem of a linear evolution partial differential equation, by using the Fokas method. We present numerical examples which correspond to specific physical rainfall problems. Based on this formalism we present an effective algorithm for the associated null-controllability problem, namely we numerically derive a family of boundary controls that steer the solution to the desired flat final state. Finally, a regularisation scheme allows the derivation of relatively small controls, in cases where this is necessary.

math.AP↗

arXiv2024

An analytical solution for vertical infiltration in bounded profiles

In this study, we derive an analytical solution to address the problem of one-dimensional vertical infiltration within bounded profiles. We consider the Richards equation together with various boundary conditions, simulating different scenarios of water application onto the surface of a homogeneous and bounded medium. To solve the corresponding initial boundary value problem over a finite interval, we apply the unified transform, commonly known as the Fokas method. Through this methodology, we obtain an integral representation that can be efficiently and directly computed numerically, yielding a convergent scheme.

math.AP↗

arXiv2023

On the numerical solution of a hyperbolic initial boundary value problem by hypersingular boundary integral equations

In this study, we consider the numerical solution of the Neumann initial boundary value problem for the wave equation in 2D domains. Employing the Laguerre transform with respect to the temporal variable, we effectively transform this problem into a series of Neumann elliptic problems. The development of a fundamental sequence for these elliptic equations provides us with the means to introduce modified double layer potentials. Consequently, we are able to derive a sequence of boundary hypersingular integral equations as a result of this transformation. To discretize the system of equations, we apply the Maue transform and implement the Nyström method with trigonometric quadrature techniques. To demonstrate the practical utility of our approach, we provide numerical examples.

math.NA↗

arXiv2023

Wave scattering in 1D: D'Alembert-type representations and a reconstruction method

We derive the extension of the classical d'Alembert formula for the wave equation, which provides the analytical solution for the direct scattering problem for a medium with constant refractive index; this is achieved by employing results obtained via the Fokas method. This methodology is further extended to a medium with piecewise constant refractive index, providing the apparatus for the solution of the associated inverse scattering problem. Hence, we provide an exact reconstruction method which is valid for both full and phaseless data.

math.AP↗

arXiv2023

Quantitative Parameter Reconstruction from Optical Coherence Tomographic Data

Quantitative tissue information, like the light scattering properties, is considered as a key player in the detection of cancerous cells in medical diagnosis. A promising method to obtain these data is optical coherence tomography (OCT). In this article, we will therefore discuss the refractive index reconstruction from OCT data, employing a Gaussian beam based forward model. We consider in particular samples with a layered structure, meaning that the refractive index as a function of depth is well approximated by a piece-wise constant function. For the reconstruction, we present a layer-by-layer method where in every step the refractive index is obtained via a discretized least squares minimization. For an approximated form of the minimization problem, we present an existence and uniqueness result. The applicability of the proposed method is then verified by reconstructing refractive indices of layered media from both simulated and experimental OCT data.

math.NA↗

arXiv2022

Development of mathematical models for quantitative OCT: A review

We review mathematical models describing how Optical Coherence Tomography works. Hereby, we focus on models based on Maxwell's equations and their simplifications. We highlight especially the effects of different modeling assumptions for the incident illumination, the medium, the light propagation, and the measurement setup and illustrate the qualitatively differing behavior in numerical simulations of the OCT data and compare them with real data from OCT measurements.

physics.optics↗

arXiv2022

On the numerical solution of a hyperbolic inverse boundary value problem in bounded domains

We consider the inverse problem of reconstructing the boundary curve of a cavity embedded in a bounded domain. The problem is formulated in two dimensions for the wave equation. We combine the Laguerre transform with the integral equation method and we reduce the inverse problem to a system of boundary integral equations. We propose an iterative scheme that linearizes the equation using the Fréchet derivative of the forward operator. The application of special quadrature rules results to an ill-conditioned linear system which we solve using Tikhonov regularization. The numerical results show that the proposed method produces accurate and stable reconstructions.

math.NA↗

arXiv2021

A Quantitative Model for Optical Coherence Tomography

Optical coherence tomography (OCT) is a widely used imaging technique in the micrometer regime, which gained accelerating interest in medical imaging in the last twenty years. In up-to-date OCT literature [5,6] certain simplifying assumptions are made for the reconstructions, but for many applications a more realistic description of the OCT imaging process is of interest. In mathematical models, for example, the incident angle of light onto the sample is usually neglected or a plane wave description for the light-sample interaction in OCT is used, which ignores almost completely the occurring effects within an OCT measurement process. In this article, we make a first step to a quantitative model by considering the measured intensity as a combination of back-scattered Gaussian beams affected by the system. In contrast to the standard plane wave simplification, the presented model includes system relevant parameters such as the position of the focus and the spot size of the incident laser beam, which allow a precise prediction of the OCT data and therefore ultimately serves as a forward model. The accuracy of the proposed model - after calibration of all necessary system parameters - is illustrated by simulations and validated by a comparison with experimental data obtained from a 1300 nm swept-source OCT system.

physics.med-ph↗

arXiv2020

Reconstructing the Optical Parameters of a Layered Medium with Optical Coherence Elastography

In this work we consider the inverse problem of reconstructing the optical properties of a layered medium from an elastography measurement where optical coherence tomography is used as the imaging method. We hereby model the sample as a linear dielectric medium so that the imaging parameter is given by its electric susceptibility, which is a frequency- and depth-dependent parameter. Additionally to the layered structure (assumed to be valid at least in the small illuminated region), we allow for small scatterers which we consider to be randomly distributed, a situation which seems more realistic compared to purely homogeneous layers. We then show that a unique reconstruction of the susceptibility of the medium (after averaging over the small scatterers) can be achieved from optical coherence tomography measurements for different compression states of the medium.

math.NA↗

arXiv2020

Computed Origami Tomography

In this paper, we provide assembly instructions for an easy to build experimental setup in order to gain practical experience with tomography. In view of this, this paper can be seen as a complementary work to excellent mathematical textbooks at an undergraduate level concerned with the basic mathematical principles of tomography. Since the setup uses light for tomographic imaging, the investigated objects need to be light transparent, like origami figures. In case the reader wants to experiment with computational tomographic reconstructions without assembling the device, we provide a database of several objects together with their tomographic measurements and a publicly available software. Moreover, recent advances in Cryo-imaging enabled three-dimensional high-resolution visualization of single particles, such as for instance viruses. To exemplify, and experience, single particle Cryo-electron microscopy we provide an advanced assembly, which we use to generate data simulating a Cryo-recording. We also discuss some of the major practical difficulties for reconstructing particles from Cryo-microscopic data.

eess.IV↗

arXiv2020

The direct electromagnetic scattering problem by a piecewise constant inhomogeneous cylinder at oblique incidence

We consider the solvability of the direct scattering problem of an obliquely incident time-harmonic electromagnetic wave by a piecewise constant inhomogeneous, penetrable and infinitely long cylinder. We prove the existence and uniqueness of the solution using properties of the boundary value operators and the integral equation method. For the numerical solution, we apply a collocation method and we approximate the singular integral operators using quadrature rules. We show convergence of the numerical scheme for the interior and the scattered fields both in the near- and far-field regime.

math.AP↗

arXiv2019

Quantitative OCT reconstructions for dispersive media

We consider the problem of reconstructing the position and the time-dependent optical properties of a linear dispersive medium from OCT measurements. The medium is multi-layered described by a piece-wise inhomogeneous refractive index. The measurement data are from a frequency-domain OCT system and we address also the phase retrieval problem. The parameter identification problem can be formulated as an one-dimensional inverse problem. Initially, we deal with a non-dispersive medium and we derive an iterative scheme that is the core of the algorithm for the frequency-dependent parameter. The case of absorbing medium is also addressed.

math.NA↗

arXiv2019

On the Non-Linear Integral Equation Approach for an Inverse Boundary Value Problem for the Heat Equation

We consider the inverse problem of reconstructing the interior boundary curve of a doubly connected domain from the knowledge of the temperature and the thermal flux on the exterior boundary curve. The use of the Laguerre transform in time leads to a sequence of stationary inverse problems. Then, the application of the modified single-layer ansatz, reduces the problem to a sequence of systems of non-linear boundary integral equations. An iterative algorithm is developed for the numerical solution of the obtained integral equations. We find the Fréchet derivative of the corresponding integral operator and we show the unique solvability of the linearized equation. Full discretization is realized by a trigonometric quadrature method. Due to the inherited ill-possedness of the derived system of linear equations we apply the Tikhonov regularization. The numerical results show that the proposed method produces accurate and stable reconstructions.

math.NA↗