Efficient spectrally accurate global basis methods for high frequency wave scattering, chaotic eigenmodes, and photonics

复杂几何中亥姆霍兹方程和相关偏微分方程的精确、快速数值解是器件设计、成像和基础科学未来发展的关键。然而，在高频率（整个系统的许多波长）下，由于问题的多尺度性质，使用直接离散化变得非常具有挑战性。研究者试图建立基于边界的方法，这些方法在解决二维光谱精度的数百波长的特征模式问题方面取得了独特的成功（比竞争对手快1000倍），并将其扩展到散射问题，更一般的介质和周期性边界条件，以及三维。这两种方法分别是特解基集的全局逼近法和狄利克雷特征模的标度法。建议的扩展包括：1)使用基本解基集，并通过奇点在波场解析延拓中的作用对其进行分析；2)利用一个鲜为人知的线性梯度折射率材料的基本解解析公式，使非分段常数介质能够在边界上求解；3)通过域的Dirichlet-to-Neumann映射对标度方法的重新表述进行误差分析。4)将这些方法应用于介电光子晶体带结构的光谱精确解，以及“量子混沌”（具有遍历射线动力学的空腔的波和光谱特性）。我们的技术，如雷达、微波通信（如手机）、光学和激光、声学、医学超声成像和小型化量子设备，已经并将继续产生深远的影响。为了设计所有这些装置，必须计算它们如何反射、引导和捕获波，这是一个耗时、困难、有时不可靠的计算。研究人员提出的计算机算法将使这种计算更快、更准确，特别是当物体很大或形状复杂时。这有望导致设计上的改进，例如，光信号处理设备（依赖于光波长大小的微观周期结构），有望成为下一代快速（后硅）计算机的候选者。更深入地掌握量子混沌（被困在空腔中的波的行为，导致光线的混沌反弹）将影响纳米级量子波设备，如量子点、超高速量子计算机，以及纯数学和物理理论领域。该提案还为研究生和本科生提供应用数学和计算数学的培训，并开设一门名为“音乐和声音的数学”的课程，向非专业学生介绍波、模态和共振。

Accurate and rapid numerical solution of the Helmholtz and related partial differential equations in complex geometries is key to future progress in device design, in imaging, and in basic science. However, at high frequencies (many wavelengths across the system) this becomes prohibitively challenging using direct discretization, due to the multiscale nature of the problem. The investigator seeks to build upon boundary-based methods which have been uniquely successful (up to a thousand times faster than the competition) in solving eigenmode problems hundreds of wavelength in size with spectral accuracy in two dimensions, and to extend them to the scattering problem, to more general media and periodic boundary conditions, and to three dimensions. These methods are global approximation by particular solution basis sets, and the scaling method for Dirichlet eigenmodes.Proposed extensions include: 1) use of fundamental solutions basis sets, and their analysis via the role of singularities in the analytic continuation of the wave field, 2) exploiting a little-known analytic formula for the fundamental solution in linear graded-index materials, enabling non-piecewise-constant media to be solved on the boundary, 3) error analysis of a reformulation of the scaling method via the Dirichlet-to-Neumann map for the domain, 4) application of such methods to the spectrally accurate solution of dielectric photonic crystal band structure, and to `quantum chaos' (the wave and spectral properties of cavities with ergodic ray dynamics).The impact of our technology such as radar, microwave communication (eg cellphones), optics and lasers, acoustics, medical ultrasound imaging, and miniaturized quantum devices has been, and will continue to be, profound and far-reaching. To design all such devices, one must calculate how they will reflect, guide and trap waves, and this is a time-intensive, difficult and sometimes unreliable computation.The computer algorithms proposed by the investigator will make such calculations faster and more accurate, particularly when the objects are large or complicated in shape. This is expected to lead to improvements in the design of, for example, optical signal-processing devices (which rely on microscopic periodic structures the size of the wavelength of light), promising candidates for the next generation of fast (post-silicon) computers. A deeper grasp of quantum chaos (the behavior of waves trapped in cavities which cause chaotic bouncing ofrays) would impact nanoscale quantum wave devices such as quantum dots, super-fast quantum computers, as well as areas of pure mathematics and physics theory. The proposal also provides training in applied and computational mathematics at both graduate and undergraduate levels, and a course on the ``Mathematics of Music and Sound'' introducing non-majors to waves, modes, and resonance.