Ordered upwind methods for static Hamilton–Jacobi equations

Abstract

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.

A second order discontinuous Galerkin fast sweeping method for Eikonal equations

“…Among such methods are the fast marching method and the fast sweeping method. The fast marching method [48,43,22,44,45] is based on the Dijkstra's algorithm [18]. The solution is updated by following the causality in a sequential way; i.e., the solution is updated pointwise in the order that the solution is strictly increasing (decreasing); hence two essential ingredients are needed in the algorithm: an upwind difference scheme and a heap-sort algorithm.…”

White matter tractography by anisotropic wavefront evolution and diffusion tensor imaging

“…As we have pointed out earlier, the resulting PDE becomes anisotropic. Since the FMM only handles speed functions that depend on position (Sethian and Vladimirsky, 2003), it is interesting to compare the two methods when measuring connectivity.…”

“…This smoothing term is a function of the second derivatives of the equation and prevents the developments of such discontinuities. Numerical approximations of the viscosity solution have been studied (Kao et al, 2002Sethian and Vladimirsky, 2003).…”

“…While numerical methods for obtaining viscous solutions to static anisotropic propagation equations exist, their implementation tends to be more involved than those used in isotropic problems. Single-pass (Sethian and Vladimirsky, 2003) and iterative methods (Kao et al, 2002 have been devised to construct accurate solutions for such equations. Single-pass algorithms are based on the monotonicity of the solution along the characteristic directions in order to compute the arrival time T(x).…”

Convergence of a Generalized Fast-Marching Method for an Eikonal Equation with a Velocity-Changing Sign

“…This FMM scheme has been proved to be convergent, using a relation between the FMM solution and the numerical solution to finite difference schemes for the Level Sets formulation, for which it is known that these schemes are convergent (see Sethian, Vladimirsky [24] and Cristiani, Falcone [11]). More recently, the method has been extended to more general Hamilton-Jacobi equation by Sethian and Vladimirsky [24,25] and it has been also adapted to the case of time-dependent non-negative velocities c(x, t) ≥ 0 by Vladimirsky [28]. We also refer to Chopp [9] for some results for non-monotone propagation but with time-independent velocity.…”