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.

Abstract. The fast marching method (FMM) has proved to be a very efficient algorithm for solving the isotropic Eikonal equation. Because it is a minor modification of Dijkstra's algorithm for finding the shortest path through a discrete graph, FMM is also easy to implement. In this paper we describe a new class of Hamilton-Jacobi (HJ) PDEs with axis-aligned anisotropy which satisfy a causality condition for standard finite-difference schemes on orthogonal grids and can hence be solved using the FMM; the only modification required to the algorithm is in the local update equation for a node. This class of HJ PDEs has applications in anelliptic wave propagation and robotic path planning, and brief examples are included. Since our class of HJ PDEs and grids permit asymmetries, we also examine some methods of improving the efficiency of the local update that do not require symmetric grids and PDEs. Finally, we include explicit update formulas for variations of the Eikonal equation that use the Manhattan, Euclidean, and infinity norms on orthogonal grids of arbitrary dimension and with variable node spacing.

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Abstract-We present an efficient dynamic programming algorithm to solve the problem of optimal multi-location robot rendezvous. The rendezvous problem considered can be structured as a tree, with each node representing a meeting of robots, and the algorithm computes optimal meeting locations and connecting robot trajectories. The tree structure is exploited by using dynamic programming to compute solutions in two passes through the tree: an upwards pass computing the cost of all potential solutions, and a downwards pass computing optimal trajectories and meeting locations. The correctness and efficiency of the algorithm are analyzed theoretically, while a continuous robot arm problem demonstrates the algorithm's practicality.