Jianliang Qian

High Order Fast Sweeping Methods for Static Hamilton---Jacobi Equations

We construct high order fast sweeping numerical methods for computing viscosity solutions of static Hamilton–Jacobi equations on rectangular grids. These methods combine high order weighted essentially non-oscillatory (WENO) approximations to derivatives, monotone numerical Hamiltonians and Gauss–Seidel iterations with alternating-direction sweepings. Based on well-developed first order sweeping methods, we design a novel approach to incorporate high order approximations to derivatives into numerical Hamiltonians such that the resulting numerical schemes are formally high order accurate and inherit the fast convergence from the alternating sweeping strategy. Extensive numerical examples verify efficiency, convergence and high order accuracy of the new methods.

Fast Sweeping Methods for Eikonal Equations on Triangular Meshes

The original fast sweeping method, which is an efficient iterative method for stationary Hamilton-Jacobi equations, relies on natural ordering provided by a rectangular mesh. We propose novel ordering strategies so that the fast sweeping method can be extended efficiently and easily to any unstructured mesh. To that end we introduce multiple reference points and order all the nodes according to their

A Fast Sweeping Method for Static Convex Hamilton-Jacobi Equations

l^p

An adaptive finite-difference method for traveltimes and amplitudes

-metrics to those reference points. We show that these orderings satisfy the two most important properties underlying the fast sweeping method: (1) these orderings can cover all directions of information propagating efficiently; (2) any characteristic can be decomposed into a finite number of pieces and each piece can be covered by one of the orderings. We prove the convergence of the new algorithm. The computational complexity of the algorithm is nearly optimal in the sense that the total computational cost consists of

Eulerian Gaussian beams for high-frequency wave propagation

O(M)

An Efficient Neumann Series-Based Algorithm for Thermoacoustic and Photoacoustic Tomography with Variable Sound Speed

flops for iteration steps and

Mountain Waves and Gaussian Beams

O(M{\rm log}M)

Finite-difference quasi-P traveltimes for anisotropic media

flops for sorting at the predetermined initialization step which can be efficiently optimized by adopting a linear time sorting method, where

A Slowness Matching Eulerian Method for Multivalued Solutions of Eikonal Equations

M

A level set-based Eulerian approach for anisotropic wave propagation

is the total number of mesh points. Extensive numerical examples demonstrate that the new algorithm converges in a finite number of iterations independent of mesh size.