  We study the discretization of the Escape Time problem: find the length of
the shortest path joining an arbitrary point of a domain, to the domain's
boundary. Path length is measured locally via a Finsler metric, potentially
asymmetric and strongly anisotropic. This Optimal Control problem can be
reformulated as a static Hamilton Jacobi, or Anisotropic Eikonal, Partial
Differential Equation, as well as a front propagation model. It has numerous
applications, ranging from motion planning to image segmentation.
  We introduce a new algorithm, Fast Marching using Anisotropic Stencil
Refinement (FM-ASR), which addresses this problem on a two dimensional domain
discretized on a cartesian grid. The local stencils used in our discretization
are produced by arithmetic means. The complexity of the FM-ASR, in an average
sense over all grid orientations, only depends (poly-)logarithmically on the
anisotropy ratio of the metric, while most alternative approaches have a
polynomial dependence. Numerical experiments show, in several occasions, that
the accuracy/complexity compromise is improved by an order of magnitude or
more.
