Semi-Lagrangian schemes for Hamilton-Jacobi equations, discrete representation formulae and Godunov methods

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.

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Semi-Lagrangian Schemes for Hamilton-Jacobi Equations, Discrete Representation Formulae

Computational convex analysis algorithms have been rediscovered several times in the past by researchers from different fields. To further communications between practitioners, we review the field of computational convex analysis, which focuses on the numerical computation of fundamental transforms arising from convex analysis. Current models use symbolic, numeric, and hybrid symbolic-numeric algorithms. Our objective is to disseminate widely the most efficient numerical algorithms useful for applications in image processing (computing the distance transform, the generalized distance transform, and mathematical morphology operators), partial differential equations (solving Hamilton-Jacobi equations and using differential equations numerical schemes to compute the convex envelope), max-plus algebra (computing the equivalent of the fast Fourier transform), multifractal analysis, etc. The fields of applications include, among others, computer vision, robot navigation, thermodynamics, electrical networks, medical imaging, and network communication.

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What Shape Is Your Conjugate? A Survey of Computational Convex Analysis and Its Applications

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.

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Envelopes and nonconvex Hamilton–Jacobi equations

The $\g$ (sub)level set of the solution to $w_t+H(\g,D_xw)=0$ is the same as the $\g$ (sub)level set of the solution to $u_t+H(u,D_xu)=0$, and the solution u may be built from w. This result is applied to determining upper and lower bounds for a solution of ut+H1 (u,Du)+H2 (u,Du)=0, with H1 convex and H2 concave, as well as ut+H(u,Du)=0, but with initial data $u(0,x)=g_1(x) \vee g_2(x)$ or $g_1(x) \wedge g_2(x)$, with g1 quasi-convex and g2 quasi-concave. A differential game in $L^{\i}$ is constructed giving a new proof of the Hopf formula.

Applications of the Hopf-Lax formula for u t + H ( u,Du ) = 0

Simple explicit estimates are presented for the viscosity solution of the Cauchy problem for the Hamilton--Jacobi equation where either the Hamiltonian or the initial data are the sum of a convex and a concave function. The estimates become equalities whenever a "minmax" equals a "maxmin" and thus a representation formula for the solution is obtained, generalizing the classical Hopf formulas as well as some formulas of Kruzkov [Functional Anal. Appl., 2 (1969), pp. 128--136].

Silva Faggian

Papers

Hopf-type estimates and formulas for nonconvex nonconcave Hamilton-Jacobi equations