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Yuri Ashrafyan

Publications and source records attributed to Yuri Ashrafyan.

15 recordsLinked to original sources
arXiv2026

Bregman-projected mirror methods for regularized stationary mean-field games

We develop and analyze a Bregman-projected mirror iteration for low-order regularizations of stationary mean-field game (MFG) systems in their natural Banach space setting. For separable Hamiltonians of the form \(H(x,p,m)=H_0(x,p)-g(m)\), with quadratic or super-quadratic Hamiltonian growth and linear or super-linear density couplings, we formulate a low-order \(\bar\gamma\)-Laplacian regularization of the stationary MFG system as a variational inequality on \(L^{\bar\beta}(\mathbb T^d)\times W^{1,\bar\gamma}(\mathbb T^d)\). To approximate solutions of this regularized variational inequality, we introduce a Bregman geometry matched to the mixed Lebesgue--Sobolev exponents of the problem and analyze a constrained two-step mirror method with frozen operator evaluation. For the exact constrained iteration and each fixed regularization parameter \(\epsi>0\), we derive a one-step Bregman inequality and use it to prove that the constrained iteration converges strongly to the unique solution of the regularized variational inequality under natural summability conditions on the step sizes. Numerical experiments on one- and two-dimensional models, validated against exact test solutions, illustrate residual decay under mesh refinement and suggest improved practical performance of the two-step implementation in the tested discretizations.

math.NA↗
arXiv2026

A First-Order Mean-Field Game on a Bounded Domain with Mixed Boundary Conditions

Entry-exit dynamics are crucial in modeling crowd movement. Here, we present a novel first-order, stationary mean-field game model on a bounded domain that accurately captures these dynamics. The interior dynamics of the system are governed by a standard first-order stationary MFG system consisting of a Hamilton-Jacobi equation coupled with a transport equation. The model incorporates nonstandard mixed boundary conditions corresponding to an entry region $Γ_N$, where a Neumann condition prescribes agent inflow, and an exit region $Γ_D$, where a no-entry condition prevents inward flow. Additionally, we impose an upper bound on the exit cost through $Γ_D$, combined with a complementary contact-set condition. The contact-set condition distinguishes boundary contact points, where the exit cost is attained and exit is permitted, from non-contact points, where a strict no-penetration condition is enforced. This mixed approach overcomes the limitations of classical Dirichlet conditions, which can artificially force boundary points to serve as both entry and exit locations. We analyze the system through a variational formulation, applying the direct method of the calculus of variations to establish the existence of solutions under minimal regularity assumptions. Furthermore, we prove a partial uniqueness result for the gradient of the value function (particularly in regions with positive agent density) and establish the uniqueness of the density function. Several examples, including one- and two-dimensional cases, illustrate the proper assignment of entry and exit roles and demonstrate that contact does not necessarily enforce exit. Additionally, they showcase first-order MFG phenomena, such as the formation of empty regions, where agent density vanishes. These results provide a rigorous mathematical foundation for modeling realistic entry-exit scenarios.

math.AP↗
arXiv2025

A Fully-discrete Semi-Lagrangian scheme for a price formation MFG model

Here, we examine a fully-discrete Semi-Lagrangian scheme for a mean-field game price formation model. We show the existence of the solution of the discretized problem and that it is monotone as a multivalued operator. Moreover, we show that the limit of the discretization converges to the weak solution of the continuous price formation mean-field game using monotonicity methods. Numerical simulations demonstrate that this scheme can provide results efficiently, comparing favorably with other methods in the examples we tested.

math.NA↗
arXiv2024

Spectral Theory of Dirac Operators

The main issues of the spectral theory of Dirac operators are presented, namely: transformation operators, asymptotics of eigenvalues and eigenfunctions, description of symmetric and self-adjoint operators in Hilbert space, expansion in eigenfunctions, uniqueness theorems in inverse problems, constructive solution of inverse problems, description of isospectral operators, and some other questions. This book is aimed at specialists in spectral theory and graduate students of mathematics at universities.

math.SP↗
arXiv2022

A Variational Approach For Price Formation Models In One Dimension

In this paper, we study a class of first-order mean-field games (MFGs) that model price formation. Using Poincar{é} Lemma, we eliminate one of the equations and obtain a variational problem for a single function. This variational problem offers an alternative approach for the numerical solution of the original MFGs system. We show a correspondence between solutions of the MFGs system and the variational problem. Moreover, we address the existence of solutions for the variational problem using the direct method in the calculus of variations. We end the paper with numerical results for a linear-quadratic model.

math.AP↗
arXiv2022

The Potential Method For Price-Formation Models

We consider the mean-field game price formation model introduced by Gomes and Saúde. In this MFG model, agents trade a commodity whose supply can be deterministic or stochastic. Agents maximize profit, taking into account current and future prices. The balance between supply and demand determines the price. We introduce a potential function that converts the MFG into a convex variational problem. This variational formulation is particularly suitable for machine learning approaches. Here, we use a recurrent neural network to solve this problem. In the last section of the paper, we compare our results with known analytical solutions.

math.NA↗
arXiv2021

A duality approach to a price formation MFG model

We study the connection between the Aubry-Mather theory and a mean-field game (MFG) price-formation model. We introduce a framework for Mather measures that is suited for constrained time-dependent problems in R. Then, we propose a variational problem on a space of measures, from which we obtain a duality relation involving the MFG problem examined in [36].

math.AP↗
arXiv2020

On Ambarzumyan-type Inverse Problems of Vibrating String Equations

We consider the inverse spectral theory of vibrating string equations. In this regard, first eigenvalue Ambarzumyan-type uniqueness theorems are stated and proved subject to separated, self-adjoint boundary conditions. More precisely, it is shown that there is a curve in the boundary parameters' domain on which no analog of it is possible. Necessary conditions of the $n$-th eigenvalue are identified, which allows to state the theorems. In addition, several properties of the first eigenvalue are examined. Lower and upper bounds are identified, and the areas are described in the boundary parameters' domain on which the sign of the first eigenvalue remains unchanged. This paper contributes to inverse spectral theory as well as to direct spectral theory.

math.SP↗
arXiv2018

Inverse Sturm-Liouville problems with summable potential

We describe the necessary and sufficient conditions for two sequences {μ_n}^\infty_n=0 and {a_n}^\infty_n=0 to be correspondingly the set of eigenvalues and the set of norming constants of a Sturm-Liouville problem with real summable potential q and in advance fixed separated boundary conditions.

math.SP↗
arXiv2017

Gradient of eigenvalues of Dirac operators and its applications

For Dirac operators, which have discrete spectra, the concept of eigenvalues gradient is given and formulae for this gradients are obtained in terms of normalized eigenfunctions. It is shown how the gradient is being used to describe isospectral operators or when finite number of spectral data is changed.

math.CA↗
arXiv2016

Inverse Sturm-Liouville problems with fixed boundary conditions

Necessary and sufficient conditions for two sequences $\{μ_n\}_{n=0}^\infty$ and $\{ a_n\}_{n=0}^\infty$ to be the spectral data for a certain Sturm-Liouville problem are well known. We add two more conditions so that the same two sequences become necessary and sufficient for being the spectral data for a Sturm-Liouville problem with fixed boundary conditions.

math.SP↗