Abstract
We develop a single pass method for approximating the solution to an anisotropic eikonal equation related to the anisotropic min-time optimal trajectory problem. Ordered Upwind Method (OUM) solves this equation, which is a single-pass method with an asymptotic complexity. OUM uses the search along the accepted front (SAAF) to update the value at considered nodes. Our technique, which we refer to as “Neighbor-Gradient Single-Pass Method”, uses the minimizer of the Hamiltonian, in which the gradient is substituted with neighbor gradient information, to avoid SAAF. Our technique is considered in the context of control-theoretic problem. We begin by discussing SAAF of OUM. We then prove some properties of the value function and its gradient, which provide the key motivation for constructing our method. Based on these discussions, we present a new single-pass method, which is fast since it does not require SAAF. We test this method with several anisotropic eikonal equations to observe that it works well while significantly reducing the computational cost.
Similar content being viewed by others
Data availability
Enquiries about data availability should be directed to the authors.
References
Osher, S., Fedkiw, R.: Level Set Methods and Dynamic Implicit Surfaces. Springer-Verlag, New York Inc (2003)
Sethian, J.A.: A fast marching level set method for monotonically advancing fronts. Proc. Natl. Acad. Sci. 93, 1591–1595 (1996)
Sethian, J. A.: Level set methods and fast marching methods: evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science,2 nd ed. In: Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press. (3) (1999)
Cristiani, E.: A fast marching method for Hamilton-Jacobi equation modeling monotone front propagations. J. Sci. Comput. 39(2), 189–205 (2009)
Chow, Y.T., Darbon, J., Osher, S., Yin, W.: Algorithm for overcoming the curse of dimensionality for time-dependent non-convex Hamilton-jacobi equations arising from optimal control and differential games problems. J. Sci. Comput. 73, 617–643 (2017)
Lelièvre, P.G., Farquharson, C.G., Hurich, C.A.: Computing first arrival seismic travel times on unstructured 3-D tetrahedral grids using the fast marching method. Geophys. J. Int. 184(2), 885–896 (2011)
Tozza, S., Falcone, M.: Analysis and approximation of some shape-from-shading models for non-lambertian surfaces. J. Math. Imaging Vis. 55(2), 153–178 (2016)
Peyré, G., Péchaud, M., Keriven, R., Cohen, LD: Geodesic methods in computer vision and graphics, foundations and trends in computer graphics and vision, 5(3–4), 197–397 (2010)
Cacace, S., Cristiani, E., Falcone, M.: Can local single-pass methods solve any stationary Hamilton–Jacobi–Bellman equation? SIAM J. Sci. Comput. 36, 570–587 (2014)
Falcone, M., Ferretti, R.: Semi-Lagrangian schemes for Hamilton-Jacobi equations, discrete representation formulae and Godunov methods. J. Comput. Phys. 175, 559–575 (2002)
Falcone, M., Ferretti, R.: Discrete time high-order schemes for viscosity solutions of Hamilton-Jacobi-Bellman equations. Numer. Math. 67, 315–344 (1994)
Cristiani, E., Falcone, M.: Fast semi-Lagrangian schemes for the Eikonal equation and applications. SIAM J. Numer. Anal. 45(5), 1979–2011 (2007)
Tsitsiklis, J.N.: Efficient algorithms for globally optimal trajectories. IEEE Trans. Automat. Control. 40, 1528–1538 (1995)
Sethian, J.A.: Fast marching methods. SIAM Rev. 41(2), 199–235 (1999)
Cacace, S., Cristiani, E., Falcone, M.: A local ordered upwind method for hamilton-jacobi and isaacs equations. In: Proceedings of 18th IFAC World Congress 2011, pp. 6800–6805
Sethian, J.A., Vladmirsky, A.: Ordered upwind methods for static Hamilton-Jacobi equations, theory and algorithms. SIAM J. Numer. Anal. 41(1), 323–363 (2003)
Sethian, J.A., Vladimirsky, A.: Ordered upwind methods for static Hamilton-Jacobi equations. Proc. Natl. Acad. Sci. 98(20), 11069–11074 (2001)
Shum, A., Morris, K., Khajepour, A.: Convergence Rate for the Ordered Upwind Method. J. Sci. Comput. 68(3), 889–913 (2016)
Alton, K.: Dijkstra-like Ordered Upwind Methods for Solving Static Hamilton-Jacobi Equations, Ph.D. thesis, University of British Columbia, Vancouver, 2010.
Alton, K., Mitchell, I.M.: Fast marching methods for stationary Hamilton-Jacobi equations with axis-aligned anisotropy. SIAM J. Numer. Anal. 43, 363–385 (2008)
Alton, K., Mitchell, I.M.: An ordered upwind method with pre-computed stencil and monotone node acceptance for solving static convex Hamilton-Jacobi equations. J. Sci. Comput. 51, 313–348 (2012)
Kao, C.Y., Osher, S., Tsai, Y.: Fast sweeping methods for static Hamilton-Jacobi equations. SIAM J. Numer. Anal. 42(6), 2004–2005 (2005)
Kao, C.Y., Osher, S., Qian, J.: Lax-Friedrichs sweeping scheme for static Hamilton-Jacobi equations. J. Comput. Phys. 196, 367–439 (2004)
Tsai, Y., Cheng, L., Osher, S., Zhao, H.: Fast sweeping algorithms for a class of Hamilton-Jacobi equations. SIAM J. Numer. Anal. 41, 673–694 (2004)
Qian, J., Zhang, Y.T., Zhao, H.: A fast sweeping method for static convex Hamilton-Jacobi equations. J. Sci. Comput. 31, 237–271 (2007)
Zhao, H.: A fast sweeping method for eikonal equations. Math. Comp. 74, 603–627 (2005)
Qian, J., Zhang, Y., Zhao, H.: Fast sweeping methods for Eikonal equations on triangulated meshes. SIAM J. Numer. Anal. 45, 83–107 (2007)
Jeong, Fu.Z., Pan, W.K., Kirby, Y., Whitaker, R.M.: A fast iterative method for solving the eikonal equation on triangulated surfaces. SIAM J. Sci. Comput. 33, A2468–A2488 (2011)
Jeong, W.K., Whitaker, R.T.: A fast iterative method for eikonal equations. SIAM J. Sci. Comput. 30, 2512–2534 (2008)
Jeong, W. K., Whitaker, R. T.: A fast iterative method for a class of Hamilton–Jacobi equations on parallel systems. University of Utah, Technical Report UUCS-07-010 (2007)
Dahiya, D., Baskar, S.: Characteristic fast marching method on triangular grids for the generalized eikonal equation in moving media. Wave Motion 59, 81–93 (2015)
Dahiya, D., Baskar, S., Coulouvrat, F.: Characteristic fast marching method for monotonically propagating fronts in a moving medium. SIAM J. Sci. Comput. 35(4), A1880–A1902 (2013)
Glowinski, R., Leung, S., Qian, J.: Operator-splitting based fast sweeping methods for isotropic wave propagation in a moving fluid. SIAM J. Sci. Comput. 38(2), A1195–A1223 (2016)
Glowinski, R., Liu, H., Leung, S., Qian, J.: A finite element/operator-splitting method for the numerical solution of the two dimensional elliptic Monge–Ampère equation, 79(1), 1–47 (2019)
Ho, M., Ri, J., Kim, S.: Improved characteristic fast marching method for the generalized eikonal equation in a moving medium. J. Sci. Comput. 81(3), 2484–2502 (2019)
Bardi, M., Capuzzo-Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhauser, Boston (1997)
Sinestrari, C.: Semiconcavity of solutions of stationary Hamilton-Jacobi equations. Nonlinear. Anal. 24(9), 1321–1326 (1995)
Vladmirsky, A.: Fast Methods for Static Hamilton-Jacobi Partial Differential Equations, Ph.D. thesis, University of California, Berkeley, 2001.
Acknowledgements
The authors would like to thank the referees for their valuable comments and suggestions, which helped the authors to improve the article significantly.
Funding
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix
Appendix
Lemma 1
The viscosity solution \(u(x)\) of (6) is semi-concave.
Proof
Let us prove that \(H(x,p)\) is Lipchitz continuous in \(x\).
For arbitrary \(x,y\), suppose that.
If \(p \cdot a_{0} f(x,a_{0} ) \ge p \cdot a_{1} f(y,a_{1} )\), then.
If \(p \cdot a_{0} f(x,a_{0} ) < p \cdot a_{1} f(y,a_{1} )\), then.
Therefore \(H(x,p)\) is Lipchitz continuous in \(x\). We also know that \(H{(}x,p)\) is convex in the second argument and thus \(u\) is semi-concave in \(\Omega\). ([31]) □
Lemma 2
Assume that \(u\) is differentiable at \(x \in \Omega\) and suppose that \(\alpha ( \cdot )\) is an optimal control for \(x\) and \(y( \cdot )\) is the optimal trajectory for \(x\) . Then \(\nabla u(y( \cdot ))\) is continuous at \([0,\; + \infty )\) .
Proof
For arbitrary \(t > 0\), \(u\) is differentiable at \(y{(}t)\) because characteristics never emanate from the shocks-non differentiable point. ([16]).
If \(\nabla u\) is defined at \(x \in \Omega \backslash \partial \Omega\), there exists constant \(L\) such that \(\left\| {\nabla u} \right\| \le L\). [16].
We fix any \(t \ge 0\). For arbitrary sequence \(t_{n} \to t(t_{n} \ge 0)\), \(y{(}t_{n} ) \to y(t)\). The \(\nabla u(y(t_{n} ))\) is bounded. Therefore, it has convergent subsequence. Since \(u\) is semi-concave, limits of all convergent subsequences are the same.i.e.,\(\nabla u(y(t))\). [36] Therefore \(\nabla u(y(t_{n} )) \to \nabla u(y(t))\) and thus \(\nabla u(y( \cdot ))\) is continuous at \([0,\; + \infty )\). □
Lemma 3
Consider a grid node \(x \in X\) and assume that u is differentiable at \(x.\) Suppose that the characteristic for \(x\) intersects the line segment \(x_{1} x_{2}\) at \(x_{0}\) , where \(x_{1} ,x_{2} \in ND(x),\;x_{1} \in N(x_{2} )\) . Then.
Proof
Suppose that \(\alpha ( \cdot )\) is an optimal control for \(x\) and \(y( \cdot )\) is the optimal trajectory for \(x\).
If \(y(t_{{0}} ) = x_{0}\), since \(\nabla u(y( \cdot ))\) is continuous at 0, we get.
By the Bellman’s optimality principle, we get \(t_{{0}} \le \frac{{x_{0} - x_{1} }}{{F_{2} }}\) and \(\frac{{x_{0} - x_{1} }}{{F_{1} }} \le t_{{0}}\).
Therefore \(O(t_{0} ) = O(h)\). From the fact that \(y(0) = x,y(t_{0} ) = x_{0}\), we conclude that.
\(\left\| {\nabla u(x_{0} ) - \nabla u(x)} \right\| = O(h) \, \).□
Lemma 4
If \(xx_{1} x_{2}\) is a sufficiently small simplex, which contains the characteristic for \(x\) , then.
Proof
By the Bellman’s optimality principle, we get.
Since \(xx_{1} x_{2}\) is sufficiently small simplex, we obtain that \(u(x) > \lambda u(x_{1} ) + \left( {1 - \lambda } \right)u(x_{2} )\).
Therefore, \(\min (u(x_{1} ),\;u(x_{2} )) < u(x)\). □
Rights and permissions
About this article
Cite this article
Ho, MS., Pak, JS., Jo, SM. et al. Neighbor-Gradient Single-Pass Method for Solving Anisotropic Eikonal Equation. J Sci Comput 91, 33 (2022). https://doi.org/10.1007/s10915-022-01773-3
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10915-022-01773-3