Avetik Arakelyan's Publications (Full profile)

In this work, we address graph based semi-supervised learning using the theory of the spatial segregation of competitive systems. First, we define a discrete counterpart over connected graphs by using direct analogue of the corresponding competitive system. This model turns out does not have a unique solution as we expected. Nevertheless, we suggest gradient projected and regularization methods to reach some of the solutions. Then we focus on a slightly different model motivated from the recent numerical results on the spatial segregation of reaction–diffusion systems. In this case we show that the model has a unique solution and propose a novel classification algorithm based on it. Finally, we present numerical experiments showing the method is efficient and comparable to other semi-supervised learning algorithms at high and low label rates.

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 convergence of the proposed algorithm is proved. Moreover, we consider the finite difference scheme for this algorithm and obtain its convergence. At the end of the paper we present and discuss computational results.

We prove a uniqueness for the positive solution to a strongly competing system of Lotka-Volterra type problem in the limiting configuration, when the competition rate tends to infinity. We give an alternate proof of uniqueness based on properties of limiting solutions

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 the corresponding finite difference scheme. Based on that scheme, we suggest an iterative algorithm and show its consistency and stability. For the special case $m=2$, we show that the problem gives rise to the generalized version of the so-called two-phase obstacle problem. In this particular case we introduce the notion of viscosity solutions and prove convergence of the difference scheme to the unique viscosity solution. At the end of the paper we present computational tests, for different internal dynamics, and discuss numerical results.