114 research outputs found
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
Rate of convergence for periodic homogenization of convex Hamilton-Jacobi equations in one dimension
Let and be viscosity solutions of the oscillatory
Hamilton-Jacobi equation and its corresponding effective equation. Given
bounded, Lipschitz initial data, we present a simple proof to obtain the
optimal rate of convergence of as for a large class of convex
Hamiltonians in one dimension. This class includes the Hamiltonians
from classical mechanics with separable potential. The proof makes use of
optimal control theory and a quantitative version of the ergodic theorem for
periodic functions in dimension .Comment: 22 pages, typos corrected, references added, final accepted versio
A junction condition by specified homogenization and application to traffic lights
Given a coercive Hamiltonian which is quasi-convex with respect to the
gradient variable and periodic with respect to time and space at least "far
away from the origin", we consider the solution of the Cauchy problem of the
corresponding Hamilton-Jacobi equation posed on the real line. Compact
perturbations of coercive periodic quasi-convex Hamiltonians enter into this
framework for example. We prove that the rescaled solution converges towards
the solution of the expected effective Hamilton-Jacobi equation, but whose
"flux" at the origin is "limited" in a sense made precise by the authors in
\cite{im}. In other words, the homogenization of such a Hamilton-Jacobi
equation yields to supplement the expected homogenized Hamilton-Jacobi equation
with a junction condition at the single discontinuous point of the effective
Hamiltonian. We also illustrate possible applications of such a result by
deriving, for a traffic flow problem, the effective flux limiter generated by
the presence of a finite number of traffic lights on an ideal road. We also
provide meaningful qualitative properties of the effective limiter.Comment: 41 page
Aubry sets for weakly coupled systems of Hamilton--Jacobi equations
We introduce a notion of Aubry set for weakly coupled systems of
Hamilton--Jacobi equations on the torus and characterize it as the region where
the obstruction to the existence of globally strict critical subsolutions
concentrates. As in the case of a single equation, we prove the existence of
critical subsolutions which are strict and smooth outside the Aubry set. This
allows us to derive in a simple way a comparison result among critical sub and
supersolutions with respect to their boundary data on the Aubry set, showing in
particular that the latter is a uniqueness set for the critical system. We also
highlight some rigidity phenomena taking place on the Aubry set.Comment: 35 pages v.2 the introduction has been rewritten and shortened. Some
proofs simplified. Corrections and references added. Corollary 5.3 added
stating antisymmetry of the Ma\~n\'e matrix on points of the Aubry set.
Section 6 contains a new example
Error estimates and convergence rates for the stochastic homogenization of Hamilton-Jacobi equations
We present exponential error estimates and demonstrate an algebraic
convergence rate for the homogenization of level-set convex Hamilton-Jacobi
equations in i.i.d. random environments, the first quantitative homogenization
results for these equations in the stochastic setting. By taking advantage of a
connection between the metric approach to homogenization and the theory of
first-passage percolation, we obtain estimates on the fluctuations of the
solutions to the approximate cell problem in the ballistic regime (away from
flat spot of the effective Hamiltonian). In the sub-ballistic regime (on the
flat spot), we show that the fluctuations are governed by an entirely different
mechanism and the homogenization may proceed, without further assumptions, at
an arbitrarily slow rate. We identify a necessary and sufficient condition on
the law of the Hamiltonian for an algebraic rate of convergence to hold in the
sub-ballistic regime and show, under this hypothesis, that the two rates may be
merged to yield comprehensive error estimates and an algebraic rate of
convergence for homogenization.
Our methods are novel and quite different from the techniques employed in the
periodic setting, although we benefit from previous works in both first-passage
percolation and homogenization. The link between the rate of homogenization and
the flat spot of the effective Hamiltonian, which is related to the
nonexistence of correctors, is a purely random phenomenon observed here for the
first time.Comment: 57 pages. Revised version. To appear in J. Amer. Math. So
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