4,396 research outputs found
Conservation laws arising in the study of forward-forward Mean-Field Games
We consider forward-forward Mean Field Game (MFG) models that arise in
numerical approximations of stationary MFGs. First, we establish a link between
these models and a class of hyperbolic conservation laws as well as certain
nonlinear wave equations. Second, we investigate existence and long-time
behavior of solutions for such models
The Hessian Riemannian flow and Newton's method for Effective Hamiltonians and Mather measures
Effective Hamiltonians arise in several problems, including homogenization of
Hamilton--Jacobi equations, nonlinear control systems, Hamiltonian dynamics,
and Aubry--Mather theory. In Aubry--Mather theory, related objects, Mather
measures, are also of great importance. Here, we combine ideas from mean-field
games with the Hessian Riemannian flow to compute effective Hamiltonians and
Mather measures simultaneously. We prove the convergence of the Hessian
Riemannian flow in the continuous setting. For the discrete case, we give both
the existence and the convergence of the Hessian Riemannian flow. In addition,
we explore a variant of Newton's method that greatly improves the performance
of the Hessian Riemannian flow. In our numerical experiments, we see that our
algorithms preserve the non-negativity of Mather measures and are more stable
than {related} methods in problems that are close to singular. Furthermore, our
method also provides a way to approximate stationary MFGs.Comment: 24 page
Exponential decay of correlation for the Stochastic Process associated to the Entropy Penalized Method
In this paper we present an upper bound for the decay of correlation for the
stationary stochastic process associated with the Entropy Penalized Method. Let
L(x, v):\Tt^n\times\Rr^n\to \Rr be a Lagrangian of the form
L(x,v) = {1/2}|v|^2 - U(x) + .
For each value of and , consider the operator
\Gg[\phi](x):= -\epsilon h {ln}[\int_{\re^N} e
^{-\frac{hL(x,v)+\phi(x+hv)}{\epsilon h}}dv], as well as the reversed operator
\bar \Gg[\phi](x):= -\epsilon h {ln}[\int_{\re^N}
e^{-\frac{hL(x+hv,-v)+\phi(x+hv)}{\epsilon h}}dv], both acting on continuous
functions \phi:\Tt^n\to \Rr. Denote by the solution of
\Gg[\phi_{\epsilon,h}]=\phi_{\epsilon,h}+\lambda_{\epsilon,h}, and by the solution of \bar \Gg[\phi_{\epsilon,h}]=\bar
\phi_{\epsilon,h}+\lambda_{\epsilon,h}. In order to analyze the decay of
correlation for this process we show that the operator has a maximal
eigenvalue isolated from the rest of the spectrum
Chitosan-Hyaluronic acid hybrid vectors for retinal gene therapy
info:eu-repo/semantics/publishedVersio
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