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Pierre Cardaliaguet - One of the best experts on this subject based on the ideXlab platform.
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HÖLDER REGULARITY OF Hamilton-Jacobi Equations WITH STOCHASTIC FORCING
2020Co-Authors: Pierre Cardaliaguet, Benjamin SeegerAbstract:We obtain space-time Hölder regularity estimates for solutions of first-and second-order Hamilton-Jacobi Equations perturbed with an additive stochastic forcing term. The bounds depend only on the growth of the Hamiltonian in the gradient and on the regularity of the stochastic coefficients, in a way that is invariant with respect to a hyperbolic scaling.
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Quantitative stochastic homogenization of viscous Hamilton-Jacobi Equations
Communications in Partial Differential Equations, 2015Co-Authors: Scott N. Armstrong, Pierre CardaliaguetAbstract:We prove explicit estimates for the error in random homogenization of degenerate, second-order Hamilton-Jacobi Equations, assuming the coefficients satisfy a finite range of dependence. In particular, we obtain an algebraic rate of convergence with overwhelming probability under certain structural conditions on the Hamiltonian.
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Hölder regularity for viscosity solutions of fully nonlinear, local or nonlocal, Hamilton-Jacobi Equations with super-quadratic growth in the gradient
SIAM Journal on Control and Optimization, 2011Co-Authors: Pierre Cardaliaguet, Catherine RainerAbstract:Viscosity solutions of fully nonlinear, local or non local, Hamilton-Jacobi Equations with a super-quadratic growth in the gradient variable are proved to be Hölder continuous, with a modulus depending only on the growth of the Hamiltonian. The proof involves some representation formula for nonlocal Hamilton-Jacobi Equations in terms of controlled jump processes and a weak reverse inequality.
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Hölder estimates in space-time for viscosity solutions of Hamilton-Jacobi Equations
Communications on Pure and Applied Mathematics, 2010Co-Authors: Piermarco Cannarsa, Pierre CardaliaguetAbstract:It is well-known that solutions to the basic problem in the calculus of variations may fail to be Lipschitz continuous when the Lagrangian depends on t. Similarly, for viscosity solutions to time-dependent Hamilton-Jacobi Equations one cannot expect Lipschitz bounds to hold uniformly with respect to the regularity of coefficients. This phenomenon raises the question whether such solutions satisfy uniform estimates in some weaker norm. We will show that this is the case for a suitable Hölder norm, obtaining uniform estimates in (x,t) for solutions to first and second order Hamilton-Jacobi Equations. Our results apply to degenerate parabolic Equations and require superlinear growth at infinity, in the gradient variables, of the Hamiltonian. Proofs are based on comparison arguments and representation formulas for viscosity solutions, as well as weak reverse Hölder inequalities.
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H\"older regularity for viscosity solutions of fully nonlinear, local or nonlocal, Hamilton-Jacobi Equations with super-quadratic growth in the gradient
arXiv: Optimization and Control, 2010Co-Authors: Pierre Cardaliaguet, Catherine RainerAbstract:Viscosity solutions of fully nonlinear, local or non local, Hamilton-Jacobi Equations with a super-quadratic growth in the gradient variable are proved to be H\"older continuous, with a modulus depending only on the growth of the Hamiltonian. The proof involves some representation formula for nonlocal Hamilton-Jacobi Equations in terms of controlled jump processes and a weak reverse inequality.
Hitoshi Ishii - One of the best experts on this subject based on the ideXlab platform.
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An example in the vanishing discount problem for monotone systems of Hamilton-Jacobi Equations
arXiv: Analysis of PDEs, 2020Co-Authors: Hitoshi IshiiAbstract:In recent years, there have been many contributions to the vanishing discount problem for Hamilton-Jacobi Equations. In the case of the scalar equation, B. Ziliotto [Convergence of the solutions of the discounted Hamilton-Jacobi equation: a counterexample. J. Math. Pures Appl. (9) 128 (2019), 330-338] has shown an example of the Hamilton-Jacobi equation having non-convex Hamiltonian in the gradient variable, for which the full convergence of the solutions does not hold as the discount factor tends to zero. We give an example of the nonlinear monotone system of Hamilton-Jacobi Equations having convex Hamiltonians in the gradient variable, for which the whole family convergence of the solutions does not hold.
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Averaging of Hamilton-Jacobi Equations over Hamiltonian flows
arXiv: Analysis of PDEs, 2019Co-Authors: Hitoshi Ishii, Taiga KumagaiAbstract:We study the asymptotic behavior of solutions to the Dirichlet problem for Hamilton-Jacobi Equations with large drift terms, where the drift terms are given by the Hamiltonian vector fields of Hamiltonian $H$. This is an attempt to understand the averaging effect for fully nonlinear degenerate elliptic Equations. In this work, we restrict ourselves to the case of Hamilton-Jacobi Equations. The second author has already established averaging results for Hamilton-Jacobi Equations with convex Hamiltonians ($G$ below) under the classical formulation of the Dirichlet condition. Here we treat the Dirichlet condition in the viscosity sense, and establish an averaging result for Hamilton-Jacobi Equations with relatively general Hamiltonian $G$.
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Averaging of Hamilton-Jacobi Equations along divergence-free vector fields
Discrete & Continuous Dynamical Systems - A, 2019Co-Authors: Hitoshi Ishii, Taiga KumagaiAbstract:We study the asymptotic behavior of solutions to the Dirichlet problem for Hamilton-Jacobi Equations with large drift terms, where the drift terms are given by divergence-free vector fields. This is an attempt to understand the averaging effect for fully nonlinear degenerate elliptic Equations. In this work, we restrict ourselves to the case of Hamilton-Jacobi Equations. The second author has already established averaging results for Hamilton-Jacobi Equations with convex Hamiltonians ( \begin{document}$ G $\end{document} below) under the classical formulation of the Dirichlet condition. Here we treat the Dirichlet condition in the viscosity sense and establish an averaging result for Hamilton-Jacobi Equations with relatively general Hamiltonian \begin{document}$ G $\end{document} .
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Relaxation of Hamilton-Jacobi Equations
Archive for Rational Mechanics and Analysis, 2003Co-Authors: Hitoshi Ishii, Paola LoretiAbstract:We study the relaxation of Hamilton-Jacobi Equations. The relaxation in our terminology is the following phenomenon: the pointwise supremum over a certain collection of subsolutions, in the almost everywhere sense, of a Hamilton-Jacobi equation yields a viscosity solution of the ``convexified'' Hamilton-Jacobi equation. This phenomenon has recently been observed in [13] in eikonal Equations. We show in this paper that this relaxation is a common phenomenon for a wide range of Hamilton-Jacobi Equations.
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Homogenization of the Cauchy Problem for Hamilton-Jacobi Equations
Stochastic Analysis Control Optimization and Applications, 1999Co-Authors: Hitoshi IshiiAbstract:We study the asymptotic behavior of solutions of the Cauchy problem for Hamilton-Jacobi Equations with periodic coefficients as the frequency of periodicity tends to infinity. The limit functions are characterized as unique solutions of Hamilton-Jacobi Equations with the Hamiltonians determined by the corresponding cell problems. Our result applies to the case where the initial data oscillate periodically and so does the Hamiltonian both in the spatial and time variables.
Hung V. Tran - One of the best experts on this subject based on the ideXlab platform.
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state constraint static hamilton jacobi Equations in nested domains
Siam Journal on Mathematical Analysis, 2020Co-Authors: Yeoneung Kim, Hung V. TranAbstract:We study state-constraint static Hamilton--Jacobi Equations in a sequence of domains $\{\Omega_k\}_{k \in \Bbb{N}}$ in $\Bbb{R}^n$ such that $\Omega_k \subset \Omega_{k+1}$ for all $k\in \Bbb{N}$. ...
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state constraint static hamilton jacobi Equations in nested domains
arXiv: Analysis of PDEs, 2019Co-Authors: Yeoneung Kim, Hung V. TranAbstract:We study state-constraint static Hamilton-Jacobi Equations in a sequence of domains $\{\Omega_k\}_{k \in \mathbb{N}}$ in $\mathbb{R}^n$ such that $\Omega_k \subset \Omega_{k+1}$ for all $k\in \mathbb{N}$. We obtain rates of convergence of $u_k$, the solution to the state-constraint problem in $\Omega_k$, to $u$, the solution to the corresponding problem in $\Omega = \bigcup_{k \in \mathbb{N}} \Omega_k$. In many cases, the rates obtained are proven to be optimal. Various new examples and discussions are provided at the end of the paper.
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Remarks on viscous Hamilton-Jacobi Equations
arXiv: Analysis of PDEs, 2013Co-Authors: Scott N. Armstrong, Hung V. TranAbstract:We present comparison principles, Lipschitz estimates and study state constraints problems for degenerate, second-order Hamilton-Jacobi Equations.
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Adjoint methods for static Hamilton–Jacobi Equations
Calculus of Variations and Partial Differential Equations, 2011Co-Authors: Hung V. TranAbstract: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.
Roberto Ferretti - One of the best experts on this subject based on the ideXlab platform.
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Value iteration convergence of ε-monotone schemes for stationary Hamilton-Jacobi Equations
Discrete and Continuous Dynamical Systems, 2015Co-Authors: Olivier Bokanowski, Maurizio Falcone, Roberto Ferretti, Lars Grüne, Dante Kalise, Hasnaa ZidaniAbstract:We present an abstract convergence result for the xed point approximation of stationary Hamilton{Jacobi Equations. The basic assumptions on the discrete operator are invariance with respect to the addition of constants, "-monotonicity and consistency. The result can be applied to various high-order approximation schemes which are illustrated in the paper. Several applications to Hamilton{Jacobi Equations and numerical tests are presented.
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Value iteration convergence of "-monotone schemes for stationary Hamilton-Jacobi Equations
Discrete and Continuous Dynamical Systems - Series A, 2015Co-Authors: Olivier Bokanowski, Maurizio Falcone, Roberto Ferretti, Lars Grüne, Dante Kalise, Hasnaa ZidaniAbstract:We present an abstract convergence result for the xed point approximation of stationary Hamilton{Jacobi Equations. The basic assumptions on the discrete operator are invariance with respect to the addition of constants, "-monotonicity and consistency. The result can be applied to various high-order approximation schemes which are illustrated in the paper. Several applications to Hamilton{Jacobi Equations and numerical tests are presented.
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semi lagrangian approximation schemes for linear and hamilton jacobi Equations
2014Co-Authors: Maurizio Falcone, Roberto FerrettiAbstract:This largely self-contained book provides a unified framework of semi-Lagrangian strategy for the approximation of hyperbolic PDEs, with a special focus on Hamilton-Jacobi Equations. The authors provide a rigorous discussion of the theory of viscosity solutions and the concepts underlying the construction and analysis of difference schemes; they then proceed to high-order semi-Lagrangian schemes and their applications to problems in fluid dynamics, front propagation, optimal control, and image processing. The developments covered in the text and the references come from a wide range of literature.
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semi lagrangian schemes for hamilton jacobi Equations discrete representation formulae and godunov methods
Journal of Computational Physics, 2002Co-Authors: Maurizio Falcone, Roberto FerrettiAbstract:We study a class of semi-Lagrangian schemes which can be interpreted as a discrete version of the Hopf-Lax-Oleinik representation formula for the exact viscosity solution of first order evolutive Hamilton-Jacobi Equations. That interpretation shows that the scheme is potentially accurate to any prescribed order. We discuss how the method can be implemented for convex and coercive Hamiltonians with a particular structure and how this method can be coupled with a discrete Legendre trasform. We also show that in one dimension, the first-order semi-Lagrangian scheme coincides with the integration of the Godunov scheme for the corresponding conservation laws. Several test illustrate the main features of semi-Lagrangian schemes for evolutive Hamilton-Jacobi Equations.
Fabio Camilli - One of the best experts on this subject based on the ideXlab platform.
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Approximation of Hamilton-Jacobi Equations with Caputo time-fractional derivative
arXiv: Numerical Analysis, 2019Co-Authors: Fabio Camilli, Serikbolsyn DuisembayAbstract:In this paper, we investigate the numerical approximation of Hamilton-Jacobi Equations with the Caputo time-fractional derivative. We introduce an explicit in time discretization of the Caputo derivative and a finite difference scheme for the approximation of the Hamiltonian. We show that the approximation scheme so obtained is stable under an appropriate condition on the discretization parameters and converges to the unique viscosity solution of the Hamilton-Jacobi equation.
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Hamilton-Jacobi Equations on networks
2014Co-Authors: Fabio CamilliAbstract:Lecture 1: A short introduction to linear differential Equations on networks. Lecture 2: Hamilton-Jacobi Equations on networks. Lecture 3: Vanishing viscosity, comparison among various definitions of viscosity solution on network and numerical methods Lecture 4: Eikonal Equations on Sierpinski gasket
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Hypercontractivity of a semi-Lagrangian scheme for Hamilton-Jacobi Equations
arXiv: Numerical Analysis, 2013Co-Authors: Fabio Camilli, Paola Loreti, Cristina PocciAbstract:The equivalence between logarithmic Sobolev inequalities and hypercontractivity of solutions of Hamilton-Jacobi Equations has been proved in [5]. We consider a semi-Lagrangian approximation scheme for the Hamilton-Jacobi equation and we prove that the solution of the discrete problem satisfies a hypercontractivity estimate. We apply this property to obtain an error estimate of the set where the truncation error is concentrated.
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Hamilton-Jacobi Equations constrained on networks
Nonlinear Differential Equations and Applications, 2013Co-Authors: Yves Achdou, Fabio Camilli, Alessandra Cutrì, Nicoletta TchouAbstract:We consider continuous-state and continuous-time control problems where the admissible trajectories of the system are constrained to remain on a network. In our setting, the value function is continuous. We de ne a notion of constrained viscosity solution of Hamilton-Jacobi Equations on the network and we study related comparison principles. Under suitable assumptions, we prove in particular that the value function is the unique constrained viscosity solution of the Hamilton-Jacobi equation on the network.
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A comparison among various notions of viscosity solutions for Hamilton-Jacobi Equations on networks
Journal of Mathematical Analysis and Applications, 2013Co-Authors: Fabio Camilli, Claudio MarchiAbstract:Three definitions of viscosity solutions for Hamilton-Jacobi Equations on networks recently appeared in literature ([1,4,6]). Being motivated by various applications, they appear to be considerably different. Aim of this note is to establish their equivalence.
