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

Abstract

Abstract. We define a -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 . We develop a Monotone Acceptance Ordered Upwind Method (MAOUM) that first determines a consistent, -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 > 0, MAOUM can be converted to a Dial-like algorithm that sorts and accepts values using buckets of width . We present three hierarchical criteria for -causality of a node value update from a simplex of nodes in the stencil. The asymptotic complexity of MAOUM is found to be ((Ψ ) log ), where is the dimension,Ψ is a measure of anisotropy in the equation, and is a measure of the degree of nonuniformity in the grid. This complexity is a constant factor (Ψ ) greater than that of the Dijkstra-like Fast Marching Method, but MAOUM solves a much more general class of static HJ PDEs. Although 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 (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.

Efficient fast marching with Finsler metrics

“…The stencils are usually extremely simple, see Figure 2, left and center left. The complexity of these methods is linear in N := #(Ω * ) in the special case of an Isotropic metric, O(λ(F)N ) for the Fast Sweeping 2 The stencil construction of the AGSI and of the MAOUM requires a mesh of the underlying discrete domain Ω * , here a subset of hZZ 2 for some h > 0. We triangulated this grid with rescaled translates of the triangle of vertices (0, 0), (1, 0), (0, 1), and of its symmetric with respect to the origin.…”

Local Minimization Algorithms for Dynamic Programming Equations

“…Furthermore, let There exists an extensive literature concerning the construction of numerical schemes for static HJB equations. The spectrum of numerical techniques includes ordered upwind methods [20,3], high-order schemes [24], domain decomposition techniques [7] and geometric approaches [6], among many others (we refer to [13,Chapter 5,p.145] for a review of classical approximation methods). In this paper we follow a semi-Lagrangian approach [11], which is broadly used to approximate HJB equations arising in optimal control problems, see,e.g., [13].…”

Anisotropic challenges in pedestrian flow modeling

“…Many of the issues we discuss are relevant in that context as well, but we focus on the stroboscopic models for the sake of simplicity. 2 The equation introduced by Hughes was slightly more general than the time-optimal path planning -it allowed for an additional high-density-discomfort/penalty factor in the Eikonal equation. We omit it here for the sake of simplicity.…”

“…In Section 4, we discussed uniqueness of Nash Equilibria for the system of coupled Hamilton-Jacobi-Isaacs PDEs (4.1) and (4.2). For numerical simulations, the semi-Lagrangian discretization of these PDEs on the grid results in a pair of coupled opti-mization problems at each gridpoint (i,j): 2) where N ϕ a ij refers to the values of ϕ a at the neighboring gridpoints to (i,j). Ideally, we would like to know that the system (5.1-5.2) has a unique Nash Equilibrium regardless of values of ρ a ij ,ρ b ij ,N ϕ a ij , and N ϕ b ij whenever the same is true for (4.1-4.2) regardless of the vectors ∇ϕ a and ∇ϕ b .…”