Ordered upwind methods for static Hamilton–Jacobi equations

We introduce a family of fast ordered upwind methods for approximating solutions to a wide class of static Hamilton–Jacobi equations with Dirichlet boundary conditions. Standard techniques often rely on iteration to converge to the solution of a discretized version of the partial differential equation. Our fast methods avoid iteration through a careful use of information about the characteristic directions of the underlying partial differential equation. These techniques are of complexity O(M log M), where M is the total number of points in the domain. We consider anisotropic test problems in optimal control, seismology, and paths on surfaces.

[1]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[2]  G. W. Postma.  Wave propagation in a stratified medium , 1955 .

[3]  Edsger W. Dijkstra,et al.  A note on two problems in connexion with graphs , 1959, Numerische Mathematik.

[4]  Richard Bellman,et al.  Introduction to the mathematical theory of control processes , 1967 .

[5]  P. Lions,et al.  Viscosity solutions of Hamilton-Jacobi equations , 1983 .

[6]  R. González,et al.  On Deterministic Control Problems: An Approximation Procedure for the Optimal Cost I. The Stationary Problem , 1985 .

[7]  E. Rouy,et al.  A viscosity solutions approach to shape-from-shading , 1992 .

[8]  M. Falcone,et al.  Level Sets of Viscosity Solutions: some Applications to Fronts and Rendez-vous Problems , 1994, SIAM J. Appl. Math..

[9]  Tyrone E. Duncan,et al.  Numerical Methods for Stochastic Control Problems in Continuous Time (Harold J. Kushner and Paul G. Dupuis) , 1994, SIAM Rev..

[10]  J. Tsitsiklis.  Efficient algorithms for globally optimal trajectories , 1995, IEEE Trans. Autom. Control..

[11]  Phillip Colella,et al.  Two new methods for simulating photolithography development in 3D , 1996, Advanced Lithography.

[12]  James A. Sethian,et al.  Fast-marching level-set methods for three-dimensional photolithography development , 1996, Advanced Lithography.

[13]  J. Sethian,et al.  An O(N log N) algorithm for shape modeling. , 1996, Proceedings of the National Academy of Sciences of the United States of America.

[14]  J A Sethian,et al.  A fast marching level set method for monotonically advancing fronts. , 1996, Proceedings of the National Academy of Sciences of the United States of America.

[15]  J A Sethian,et al.  Computing geodesic paths on manifolds. , 1998, Proceedings of the National Academy of Sciences of the United States of America.

[16]  Alex M. Andrew,et al.  Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science (2nd edition) , 2000 .

[17]  H. Kushner.  Numerical Methods for Stochastic Control Problems in Continuous Time , 2000 .

[18]  J. Sethian,et al.  Fast methods for the Eikonal and related Hamilton- Jacobi equations on unstructured meshes. , 2000, Proceedings of the National Academy of Sciences of the United States of America.

[19]  Marko Subasic,et al.  Level Set Methods and Fast Marching Methods , 2003 .