Convergent difference schemes for nonlinear parabolic equations and mean curvature motion

Crandall, Michael G.; Lions, Pierre-Louis

doi:10.1007/s002110050228

Cited by 62 publications

(77 citation statements)

References 20 publications

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“…Indeed, Crandall and Lions in [5] gave an explicit finite difference scheme which is both monotone and consistent by introducing a scheme for (2) and by using the convergence u ε → u as ε → 0 rather than discretizing (1) directly. Deckelnick in [6] derived an L ∞ -error estimate between the numerical solution and the viscosity solution of (1) by using the convergence rate of u ε → u. Deckelnick has already obtained essentially the same convergence rate result as that in Theorem 1, but we emphasize that the proof in [6] seems to be more technical and we give a simple proof in this paper.…”

Section: Introductionmentioning

confidence: 99%

On convergence rates for solutions of approximate mean curvature equations

Mitake

2011

Proc. Amer. Math. Soc.

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Abstract. Evans and Spruck (1991) considered an approximate equation for the level-set equation of the mean curvature flow and proved the convergence of solutions. Deckelnick (2000) established a rate for the convergence. In this paper, we will provide a simple proof for the same result as that of Deckelnick. Moreover, we consider generalized mean curvature equations and introduce approximate equations for them and then establish a rate for the convergence.

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Section: Introductionmentioning

confidence: 99%

On convergence rates for solutions of approximate mean curvature equations

Mitake

2011

Proc. Amer. Math. Soc.

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“…Yet, direct computations of curvatures on raw images were still unreliable, as they were based on difference approximations on finite grids, see for example Crandall and Lions [13] or Alvarez and Morel [2]. Although these schemes are somewhat efficient for simulating image motion by mean curvature, they are not reliable when it comes to visualizing the actual curvature map.…”

Section: Introductionmentioning

confidence: 99%

The Image Curvature Microscope: Accurate Curvature Computation at Subpixel Resolution

Ciomaga¹,

Monasse²,

Morel³

2017

Image Processing on Line

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We detail in this paper the numerical implementation of the so-called image curvature microscope, an algorithm that computes accurate image curvatures at subpixel resolution, and yields a curvature map conforming with our visual perception. In contrast to standard methods, which would compute the curvature by a finite difference scheme, the curvatures are evaluated directly on the level lines of the bilinearly interpolated image, after their independent smoothing, a step necessary to remove pixelization artifacts. The smoothing step consists in the affine erosion of the level lines through a geometric scheme, and can be applied in parallel to all level lines. The online algorithm allows the user to visualize the image of curvatures at different resolutions, as well as the set of level lines before and after smoothing. Source CodeThe ANSI C++ implementation reviewed by IPOL is given in the file curv.cpp. The complete source code, code documentation, and online demo are accessible at the IPOL web page of this article 1 . The level lines extraction (files levelLine.cpp, lltree.cpp) and level lines smoothing (file gass.cpp) are independent algorithms, out of the scope of this article, and details for their implementation are planned for separate IPOL publications.

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“…In the framework of viscosity solutions, semi-Lagrangian schemes for first order HJB equations have been studied in [20]. Extensions to the second order case can be found in [35,19,23,24]. In our context, the numerical scheme that will be studied couples the classical semi-Lagrangian scheme with an additional projection step on the boundary.…”

Section: Introductionmentioning

confidence: 99%

Dynamic Programming and Error Estimates for Stochastic Control Problems with Maximum Cost

2014

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Abstract. This work is concerned with stochastic optimal control for a running maximum cost. A direct approach based on dynamic programming techniques is studied leading to the characterization of the value function as the unique viscosity solution of a second order HamiltonJacobi-Bellman (HJB) equation with an oblique derivative boundary condition. A general numerical scheme is proposed and a convergence result is provided. Error estimates are obtained for the semi-Lagrangian scheme. These results can apply to the case of lookback options in finance. Moreover, optimal control problems with maximum cost arise in the characterization of the reachable sets for a system of controlled stochastic differential equations. Some numerical simulations on examples of reachable analysis are included to illustrate our approach.

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Convergent difference schemes for nonlinear parabolic equations and mean curvature motion

Cited by 62 publications

References 20 publications

On convergence rates for solutions of approximate mean curvature equations

On convergence rates for solutions of approximate mean curvature equations

The Image Curvature Microscope: Accurate Curvature Computation at Subpixel Resolution

Dynamic Programming and Error Estimates for Stochastic Control Problems with Maximum Cost

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