Numerical methods for static Hamilton-Jacobi equations

Crandall and Lions [23] introduced the concept of viscosity solutions which provides a foundation for studying the Hamilton-Jacobi equations both theoretically and numerically. Ever since then, computing the viscosity solutions numerically has become very important in a variety of applications. A lot of numerical methods have been developed to compute the viscosity solutions.;We study the convergence of classical monotone upwind schemes, for example the fast sweeping method, for static convex Hamilton-Jacobi equations by analyzing a contraction property of such schemes. Heuristic error estimate is discussed, and the convergence proof through the Hopf formula in control theory is also studied. Monotone upwind schemes are at most first order [51]. In order to improve the accuracy when there is source singularity, we introduce a new fast sweeping method for the factored Eikonal equation, which improves the accuracy of original fast sweeping method on the Eikonal equation by resolving the source singularity with an underlying correction function. This new factorization idea comes from problems in geosciences. And it provides a possible procedure for source singularity resolution in other problems.;Furthermore, high order schemes are also important in many applications, for example the high frequency wave propagation. The ENO or WENO technique seems to be the popular one. But methods based on ENO or WENO are often slower to converge. They are based on direction by direction approximations with wide stencils to capture smoother approximations of second derivatives. We develop a compact upwind second order scheme for the Eikonal equations by observing a superconvergence phenomena of classical monotone upwind schemes: the numerical gradient of such first order schemes is also first order. The new second order scheme combines this phenomena with the Lagrangian structure of the equations. The stencil can be reduced, and it is upwind.;As an application of the fast sweeping method, we apply the method in computer vision by introducing a distance-ordered-homotopic thinning algorithm for computing the skeleton of an object represented by point clouds. This algorithm uses the closest point information calculated efficiently by the fast sweeping method.;Further possible ideas on developing fast sweeping methods for static non-convex Hamilton-Jacobi equations are also discussed in the conclusion.