TY - JOUR

T1 - STOCHASTIC AND VARIATIONAL APPROACH TO FINITE DIFFERENCE APPROXIMATION OF HAMILTON-JACOBI EQUATIONS

AU - SOGA, KOHEI

N1 - Funding Information:
Received by the editor March 22, 2018, and, in revised form, February 12, 2019, March 16, 2019, and July 8, 2019. 2010 Mathematics Subject Classification. Primary 65M06, 35F21, 49L25, 60G50. Key words and phrases. Finite difference scheme, Hamilton-Jacobi equation, viscosity solution, calculus of variations, random walk, law of large numbers. The main part of this work was done, when the author belonged to Unitéde mathématiques pures et appliquées, CNRS UMR 5669 & École Normale Supérieure de Lyon, being supported by ANR-12-BS01-0020 WKBHJ as a researcher for academic year 2014–2015, hosted by Albert Fathi. The author was partially supported by JSPS Grant-in-aid for Young Scientists (B) #15K21369.

PY - 2020/5

Y1 - 2020/5

N2 - Previously, the author presented a stochastic and variational ap- proach to the Lax-Friedrichs finite difference scheme applied to hyperbolic scalar conservation laws and the corresponding Hamilton-Jacobi equations with convex and superlinear Hamiltonians in the one-dimensional periodic set-ting, showing new results on the stability and convergence of the scheme [Soga, Math. Comp. 84 (2015), 629–651]. In the current paper, we extend these re-sults to the higher dimensional setting. Our framework with a deterministic scheme provides approximation of viscosity solutions of Hamilton-Jacobi equa-tions, their spatial derivatives and the backward characteristic curves at the same time, within an arbitrary time interval. The proof is based on stochastic calculus of variations with random walks, a priori boundedness of minimizers of the variational problems that verifies a CFL type stability condition, and the law of large numbers for random walks under the hyperbolic scaling limit. Convergence of approximation and the rate of convergence are obtained in terms of probability theory. The idea is reminiscent of the stochastic and vari-ational approach to the vanishing viscosity method introduced in [Fleming, J. Differ. Eqs 5 (1969) 515–530].

AB - Previously, the author presented a stochastic and variational ap- proach to the Lax-Friedrichs finite difference scheme applied to hyperbolic scalar conservation laws and the corresponding Hamilton-Jacobi equations with convex and superlinear Hamiltonians in the one-dimensional periodic set-ting, showing new results on the stability and convergence of the scheme [Soga, Math. Comp. 84 (2015), 629–651]. In the current paper, we extend these re-sults to the higher dimensional setting. Our framework with a deterministic scheme provides approximation of viscosity solutions of Hamilton-Jacobi equa-tions, their spatial derivatives and the backward characteristic curves at the same time, within an arbitrary time interval. The proof is based on stochastic calculus of variations with random walks, a priori boundedness of minimizers of the variational problems that verifies a CFL type stability condition, and the law of large numbers for random walks under the hyperbolic scaling limit. Convergence of approximation and the rate of convergence are obtained in terms of probability theory. The idea is reminiscent of the stochastic and vari-ational approach to the vanishing viscosity method introduced in [Fleming, J. Differ. Eqs 5 (1969) 515–530].

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U2 - 10.1090/MCOM/3437

DO - 10.1090/MCOM/3437

M3 - Article

AN - SCOPUS:85094656294

VL - 89

SP - 1135

EP - 1159

JO - Mathematics of Computation

JF - Mathematics of Computation

SN - 0025-5718

IS - 323

ER -