Showing papers by "Hung V. Tran published in 2011"
TL;DR: In this paper, the speed of convergence of the Hamilton-Jacobi equations was shown to be polynomial in the time complexity of the solutions of the regularized equations of vanishing viscosity type.
Abstract: We use the adjoint methods to study the static Hamilton–Jacobi equations and to prove the speed of convergence for those equations. The main new ideas are to introduce adjoint equations corresponding to the formal linearizations of regularized equations of vanishing viscosity type, and from the solutions σe of those we can get the properties of the solutions u of the Hamilton–Jacobi equations. We classify the static equations into two types and present two new ways to deal with each type. The methods can be applied to various static problems and point out the new ways to look at those PDE.
35 citations
TL;DR: In this article, a general construction of probability measures, which in the convex setting agree with Mather measures, is provided, which is used to construct analogues to the Aubry-Mather measures for nonconvex Hamiltonians.
Abstract: The adjoint method, introduced in [L. C. Evans, Arch. Ration. Mech. Anal., 197 (2010), pp. 1053–1088] and [H. V. Tran, Calc. Var. Partial Differential Equations, 41 (2011), pp. 301–319], is used to construct analogues to the Aubry–Mather measures for nonconvex Hamiltonians. More precisely, a general construction of probability measures, which in the convex setting agree with Mather measures, is provided. These measures may fail to be invariant under the Hamiltonian flow and a dissipation arises, which is described by a positive semidefinite matrix of Borel measures. However, in the case of uniformly quasiconvex Hamiltonians the dissipation vanishes, and as a consequence the invariance is guaranteed.
27 citations
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TL;DR: A convergence result to asymptotic solutions as time goes to infinity as well as rather restricted assumptions are established on the n-dimensional torus.
Abstract: We investigate the large-time behavior of viscosity solutions of quasi-monotone weakly coupled systems of Hamilton--Jacobi equations on the $n$-dimensional torus. We establish a convergence result to asymptotic solutions as time goes to infinity under rather restricted assumptions.
22 citations
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TL;DR: In this paper, a numerical scheme for the one dimensional time dependent Hamilton-Jacobi equation in the periodic setting was proposed and a new and simple proof of the rate of convergence of the approximations based on the adjoint method was presented.
Abstract: We consider a numerical scheme for the one dimensional time dependent Hamilton-Jacobi equation in the periodic setting We present a new and simple proof of the rate of convergence of the approximations based on the adjoint method recently introduced by Evans
1 citations