arXiv subjects

Henok Mawi

Publications and source records attributed to Henok Mawi.

5 recordsLinked to original sources

arXiv2020

Continuous time approximation of Nash equilibria

We consider the problem of approximating Nash equilibria of $N$ functions $f_1,\dots, f_N$ of $N$ variables. In particular, we deduce conditions under which systems of the form $$ \dot u_j(t)=-\nabla_{x_j}f_j(u(t)) $$ $(j=1,\dots, N)$ are well posed and in which the large time limits of their solutions $u(t)=(u_1(t),\dots, u_N(t))$ are Nash equilibria for $f_1,\dots, f_N$. To this end, we will invoke the theory of maximal monotone operators. We will also identify an application of these ideas in game theory and show how to approximate equilibria in some game theoretic problems in function spaces.

math.AP↗

arXiv2019

On the Numerical Solution of the Near Field Refractor Problem

A numerical scheme is presented to solve the one source near field refractor problem to arbitrary precision and it is proved that the scheme terminates in a finite number of iterations. The convergence of the algorithm depends upon proving appropriate Lipschitz estimates for the refractor measure. The algorithm is presented in general terms and has independent interest.

math.NA↗

arXiv2018

Refractors in anisotropic media associated with norms

We show existence of interfaces between two anisotropic materials so that light is refracted in accordance with a given pattern of energy. To do this we formulate a vector Snell law for anisotropic media when the wave fronts are given by norms for which the corresponding unit spheres are strictly convex.

math.AP↗

arXiv2016

On the Numerical Solution of the Far Field Refractor Problem

The far field refractor problem with a discrete target is solved with a numerical scheme that uses and simplify ideas from Caffarelli, Kochengin and Oliker. A numerical implementation is carried out and examples are shown.

math.NA↗

arXiv2015

On Hamilton-Jacobi-Bellman equations with convex gradient constraints

We study PDE of the form $\max\{F(D^2u,x)-f(x), H(Du)\}=0$ where $F$ is uniformly elliptic and convex in its first argument, $H$ is convex, $f$ is a given function and $u$ is the unknown. These equations are derived from dynamic programming in a wide class of stochastic singular control problems. In particular, examples of these equations arise in mathematical finance models involving transaction costs, in queuing theory, and spacecraft control problems. The main aspects of this work are to identify conditions under which solutions are uniquely defined and have Lipschitz continuous gradients. We also generalize previous results known for the case where $M\mapsto F(M,x)$ is the maximum of finitely many linear functions.

math.AP↗