In some applications, it is reasonable to assume that geodesics (rays) have a consistent orientation so that the Helmholtz equation may be viewed as an evolution equation in one of the spatial directions. With such applications in mind, we propose a new Eulerian computational geometrical-optics method, dubbed the fast Huygens sweeping method, for computing Green's functions of the Helmholtz equations in inhomogeneous media in the high-frequency regime and in the presence of caustics. The first novelty of the new method is that the Huygens-Kirchhoff secondary source principle is used to integrate many locally valid asymptotic solutions to yield a globally valid asymptotic solution so that caustics associated with the usual geometrical-optics ansatz can be treated automatically. The second novelty is that a butterfly algorithm is adapted to carry out the matrix-vector products induced by the Huygens-Kirchhoff integration in O(Nlog N) operations, where N is the total number of mesh points, and the proportionality constant depends on the desired accuracy and is independent of the frequency parameter. Both two-dimensional (2-D) and three-dimensional (3-D) numerical experiments are presented to demonstrate the performance and accuracy of the new method.

Dr. Jianliang Qian currently is a Professor of Mathematics at Michigan State University in East Lansing, Michigan. He earned his Ph.D. degree at Rice University under the supervision of William W. Symes in May 2000. From September 2000 to August 2002, he was a Postdoctoral Research Associate at the Institute for Mathematics and Applications (IMA), University of Minnesota. From September 2002 to July 2005, he was a CAM Assistant Professor at UCLA. From August 2005 to July 2007, he was an Assistant Professor at Wichita State University. From August 2007 to June 2010, he was an Assistant Professor at Michigan State University. He was promoted to be an Associate Professor with tenure at Michigan State University in July 2010 and was promoted to be a Full Professor in 2015. He has published more than 70 journal papers. He has organized the 2010 IMA Participating Institute Graduate Summer School on Computational Wave Propagation. His research interest is on high frequency wave propagation and related applications in inverse problems, seismic imaging and medical imaging. His research is mainly supported by NSF, and his cumulative research budget is of almost $2M.