Asymptotically Improved Solvers for the Helmholtz Equation

用计算机对物理现象进行数值模拟的重要性怎么强调都不为过。这样的计算已经成为科学研究和工业研究和发展中必不可少的工具。本研究是关于海浪散射的数值模拟。这种模拟在声纳和雷达以及医学成像、地球物理和许多其他领域都有应用。随着波浪频率的增加，波浪现象的建模变得更加复杂，而我们目前对高频波进行精确建模的能力相当有限。这个项目寻求开发新的方法来有效地和高精度地模拟高频波。该项目为博士后研究人员、研究生和本科生提供了研究培训机会。在许多感兴趣的情况下，波的散射可以通过求解变系数亥姆霍兹方程来进行数值模拟。这个方程的解是振荡的，使用传统方法计算它们的难度随着振荡频率的增加而迅速增加。最近，研究人员为变系数Helmholtz方程开发了一类新的求解器，它们实现了极高的精度，并且运行时间随着频率的增加而变得比传统求解器慢得多。它们通过求解由Helmholtz方程的解的对数所满足的非线性Riccati方程来操作。目前，这些解算器仅适用于特殊情况。这一研究计划的主要目的是将它们推广到一般情况，以发展一种比现有技术快得多的变系数Helmholtz方程的方法。

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.