Collaborative Research: Nonoscillatory Phase Methods for the Variable Coefficient Helmholtz Equation in the High-Frequency Regime

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。本项目涉及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应用中都有应用。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。该项目为研究生提供了研究培训机会。非均匀介质中波的散射的数值模拟在雷达和声纳、医学成像、地球物理以及许多其他科学应用中有着重要的应用。在许多感兴趣的情况下，这样的模拟可以通过求解变系数亥姆霍兹方程来执行。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，其中一位研究人员为变系数亥姆霍兹方程开发了一种新的求解器，这种求解器实现了极高的精度，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。该项目旨在将其扩展到一般情况，以开发一种比当前技术快得多的变系数亥姆霍兹方程的方法。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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 project 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 applications. 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 graduate students. The numerical simulation of the scattering of waves from inhomogeneous media has important applications in radar and sonar, medical imaging, geophysics, and a host of other scientific applications. In many cases of interest, such simulations 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, one of the investigators 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. This project aims to extend them to the general case to develop a method for the variable coefficient Helmholtz equation that is significantly faster than current techniques.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.