Nonconvex Hamilton Jacobi Equations Research Articles

English

We propose a method of constructing convergent high order schemes for Hamilton-Jacobi equations on triangular meshes, which is based on combining a high order scheme with a first order monotone scheme. According to this methodology, we construct adaptive schemes of weighted essentially non-oscillatory type on triangular meshes for non-convex Hamilton-Jacobi equations in which the first order monotone approximations are occasionally applied near singular points of the solution (discontinuities of the derivative) instead of weighted essentially non-oscillatory approximations. Through detailed numerical experiments, the convergence and effectiveness of the proposed adaptive schemes are demonstrated.

Non-convex Hamilton–Jacobi equations with gradient constraints

We study non-convex Hamilton–Jacobi equations in the presence of gradient constraints and produce new, optimal, regularity results for the solutions. A distinctive feature of those equations regards the existence of a lower bound to the norm of the gradient; it competes with the elliptic operator governing the problem, affecting the regularity of the solutions. This class of models relates to various important questions and finds applications in several areas; of particular interest is the modeling of optimal dividends problems for multiple insurance companies in risk theory and singular stochastic control in reversible investment models.

Stochastic homogenization of certain nonconvex Hamilton–Jacobi equations

In this paper, we prove the stochastic homogenization of certain nonconvex Hamilton–Jacobi equations. The nonconvex Hamiltonians, which are generally uneven and inseparable, are generated by a sequence of quasiconvex Hamiltonians and a sequence of quasiconcave Hamiltonians through the min-max formula. We provide a monotonicity assumption on the contact values between those stably paired Hamiltonians so as to guarantee the stochastic homogenization.

Stochastic homogenization of quasilinear Hamilton–Jacobi equations and geometric motions

We study random homogenization of second-order, degenerate and quasilinear Hamilton–Jacobi equations which are positively homogeneous in the gradient. Included are the equations of forced mean curvature motion and others describing geometric motions of level sets as well as a large class of viscous, non-convex Hamilton–Jacobi equations. The main results include the first proof of qualitative stochastic homogenization for such equations. We also present quantitative error estimates which give an algebraic rate of homogenization.

The selection problem for discounted Hamilton–Jacobi equations: some non-convex cases

Here, we study the selection problem for the vanishing discount approximation of non-convex, first-order Hamilton–Jacobi equations. While the selection problem is well understood for convex Hamiltonians, the selection problem for non-convex Hamiltonians has thus far not been studied. We begin our study by examining a generalized discounted Hamilton–Jacobi equation. Next, using an exponential transformation, we apply our methods to strictly quasi-convex and to some non-convex Hamilton–Jacobi equations. Finally, we examine a non-convex Hamiltonian with flat parts to which our results do not directly apply. In this case, we establish the convergence by a direct approach.

Regularity properties of viscosity solution of nonconvex Hamilton–Jacobi equations

ABSTRACTSome properties of characteristic curves in connection with the viscosity solution of the Hamilton–Jacobi equation defined by the Hopf formula are studied. We are concerned with the points at which the solution u(t, x) is differentiable, and the strip of the form in the domain where u(t, x) is of class

Min–max formulas and other properties of certain classes of nonconvex effective Hamiltonians

This paper is the first attempt to systematically study properties of the effective Hamiltonian $$\overline{H}$$ arising in the periodic homogenization of some coercive but nonconvex Hamilton–Jacobi equations. Firstly, we introduce a new and robust decomposition method to obtain min–max formulas for a class of nonconvex $$\overline{H}$$ . Secondly, we analytically and numerically investigate other related interesting phenomena, such as “quasi-convexification” and breakdown of symmetry, of $$\overline{H}$$ from other typical nonconvex Hamiltonians. Finally, in the appendix, we show that our new method and those a priori formulas from the periodic setting can be used to obtain stochastic homogenization for the same class of nonconvex Hamilton–Jacobi equations. Some conjectures and problems are also proposed.

Homogenization and non-homogenization of certain non-convex Hamilton–Jacobi equations

We continue the study of the homogenization of coercive non-convex Hamilton–Jacobi equations in random media identifying two general classes of Hamiltonians with very distinct behavior. For the first class there is no homogenization in a particular environment while for the second homogenization takes place in environments with finite range dependence. Motivated by the recent counter-example of Ziliotto, who constructed a coercive but non-convex Hamilton–Jacobi equation with stationary ergodic random potential field for which homogenization does not hold, we show that same happens for coercive Hamiltonians which have a strict saddle-point, a very local property. We also identify, based on the recent work of Armstrong and Cardaliaguet on the homogenization of positively homogeneous random Hamiltonians in environments with finite range dependence, a new general class Hamiltonians, namely equations with uniformly strictly star-shaped sub-level sets, which homogenize.

Effective nonlinear Neumann boundary conditions for 1D nonconvex Hamilton–Jacobi equations

We study Hamilton–Jacobi equations in [0,+∞) of evolution type with nonlinear Neumann boundary conditions in the case where the Hamiltonian is not necessarily convex with respect to the gradient variable. In this paper, we give two main results. First, we prove for a nonconvex and coercive Hamiltonian that general boundary conditions in a relaxed sense are equivalent to effective ones in a strong sense. Here, we exhibit the effective boundary conditions while for a quasi-convex Hamiltonian, we already know them (Imbert and Monneau, 2016). Second, we give a comparison principle for a nonconvex and nonnecessarily coercive Hamiltonian where the boundary condition can have constant parts.

Algorithm for Overcoming the Curse of Dimensionality For Time-Dependent Non-convex Hamilton–Jacobi Equations Arising From Optimal Control and Differential Games Problems

In this paper we develop a parallel method for solving possibly non-convex time-dependent Hamilton---Jacobi equations arising from optimal control and differential game problems. The subproblems are independent so they can be implemented in an embarrassingly parallel fashion, which usually has an ideal parallel speedup. The algorithm is proposed to overcome the curse of dimensionality (Bellman in Adaptive control processes: a guided tour. Princeton University Press, Princeton, 1961; Dynamic programming. Princeton University Press, Princeton, 1957) when solving HJ PDE . We extend previous work Chow et al. (Algorithm for overcoming the curse of dimensionality for certain non-convex Hamilton---Jacobi equations, Projections and differential games, UCLA CAM report, pp 16---27, 2016) and Darbon and Osher (Algorithms for overcoming the curse of dimensionality for certain Hamilton---Jacobi equations arising in control theory and elsewhere, UCLA CAM report, pp 15---50, 2015) and apply a generalized Hopf formula to solve HJ PDE involving time-dependent and perhaps non-convex Hamiltonians. We suggest a coordinate descent method for the minimization procedure in the Hopf formula. This method is preferable when even the evaluation of the function value itself requires some computational effort, and also when we handle higher dimensional optimization problem. For an integral with respect to time inside the generalized Hopf formula, we suggest using a numerical quadrature rule. Together with our suggestion to perform numerical differentiation to minimize the number of calculation procedures in each iteration step, we are bound to have numerical errors in our computations. These errors can be effectively controlled by choosing an appropriate mesh-size in time and the method does not use a mesh in space. The use of multiple initial guesses is suggested to overcome possibly multiple local extrema in the case when non-convex Hamiltonians are considered. Our method is expected to have wide application in control theory and differential game problems, and elsewhere.

Stochastic homogenization of nonconvex Hamilton–Jacobi equations in one space dimension

We prove stochastic homogenization for a general class of coercive, nonconvex Hamilton–Jacobi equations in one space dimension. Some properties of the effective Hamiltonian arising in the nonconvex case are also discussed.

Random homogenization of coercive Hamilton–Jacobi equations in 1d

In this paper, we will prove the random homogenization of general coercive non-convex Hamilton-Jacobi equations in one dimensional case. This extends the result of Armstrong, Tran and Yu when the Hamiltonian has a separable form $H(p,x,\omega)=H(p)+V(x,\omega)$ for any coercive $H(p)$.

Stochastic homogenization of a nonconvex Hamilton–Jacobi equation

We present a proof of qualitative stochastic homogenization for a nonconvex Hamilton–Jacobi equations. The new idea is to introduce a family of “sub-equations” and to control solutions of the original equation by the maximal subsolutions of the latter, which have deterministic limits by the subadditive ergodic theorem and maximality.

Envelopes and nonconvex Hamilton–Jacobi equations

This paper introduces a new representation formula for viscosity solutions of nonconvex Hamilton–Jacobi PDE using “generalized envelopes” of affine solutions. We study as well envelope and singular characteristic constructions of equivocal surfaces and discuss also differential game theoretic interpretations. In memory of Arik A. Melikyan.

A PDE Approach to Large-Time Asymptotics for Boundary-Value Problems for Nonconvex Hamilton–Jacobi Equations

We investigate the large-time behavior of three types of initial-boundary value problems for Hamilton–Jacobi Equations with nonconvex Hamiltonians. We consider the Neumann or oblique boundary condition, the state constraint boundary condition and Dirichlet boundary condition. We establish general convergence results for viscosity solutions to asymptotic solutions as time goes to infinity via an approach based on PDE techniques. These results are obtained not only under general conditions on the Hamiltonians but also under weak conditions on the domain and the oblique direction of reflection in the Neumann case.

The Riemann problem for nonconvex scalar conservation laws and Hamilton-Jacobi equations

The Riemann problem for nonconvex scalar conservation laws and Hamilton-Jacobi equations

Algorithm for overcoming the curse of dimensionality for certain non-convex Hamilton–Jacobi equations, projections and differential games

Abstract : In this paper, we develop a method for solving a large class of non-convex Hamilton-Jacobi partial differential equations (HJ PDE). The method yields decoupled subproblems, which can be solved in an embarrassingly parallel fashion. The complexity of the resulting algorithm is polynomial in the problem dimension; hence, it overcomes the curse of dimensionality [1, 2]. We extend previous work in[6] and apply the Hopf formula to solve HJ PDE involving non-convex Hamiltonians. We propose an ADMM approach for finding the minimizer associated with the Hopf formula. Some explicit formulae of proximal maps, as well as newly-defined stretch operators, are used in the numerical solutions of ADMM subproblems. Our approach is expected to have wide applications in continuous dynamic games, control theory problems, and elsewhere.