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也不过分。这种计算已经成为科学研究和工业研究与发展的重要工具。本研究涉及波散射的数值模拟。这种模拟可以应用于声纳和雷达，以及医学成像、地球物理学和许多其他领域。随着波的频率增加，波现象的建模变得更加复杂，而我们目前精确模拟高频波的能力相当有限。该项目旨在开发高效、高精度的高频波建模新方法。本项目为博士后、研究生和本科生提供科研培训机会。在许多令人感兴趣的情况下，波散射的数值模拟可以通过求解变系数亥姆霍兹方程来进行。该方程的解是振荡的，用常规方法计算的难度随着振荡频率的增加而迅速增加。最近，研究者开发了一类新的求解变系数亥姆霍兹方程的求解器，它具有极高的精度，并且随着频率的增加，运行时间比传统求解器慢得多。它们通过求解非线性里卡第方程来运行，而里卡第方程是由亥姆霍兹方程解的对数满足的。目前，这些解算器只适用于特殊情况。本研究计划的主要目的是将它们扩展到一般情况，以开发一种比现有技术快得多的变系数亥姆霍兹方程的方法。

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.