arXiv subjects

Leonidas Mindrinos

Publications and source records attributed to Leonidas Mindrinos.

28 records · Page 2Linked to original sources

arXiv2018

On a boundary integral solution of a lateral planar Cauchy problem in elastodynamics

A boundary integral based method for the stable reconstruction of missing boundary data is presented for the governing hyperbolic equation of elastodynamics in annular planar domains. Cauchy data in the form of the solution and traction is reconstructed on the inner boundary curve from the similar data given on the outer boundary. The ill-posed data reconstruction problem is reformulated as a sequence of boundary integral equations using the Laguerre transform with respect to time and employing a single-layer approach for the stationary problem. Singularities of the involved kernels in the integrals are analysed and made explicit, and standard quadrature rules are used for discretisation. Tikhonov regularization is employed for the stable solution of the obtained linear system. Numerical results are included showing that the outlined approach can be turned into a practical working method for finding the missing data.

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arXiv2018

The electromagnetic scattering problem by a cylindrical doubly-connected domain at oblique incidence: the direct problem

We consider the direct electromagnetic scattering problem of time-harmonic obliquely incident waves by a infinitely long, homogeneous and doubly-connected cylinder in three dimensions. We apply a hybrid integral equation method (combination of the direct and indirect methods) and we transform the scattering problem to a system of singular and hypersingular integral equations. The well-posedness of the corresponding problem is proven. We use trigonometric polynomial approximations and we solve the system of the discretized integral operators by a collocation method.

math.AP↗

arXiv2017

Quantitative reconstructions in multi-modal photoacoustic and optical coherence tomography imaging

In this paper we perform quantitative reconstruction of the electric susceptibility and the Grüneisen parameter of a non-magnetic linear dielectric medium using measurement of a multi-modal photoacoustic and optical coherence tomography system. We consider the mathematical model presented in [11], where a Fredholm integral equation of the first kind for the Grüneisen parameter was derived. For the numerical solution of the integral equation we consider a Galerkin type method.

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arXiv2017

The inverse scattering problem for orthotropic media in Polarization-sensitive Optical Coherence Tomography

In this paper we provide a mathematical model for imaging an anisotropic, orthotropic medium with Polarization-Sensitive Optical Coherence Tomography (PS-OCT). The imaging problem is formulated as an inverse scattering problem in three dimensions for reconstructing the electrical susceptibility of the medium using Maxwell's equations. Our reconstruction method is based on the second-order Born-approximation of the electric field.

math.AP↗

arXiv2017

On the numerical solution of the elastodynamic problem by a boundary integral equation method

A numerical method for the Dirichlet initial boundary value problem for the elastic equation in the exterior and unbounded region of a smooth closed simply connected 2-dimensional domain, is proposed and investigated. This method is based on a combination of a Laguerre transformation with respect to the time variable and a boundary integral equation approach in the spatial variables. Using the Laguerre transformation in time reduces the time-depended problem to a sequence of stationary boundary value problems, which are solved by a boundary layer approach resulting to a sequence of boundary integral equations of the first kind. The numerical discretization and solution are obtained by a trigonometrical quadrature method. Numerical results are included.

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arXiv2017

The inverse electromagnetic scattering problem by a penetrable cylinder at oblique incidence

In this work we consider the method of non-linear boundary integral equation for solving numerically the inverse scattering problem of obliquely incident electromagnetic waves by a penetrable homogeneous cylinder in three dimensions. We consider the indirect method and simple representations for the electric and the magnetic fields in order to derive a system of five integral equations, four on the boundary of the cylinder and one on the unit circle where we measure the far-field pattern of the scattered wave. We solve the system iteratively by linearizing only the far-field equation. Numerical results illustrate the feasibility of the proposed scheme.

math.NA↗

arXiv2016

The inverse scattering problem by an elastic inclusion

In this work we consider the inverse elastic scattering problem by an inclusion in two dimensions. The elastic inclusion is placed in an isotropic homogeneous elastic medium. The inverse problem, using the third Betti's formula (direct method), is equivalent to a system of four integral equations that are non linear with respect to the unknown boundary. Two equations are on the boundary and two on the unit circle where the far-field patterns of the scattered waves lie. We solve iteratively the system of integral equations by linearising only the far-field equations. Numerical results are presented that illustrate the feasibility of the proposed method.

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arXiv2016

The direct scattering problem of obliquely incident electromagnetic waves by a penetrable homogeneous cylinder

In this paper we consider the direct scattering problem of obliquely incident time-harmonic electromagnetic plane waves by an infinitely long dielectric cylinder. We assume that the cylinder and the outer medium are homogeneous and isotropic. From the symmetry of the problem, Maxwell's equations are reduced to a system of two 2D Helmholtz equations in the cylinder and two 2D Helmholtz equations in the exterior domain where the fields are coupled on the boundary. We prove uniqueness and existence of this differential system by formulating an equivalent system of integral equations using the direct method. We transform this system into a Fredholm type system of boundary integral equations in a Sobolev space setting. To handle the hypersingular operators we take advantage of Maue's formula. Applying a collocation method we derive an efficient numerical scheme and provide accurate numerical results using as test cases transmission problems corresponding to analytic fields derived from fundamental solutions.

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arXiv2016

Inverse Problems of Combined Photoacoustic and Optical Coherence Tomography

Optical coherence tomography (OCT) and photoacoustic tomography (PAT) are emerging non-invasive biological and medical imaging techniques. It is a recent trend in experimental science to design experiments that perform PAT and OCT imaging at once. In this paper we present a mathematical model describing the dual experiment. Since OCT is mathematically modelled by Maxwell's equations or some simplifications of it, whereas the light propagation in quantitative photoacoustics is modelled by (simplifications of) the radiative transfer equation, the first step in the derivation of a mathematical model of the dual experiment is to obtain a unified mathematical description, which in our case are Maxwell's equations. As a by-product we therefore derive a new mathematical model of photoacoustic tomography based on Maxwell's equations. It is well known by now, that without additional assumptions on the medium, it is not possible to uniquely reconstruct all optical parameters from either one of these modalities alone. We show that in the combined approach one has additional information, compared to a single modality, and the inverse problem of reconstruction of the optical parameters becomes feasible.

math.AP↗

arXiv2014

Mathematical Modelling of Optical Coherence Tomography

In this chapter a general mathematical model of Optical Coherence Tomography (OCT) is presented on the basis of the electromagnetic theory. OCT produces high resolution images of the inner structure of biological tissues. Images are obtained by measuring the time delay and the intensity of the backscattered light from the sample considering also the coherence properties of light. The scattering problem is considered for a weakly scattering medium located far enough from the detector. The inverse problem is to reconstruct the susceptibility of the medium given the measurements for different positions of the mirror. Different approaches are addressed depending on the different assumptions made about the optical properties of the sample. This procedure is applied to a full field OCT system and an extension to standard (time and frequency domain) OCT is briefly presented.

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