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

合作研究：高频域下变系数亥姆霍兹方程的非振荡相法

基本信息

批准号：
2012606
负责人：
Per-Gunnar Martinsson
金额：
$ 11.19万
依托单位：
University of Texas at Austin
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-09-01 至 2023-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2012606&HistoricalAwards=false
关键词：
Collaborative Research Nonoscillatory Phase Methods

项目摘要

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 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.

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。本课题是关于波浪散射的数值模拟。这种模拟可以应用于声纳和雷达，以及医学成像，生物物理学和许多其他应用。随着波浪频率的增加，波浪现象的建模变得更加复杂，而我们目前精确建模高频波浪的能力非常有限。本项目旨在开发新的方法，以有效地模拟高频波和高精度。该项目为研究生提供研究培训机会。非均匀介质中波散射的数值模拟在雷达和声纳、医学成像、生物物理学以及许多其他科学应用中具有重要的应用。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。该方程的解是振荡的，并且使用常规方法计算它们的难度随着振荡的频率快速增长。最近，一位研究人员开发了一种新的变系数亥姆霍兹方程求解器，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Riccati方程来操作，该非线性Riccati方程由Helmholtz方程的解的矩阵来满足。目前，这些求解器仅适用于特殊情况。该项目旨在将其扩展到一般情况，以开发一种比现有技术快得多的变系数亥姆霍兹方程方法。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。