A Parallel Two-Scale Method for Eikonal Equations

Numerous applications of Eikonal equations prompted the development of many efficient numerical algorithms. The Heap-Cell Method (HCM) is a recent serial two-scale technique that has been shown to have advantages over other serial state-of-the-art solvers for a wide range of problems [A. Chacon and A. Vladimirsky, SIAM J. Sci. Comput., 34 (2012), pp. A547--A578]. This paper presents a parallelization of HCM for a shared memory architecture. The numerical experiments in $R^3$ show that the parallel HCM exhibits good algorithmic behavior and scales well, resulting in a very fast and practical solver.

[1]  Tor Gillberg,et al.  A New Parallel 3D Front Propagation Algorithm for Fast Simulation of Geological folds , 2012, ICCS.

[2]  J. A. Sethian,et al.  Fast Marching Methods , 1999, SIAM Rev..

[3]  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.

[4]  Maurizio Falcone,et al.  A Patchy Dynamic Programming Scheme for a Class of Hamilton-Jacobi-Bellman Equations , 2011, SIAM J. Sci. Comput..

[5]  S. Osher,et al.  Lax-Friedrichs sweeping scheme for static Hamilton-Jacobi equations , 2004 .

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

[7]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[8]  Rohit Chandra,et al.  Parallel programming in openMP , 2000 .

[9]  Ross T. Whitaker,et al.  A Fast Iterative Method for Eikonal Equations , 2008, SIAM J. Sci. Comput..

[10]  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.

[11]  Stefan Turek,et al.  THE EIKONAL EQUATION: NUMERICAL EFFICIENCY VS. ALGORITHMIC COMPLEXITY ON QUADRILATERAL GRIDS , 2005 .

[12]  M. Herrmann.  A Domain Decomposition Parallelization of the Fast Marching Method , 2003 .

[13]  Zhao,et al.  PARALLEL IMPLEMENTATIONS OF THE FAST SWEEPING METHOD , 2007 .

[14]  Rolf Krause,et al.  A Massive Parallel Fast Marching Method , 2016 .

[15]  J. Tsitsiklis,et al.  Efficient algorithms for globally optimal trajectories , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[16]  Alexander Vladimirsky,et al.  Fast Two-scale Methods for Eikonal Equations , 2011, SIAM J. Sci. Comput..

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

[18]  Dimitri P. Bertsekas,et al.  A simple and fast label correcting algorithm for shortest paths , 1993, Networks.

[19]  Stanley Bak,et al.  Some Improvements for the Fast Sweeping Method , 2010, SIAM J. Sci. Comput..

[20]  Christopher M. Kuster,et al.  Computational Study of Fast Methods for the Eikonal Equation , 2005, SIAM J. Sci. Comput..

[21]  Alexander Vladimirsky,et al.  A parallel Heap-Cell Method for Eikonal equations , 2013, 1306.4743.

[22]  M. Bardi,et al.  Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations , 1997 .

[23]  Michael Breuß,et al.  An adaptive domain-decomposition technique for parallelization of the fast marching method , 2011, Appl. Math. Comput..

[24]  Hongkai Zhao,et al.  A fast sweeping method for Eikonal equations , 2004, Math. Comput..

[25]  P. Dupuis,et al.  Markov chain approximations for deterministic control problems with affine dynamics and quadratic cost in the control , 1997, Proceedings of the 36th IEEE Conference on Decision and Control.

[26]  Frédéric Gibou,et al.  A parallel fast sweeping method for the Eikonal equation , 2013, J. Comput. Phys..

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

[28]  R. Newcomb.  VISCOSITY SOLUTIONS OF HAMILTON-JACOBI EQUATIONS , 2010 .

[29]  Stanley Osher,et al.  Fast Sweeping Algorithms for a Class of Hamilton-Jacobi Equations , 2003, SIAM J. Numer. Anal..

[30]  Alexander M. Bronstein,et al.  Parallel algorithms for approximation of distance maps on parametric surfaces , 2008, TOGS.

[31]  John N. Tsitsiklis,et al.  Implementation of efficient algorithms for globally optimal trajectories , 1998, IEEE Trans. Autom. Control..

[32]  Chi-Wang Shu,et al.  A second order discontinuous Galerkin fast sweeping method for Eikonal equations , 2008, J. Comput. Phys..

[33]  J. Sethian,et al.  Ordered upwind methods for static Hamilton–Jacobi equations , 2001, Proceedings of the National Academy of Sciences of the United States of America.

[34]  Adam M. Oberman,et al.  Homogenization of Metric Hamilton-Jacobi Equations , 2009, Multiscale Model. Simul..

[35]  Seongjai Kim,et al.  An O(N) Level Set Method for Eikonal Equations , 2000, SIAM J. Sci. Comput..

[36]  D. Bertsekas,et al.  Parallel asynchronous label-correcting methods for shortest paths , 1996 .