An Ordered Upwind Method with Precomputed Stencil and Monotone Node Acceptance for Solving Static Convex Hamilton-Jacobi Equations

Abstract

We define a δ-causal discretization of static convex Hamilton-Jacobi Partial Differential Equations (HJ PDEs) such that the solution value at a grid node is dependent only on solution values that are smaller by at least δ. We develop a Monotone Acceptance Ordered Upwind Method (MAOUM) that first determines a consistent, δ-causal stencil for each grid node and then solves the discrete equation in a single-pass through the nodes. MAOUM is suited to solving HJ PDEs efficiently on highly-nonuniform grids, since the stencil size adjusts to the level of grid refinement. MAOUM is a Dijkstra-like algorithm that computes the solution in increasing value order by using a heap to sort proposed node values. If δ>0, MAOUM can be converted to a Dial-like algorithm that sorts and accepts values using buckets of width δ. We present three hierarchical criteria for δ-causality of a node value update from a simplex of nodes in the stencil.

The asymptotic complexity of MAOUM is found to be $\mathcal {O}((\hat{\Psi}\rho )^{d} N \log N)$, where d is the dimension, $\hat{\Psi}$ is a measure of anisotropy in the equation, and ρ is a measure of the degree of nonuniformity in the grid. This complexity is a constant factor $(\hat{\Psi}\rho)^{d}$ greater than that of the Dijkstra-like Fast Marching Method, but MAOUM solves a much more general class of static HJ PDEs. Although ρ factors into the asymptotic complexity, experiments demonstrate that grid nonuniformity does not have a large effect on the computational cost of MAOUM in practice. Our experiments indicate that, due to the stencil initialization overhead, MAOUM performs similarly or slightly worse than the comparable Ordered Upwind Method presented in (Sethian and Vladimirsky, SIAM J. Numer. Anal. 41:323, 2003) for two examples on uniform meshes, but considerably better for an example with rectangular speed profile and significant grid refinement around nonsmooth parts of the solution. We test MAOUM on a diverse set of examples, including seismic wavefront propagation and robotic navigation with wind and obstacles.