High Order Wave Equation Algorithms for the Frequency Domain

波的一个决定性特征是它们在不改变形状的情况下传播信息的能力。正是这种能力使波浪能够探测和成像人体、地球内部以及桥梁和隧道等工程结构。然后，这些图像可以转化为科学和工程知识，用于改进医疗诊断，防止建筑物和机械设备的故障。在这个项目中，首席研究员将开发计算模拟工具，以提高我们为共同利益利用波传播特性的能力。该项目开发的工具还可用于设计先进的材料，这些材料可以实现更好的声学、弹性和电磁元件，以及更快、更准确的传感技术。作为这项工作的一部分，学生将接受培训。该研究将进一步发展和应用先进的计算方法来求解模拟波传播的偏微分方程系统。在有效利用新兴计算体系结构的同时，近似方法将被设计得鲁棒和灵活。这项研究将极大地改进WaveHoltz方法，这是最近发现的一种思想，可以使用波动方程的时域方法来设计频域亥姆霍兹型求解器。WaveHoltz值得注意的是，它的底层线性算子对应于对称正定矩阵，并且允许解决强制问题而不是高度不确定的亥姆霍兹问题。研究将分析和开发波前预处理器和预压缩技术，用于预处理波霍兹；设计隐式和显式误差校正方法来消除WaveHoltz方法中的时间误差；并考虑多频率版本的WaveHoltz方法。该奖项反映了美国国家科学基金会的法定使命，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。

A defining feature of waves is their ability to carry information over large distances by propagating without changing their shape. It is this ability that allow waves to probe and image the human body, the interior of the earth and engineered structures like bridges and tunnels. Such images can then be turned into scientific and engineering knowledge that can be used to improve medical diagnostics and prevent failure of buildings and mechanical devices. In this project the principal investigator will develop computational simulation tools that increases our ability to exploit the properties of wave propagation for the common good. The tools developed in the project can also be used to design advanced materials that can enable better acoustic, elastic and electromagnetic components as well as faster and more accurate sensing technologies. Students will be trained as a part of this work. The research will further develop and apply advanced computational methods for solving systems of partial differential equations modeling wave propagation. The approximation methods will be designed to be robust and flexible while effectively utilizing emerging computational architectures. The research will dramatically improve the WaveHoltz method, a recently discovered idea that enables the use of time domain methods for wave equations to design frequency domain Helmholtz type solvers. WaveHoltz is remarkable in that its underlying linear operator corresponds to a symmetric positive definite matrix and allows a coercive problem to be solved rather than a highly indefinite Helmholtz problem. The research will analyze and develop wavefront preconditioners and deflation techniques for preconditioning WaveHoltz; design implicit and explicit error corrected methods for removing the temporal error in the WaveHoltz method; and consider multi-frequency versions of the WaveHoltz method.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.