arXiv subjects

L. Beilina

Publications and source records attributed to L. Beilina.

13 recordsLinked to original sources

arXiv2025

A time-adaptive optimization approach for reconstructing immune response in a mathematical model of acute HIV infection using clinical data

The paper proposes a time-adaptive optimization approach for determining the time-dependent immune response function in a mathematical model of acute HIV infection, using clinical data from four untreated patients. We formulate the problem as a parameter identification problem for an immune response system of ODE which includes novel component integrated into the third equation of the classical three-equation HIV model. Tikhonov's regularization method, Lagrangian approach, from which we derive the optimality conditions, and a numerical scheme to solve the forward and adjoint problems, as well as parameter identification problem, are presented. Three different a posteriori error estimates are derived and based on these estimates, a time adaptive optimization algorithm is formulated. Numerical experiments demonstrate the effectiveness of the proposed adaptive method in reconstructing the immune response function during the acute phase of HIV infection, using patient-specific clinical data. Computational results show improvement of reconstruction of immune response function using the local time-adaptive mesh refinement method compared to the standard conjugate gradient method applied on a uniform time mesh.

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arXiv2025

Reconstruction of source function in a parabolic equation using partial boundary measurements

In this paper, we present the analytical and numerical study of the optimization approach for determining the space-dependent source function in the parabolic inverse source problem using partial boundary measurements. The Lagrangian approach for the solution of the optimization problem is presented, and optimality conditions are derived. The proof of the Fr\'echet differentiability of the regularized Tikhonov functional and the existence result for the solution of the inverse source problem are established. A local stability estimate for the unknown source term is also presented. The numerical examples justify the theoretical investigations using the conjugate gradient method (CGM) in 2D and 3D tests with noisy data.

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arXiv2020

Numerical analysis of least squares and perceptron learning for classification problems

This work presents study on regularized and non-regularized versions of perceptron learning and least squares algorithms for classification problems. Fr'echet derivatives for regularized least squares and perceptron learning algorithms are derived. Different Tikhonov's regularization techniques for choosing the regularization parameter are discussed. Decision boundaries obtained by non-regularized algorithms to classify simulated and experimental data sets are analyzed.

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arXiv2020

Time-adaptive optimization in a parameter identification problem of HIV infection

The paper considers a time-adaptive method for determination of drug efficacy in a parameter identification problem (PIP) for system of ordinary differential equations (ODE) which describe dynamics of the primary HIV infection. Optimization approach to solve this problem is presented and a posteriori error estimates in the Tikhonov functional and Lagrangian are formulated. Based on these estimates a time adaptive algorithm is formulated and numerically tested for different scenarios of noisy observations of virus population function. Numerical results show significant improvement of reconstruction of drug efficacy parameter when using time adaptive mesh refinement compared to usual gradient method applied on a uniform time mesh.

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arXiv2019

On stabilized P1 finite element approximation fortime harmonic Maxwell's equations

One way of improving the behavior of finite element schemes for classical, time-dependent Maxwell's equations, is to render them from their hyperbolic character to elliptic form. This paper is devoted to the study of the stabilized linear finite element method for the time harmonic Maxwell's equations in a dual form obtained through the Laplace transformation in time. The model problem is for the particular case of the dielectric permittivity function which is assumed to be constant in a boundary neighborhood. For the stabilized model a coercivity relation is derived that guarantee's the existence of a unique solution for the iscrete problem. The convergence is addressed both in a priori and a posteriori settings. In the a priori error estimates we confirm the theoretical convergence of the scheme in a L2-based, gradient dependent, triple norm. The order of convergence is O(h) in weighted Sobolev space and hence optimal. We also derive, similar, optimal a posteriori error estimates controlled by a certain, weighted, norm of the residual of the computed solution. The posteriori approach is used for constructing adaptive algorithms for the computational purposes. Further, assuming a sufficiently regular solution for the dual problem, we reach the same convergence of O(h). Finally, through implementing several numerical examples, we validate the robustness of the proposed scheme.

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arXiv2019

Numerical validation of an explicit P1 finite-element scheme for Maxwell's equations in a polygon with variable permittivity away from its boundary

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.

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arXiv2017

Lipschitz stability for an inverse hyperbolic problem of determining two coefficients by a finite number of observations

We consider an inverse problem of reconstructing two spatially varying coefficients in an acoustic equation of hyperbolic type using interior data of solutions with suitable choices of initial condition. Using a Carleman estimate, we prove Lipschitz stability estimates which ensures unique reconstruction of both coefficients. Our theoretical results are justified by numerical studies on the reconstruction of two unknown coefficients using noisy backscattered data.

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arXiv2016

Uniqueness and stability of time and space-dependent conductivity in a hyperbolic cylindrical domain

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 H\"older 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.

math.AP↗

arXiv2016

A priori error estimates and computational studies for a Fermi pencil-beam equation

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 $\tilde {\mathbf v}=(\mu, \eta, \xi)$ 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,\eta, \xi)$. Hence the Fermi equation, stated in three dimensional spatial domain ${\mathbf x}=(x,y,z)$, depends only on two velocity variables ${\mathbf v}=(\eta, \xi)$. 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.

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arXiv2016

Computational design of nanophotonic structures using an adaptive finite element method

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.

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arXiv2016

Application of the Finite Element Method in a Quantitative Imaging technique

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.

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arXiv2015

Reconstruction of dielectric constants of multi-layered optical fibers using propagation constants measurements

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.

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arXiv2015

Domain decomposition finite element/finite difference method for the conductivity reconstruction in a hyperbolic equation

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.

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