L. Beilina

E. M. Karchevskii, L. Beilina, A. O. Spiridonov et al.

We present new method for the numerical reconstruction of the variable refractive index of multi-layered circular weakly guiding dielectric waveguides using the measurements of the propagation constants of their eigenwaves. Our numerical examples show stable reconstruction of the dielectric permittivity function $\varepsilon$ for random noise level using these measurements.

L. Beilina

We present the Finite Element Method (FEM) for the numerical solution of the multidimensional coefficient inverse problem (MCIP) in two dimensions. This method is used for explicit reconstruction of the coefficient in the hyperbolic equation using data resulted from a single measurement. To solve our MCIP we use approximate globally convergent method and then apply FEM for the resulted equation. Our numerical examples show quantitative reconstruction of the sound speed in small tumor-like inclusions.

M. Asadzadeh, L. Beilina, M. Naseer et al.

We derive a priori error estimates for the standard Galerkin and streamline diffusion finite element methods for the Fermi pencil-beam equation obtained from a fully three dimensional Fokker-Planck equation in space ${\mathbf x}=(x,y,z)$ and velocity $\tilde {\mathbf v}=(μ, η, ξ)$ variables. The Fokker-Planck term appears as a Laplace-Beltrami operator in the unit sphere. The diffusion term in the Fermi equation is obtained as a projection of the FP operator onto the tangent plane to the unit sphere at the pole $(1,0,0)$ and in the direction of $ {\mathbf v}_0=(1,η, ξ)$. Hence the Fermi equation, stated in three dimensional spatial domain ${\mathbf x}=(x,y,z)$, depends only on two velocity variables ${\mathbf v}=(η, ξ)$. Since, for a certain number of cross-sections, there is a closed form analytic solution available for the Fermi equation, hence an a posteriori error estimate procedure is unnecessary and in our adaptive algorithm for local mesh refinements we employ the a priori approach. Different numerical examples, in two space dimensions are justifying the theoretical results. Implementations show significant reduction of the computational error by using our adaptive algorithm.

L. Beilina, M. Cristofol, S. Li

This paper is devoted to the reconstruction of the time and space-dependent coefficient in an infinite cylindrical hyperbolic domain. Using a local Carleman estimate we prove the uniqueness and a Hölder stability in the determining of the conductivity by a single measurement on the lateral boundary. Our numerical examples show good reconstruction of the location and contrast of the conductivity function in three dimensions.

L. Beilina, L. Mpinganzima, P. Tassin

We consider the problem of the construction of the nanophotonic structures of arbitrary geometry with prescribed desired properties. We reformulate this problem as an optimization problem for the Tikhonov functional which is minimized on adaptively locally refined meshes. These meshes are refined only in places where the nanophotonic structure should be designed. Our special symmetric mesh refinement procedure allows the construction of different nanophotonic structures. We illustrate efficiency of our adaptive optimization algorithm on the construction of nanophotonic structure in two dimensions.

L. Beilina, V. Ruas

This paper is devoted to the numerical validation of an explicit finite-difference scheme for the integration in time of Maxwell's equations in terms of the sole electric field, using standard linear finite elements for the space discretization. The rigorous reliability analysis of this numerical model was the object of another authors' arXiv paper. More specifically such a study applies to the particular case where the electric permittivity has a constant value outside a sub-domain, whose closure does not intersect the boundary of the domain where the problem is defined. Our numerical experiments in two-dimension space certify that the convergence results previously derived for this approach are optimal, as long as the underlying CFL condition is satisfied.

L. Beilina

We present domain decomposition finite element/finite difference method for the solution of hyperbolic equation. The domain decomposition is performed such that finite elements and finite differences are used in different subdomains of the computational domain: finite difference method is used on the structured part of the computational domain and finite elements on the unstructured part of the domain. The main goal of this method is to combine flexibility of finite element method and efficiency of a finite difference method. An explicit discretization schemes for both methods are constructed such that finite element and finite difference schemes coincide on the common structured overlapping layer between computational subdomains. Then the resulting scheme can be considered as a pure finite element scheme which allows avoid instabilities at the interfaces. We illustrate efficiency of the domain decomposition method on the reconstruction of the conductivity function in the hyperbolic equation in three dimensions.