| We develop a family of fast methods approximating the solution to a wide class of static Hamilton-Jacobi partial differential equations. These partial differential equations are considered in the context of control-theoretic and front-propagation problems.; In general, to produce a numerical solution to such a problem, one has to solve a large system of coupled non-linear discretized equations. Our techniques use partial information about the characteristic directions to de-couple the system.; Previously known fast methods, available for isotropic problems, are discussed in detail. We introduce a family of new Ordered Upwinding Methods (OUM) for general (anisotropic) problems and prove convergence to the viscosity solution of the corresponding Hamilton-Jacobi partial differential equation. The hybrid methods introduced here are based on our analysis of the role played by anisotropy in the context of front propagation and optimal trajectory problems.; The performance of the methods is analyzed and compared to that of several other numerical approaches to these problems. Computational experiments are performed using test problems from control theory, computational geometry and seismology. |
