Asymptotically Improved Solvers for the Helmholtz Equation

点击翻译按钮获取中文摘要

The importance of the numerical simulation of physical phenomena by computers cannot be overstated. Such computations have become essential tools both in scientific research and in industrial research and development. This research concerns the numerical simulation of the scattering of waves. Such simulations have applications to sonar and radar, as well as in medical imaging, geophysics, and many other areas. Wave phenomena become more complicated to model as the frequency of the wave increases, and our current ability to accurately model high-frequency waves is quite limited. This project seeks to develop new methods for modeling high-frequency waves efficiently and to high accuracy. The project provides research training opportunities for postdoctoral researchers, graduate students and undergraduates. In many cases of interest, the numerical simulations of the scattering of waves can be performed by solving the variable coefficient Helmholtz equation. The solutions of this equation are oscillatory, and the difficulty of calculating them using conventional approaches grows quickly with the frequency of the oscillations. Recently, the investigator has developed a new class of solvers for the variable coefficient Helmholtz equation that achieve extremely high accuracy and have run times that scale much more slowly with increasing frequency than conventional solvers. They operate by solving the nonlinear Riccati equation that is satisfied by the logarithms of solutions of the Helmholtz equation. Currently, these solvers only apply in special cases. The main thrust of this research program is to extend them to the general case to develop a method for the variable coefficient Helmholtz equation that is significantly faster than current techniques.