Rafayel Barkhudaryan

Yerevan State University | YSU

In this paper we treat a non-local free boundary problem arising in financial bubbles, where the model is set in the framework of viscosity solutions. We suggest an iterative scheme which consists of a sequence of obstacle problems at each step to be solved, that in turn gives the next obstacle function in the iteration scheme. The suggested approa...

The current paper considers the problem of recovering a function using a limited number of its Fourier coefficients. Specifically, a method based on Bernoulli-like polynomials suggested and developed by Krylov, Lanczos, Gottlieb and Eckhoff is examined. Asymptotic behavior of approximate calculation of the so-called “jumps” is studied and asymptoti...

We propose an algorithm to solve the \textit{two-phase obstacle problem} by finite difference method. We obtain an error estimate for finite difference approximation and prove the convergence of proposed algorithm.

In the current work we consider the numerical solutions of equations of stationary states for a general class of the spatial segregation of reaction-diffusion systems with $m\geq 2$ population densities. We introduce a discrete multi-phase minimization problem related to the segregation problem, which allows to prove the existence and uniqueness of...

Fourier expansions employing polyharmonic–Neumann eigenfunctions have demonstrated improved convergence over those using the classical trigonometric system, due to the rapid decay of their Fourier coefficients. Building on this insight, we investigate interpolations on a finite interval that are exact for polyharmonic–Neumann eigenfunctions and exh...

Fourier expansions by the polyharmonic-Neumann eigenfunctions showed improved convergence compared to the Fourier expansions by the classical trigonometric system due to the rapid decay of the corresponding Fourier coefficients. Based on this evidence, we investigate interpolations on a finite interval that are exact for the polyharmonic-Neumann ei...

The present work is focused on exploring convergence of Physics-informed Neural Networks (PINNs) when applied to a specific class of second-order fully nonlinear Partial Differential Equations (PDEs). It is well-known that as the number of data grows, PINNs generate a sequence of minimizers which correspond to a sequence of neural networks. We show...

We consider interconnected obstacle problems and develop a numerical approximation scheme for them. These problems are given as (weakly coupled) systems of variational inequalities and model optimal decision or switching under uncertainty. For example, these problems can be regarded as a system-version of the American-type option problem with multi...

We investigate the convergence of the quasi-periodic approximations in different frameworks and reveal exact asymptotic estimates of the corresponding errors. The estimates facilitate a fair comparison of the quasi-periodic approximations to other classical well-known approaches. We consider a special realization of the approximations by the invers...

Trigonometric approximation or interpolation of a non-smooth function on a finite interval has poor convergence properties. This is especially true for discontinuous functions. The case of infinitely differentiable but non-periodic functions with discontinuous periodic extensions onto the real axis has attracted interest from many researchers. In a...

Our objective with this paper is to discuss multi-switching problems, arising as variational inequalities, that models decision under uncertainty. We prove general existence theory through monotone scheme and discuss iterative methods for numerical results. We also connect the recently developed models for asset bubbles (which is a non-local proble...

In this paper we continue to study a non-local free boundary problem arising in financial bubbles. We focus on the parabolic counterpart of the bubble problem and suggest an iterative algorithm which consists of a sequence of parabolic obstacle problems at each step to be solved, that in turn gives the next obstacle function in the iteration. The c...

In this talk, we study a non-local free boundary problem arising in financial bubbles. The model equation, studied here, is the following free boundary problem formulated as a Hamilton-Jacobi equation: \begin{equation}\label{main} \min(Lu,u(t,x)-u(t,-x)-\psi(t,x))=0,\quad (t,x)\in\mathbb{R^+}\times\Omega, \end{equation} where $\Omega\subset \mathbb...

This talk is devoted to the general class of finite difference schemes developed for a numerical approximation of solutions to a certain type of reaction-diffusion systems with m population densities. Let Ω ⊂ R 2 be a connected and bounded domain with smooth boundary and m be a fixed integer. We consider the steady-states of m competing species coe...

In the present work we deal with the numerical approximation of equations of stationary states for a general class of the spatial segregation of Reaction-diffusion system with two population densities having disjoint supports. We show that the problem gives rise the generalized version of the so-called two-phase obstacle problem and introduce the n...

Stochastic and Analytic Methods in Mathematical Physics

Presentation made at the Sixth International Scientific Conference ”Modern Methods, Problems and Applications of Operator Theory and Harmonic Analysis VI”.

We propose an algorithm to solve the two-phase obstacle problem by finite difference method. We prove the existence and uniqueness of the solution of the discrete nonlinear system and obtain an error estimate for the corresponding regularization. Also we prove the convergence of the proposed numerical algorithm. At the end of the paper we present s...

In this paper we consider the numerical approximation of the two-phase membrane (obstacle) problem by finite difference method. First, we introduce the notion of viscosity solution for the problem and construct certain discrete nonlinear approximation system. The existence and uniqueness of the solution of the discrete nonlinear system is proved. B...

We propose an algorithm to solve the two-phase obstacle problem by finite difference method. We prove the existence and uniqueness of the solution of the discrete nonlinear system and obtain an error estimate for finite difference approximation.

The paper considers expansions by eigenfunctions of the boundary problem for Dirac system. The Gibbs phenomenon for such expansions is revealed.

In this paper we consider the finite difference scheme approximation for one-phase obstacle problem and obtain an error estimate for this approximation. KeywordsFree boundary problem–obstacle problem–finite difference method

This work deals with the expansion by eigenfunctions of Dirac system. For accelerating the convergence of this kind of expansions a nonlinear method is suggested, based on Padé approximants. Asymptotic estimates of L 2 errors are derived. Some numerical examples are presented.

In this article, we investigate the method of Fourier series acceleration for smooth functions on [-1,1]. The main idea is application of Pade approximants to an asymptotic expansion of Fourier coefficients. This approach is rather efficient and, as was shown before, actually corrects the Gibbs phenomenon by means of quasi-polynomials. The efficien...

In this paper we consider and investigate a parallel algorithm for numerical solution of the second kind Fredholm integral equation with prescribed accuracy. Numerical experiments are carried down on ArmCluster. The results of numerical experiments are presented and discussed.