Discrete time high-order schemes for viscosity solutions of Hamilton-Jacobi-Bellman equations

Summary. A general method for constructing high-order approximation schemes for Hamilton-Jacobi-Bellman equations is given. The method is based on a discrete version of the Dynamic Programming Principle. We prove a general convergence result for this class of approximation schemes also obtaining, under more restrictive assumptions, an estimate in of the order of convergence and of the local truncation error. The schemes can be applied, in particular, to the stationary linear first order equation in . We present several examples of schemes belonging to this class and with fast convergence to the solution. Received July 4, 1992 / Revised version received July 7, 1993     

Large time behavior of periodic viscosity solutions for uniformly parabolic integro-differential equations

In this paper, we study the large time behavior of solutions of a class of parabolic fully nonlinear integro-differential equations in a periodic setting. In order to do so, we first solve the ergodic problem (or cell problem), i.e. we construct solutions of the form $\lambda t + v(x).$ We then prove that solutions of the Cauchy problem look like those specific solutions as time goes to infinity. We face two key difficulties to carry out this classical program: (1) the fact that we handle the case of “mixed operators” for which the required ellipticity comes from a combination of the properties of the local and nonlocal terms and (2) the treatment of the superlinear case (in the gradient variable). Lipschitz estimates previously proved by the authors (2012) and Strong Maximum principles proved by the third author (2012) play a crucial role in the analysis.     

Feynman-Kac propagators and viscosity solutions

The aim of this paper is to study viscosity solutions to the following terminal value problem on [0, t] × E:

Controlled markov processes and viscosity solutions

Controlled markov processes and viscosity solutions by W. H. Fleming and H. M. Soner. Springer-Verlag, New York (1993), 428 pp., $ 49.95. ISBN 0-387-97927-1.     

Shape-from-shading,viscosity solutions and edges

Summary This article deals with the so-called Shape-from-Shading problem which arises when recovering a shape from a single image. The general case of a distribution of light sources illuminating a Lambertian surface is considered. This involves original definitions of three types of edges, mainly the apparent contours, the grazing light edges and the shadow edges. The elevation of the shape is expressed in terms of viscosity solution of a first-order Hamilton-Jacobi equation with various boundary conditions on these edges. Various existence and uniqueness results are presented.     

Kummer function solutions of damped Burgers equations with time-dependent viscosity by exact linearization

Through a Cole-Hopf like transformation a generalized Burgers equation is linearized to Kummer's equation. The large time behaviour of solutions of the GBE is determined. Some exact closed form solutions are also found.     

Optimal stopping problems with discontinous reward: Regularity of the value function and viscosity solutions

In this paper we show that the solution of a stochastic boundary value problem with additive noise and with a completely nonlinear drift is a Markov field if only if the boundary condition is an initial or a final type condition     

Minimal solutions of the isometry equation

For a finite vector space $W$ over ${\mathbb{F}}_q$, there are described all the pairs of multisets $\{V_1,\dots ,V_{q+1}\}$ and $\{U_1,\dots ,U_{q+1}\}$ of subspaces in $W$ such that for all $w\in W$ the equality $\left|\{i\mid w\in V_i\}\right|=\left|\{i\mid w\in U_i\}\right|$ holds.     

Large time behavior of solutions for the nonlinear wave equation with viscosity in odd dimensions

We study the dissipation of solutions of the Cauchy problem for the nonlinear dissipative wave equation in odd multi-spatial dimensions. Pointwise estimates of the time-asymptotic shape of the solutions are obtained and shown to exhibit the generalized Huygens principle. Our approach is based on the detailed analysis of the Green function of the linearized system. This is used to study the coupling of nonlinear diffusion waves.     

Semicontinuous viscosity solutions for quasiconvex Hamiltonians

The main theorem connecting convex Hamiltonians and semicontinuous viscosity solutions due to Barron and Jensen is extended to quasiconvex Hamiltonians. Some applications are indicated.     

Regularity of the value function and viscosity solutions in optimal stopping problems for general Markov processes

We consider optimal stopping problems for Markov processes with a semicontinuous reward function g , and we show that under suitable conditions the value function w = w [ g ] is itself semicontinuous and is a viscosity solution of the associated variational inequality.     

A new stability result for viscosity solutions of nonlinear parabolic equations with weak convergence in time

We present a new stability result for viscosity solutions of fully nonlinear parabolic equations which allows to pass to the limit when one has only weak convergence in time of the nonlinearities. To cite this article: G. Barles, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Minimal perfect hashing in polynomial time

We present a universally applicable algorithm for generating minimal perfect hashing functions. The method has (worst case) polynomial time complexity in units of bit operations. An adjunct algorithm for reducing parameter magnitudes in the generated hash functions is given. This probabilistic method makes hash function parameter magnitudes independent of argument (input key) magnitudes.     

Regularity of viscosity solutions near KAM torus

In this paper, we give weak regularity theorems on P of u~ε(x, P), where u~ε(x, P)is the viscosity solution of the cell problem H_ε(P D_xu~ε, x)=H_ε(P).     

On viscosity solutions of irregular Hamilton-Jacobi equations

It is proved that the initial-value problem for admits a unique continuous viscosity solution under certain conditions which do not exclude that H(x, p) is discontinuous in x. Particular attention is devoted to the linear transport equation , where a may be discontinuous. Received: 21 October 2002     

Good and viscosity solutions of fully nonlinear elliptic equations

We introduce the notion of a ``good" solution of a fully nonlinear uniformly elliptic equation. It is proven that ``good" solutions are equivalent to -viscosity solutions of such equations. The main contribution of the paper is an explicit construction of elliptic equations with strong solutions that approximate any given fully nonlinear uniformly elliptic equation and its -viscosity solution. The results also extend some results about ``good" solutions of linear equations.