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

准确和快速的数值解的亥姆霍兹和相关的偏微分方程在复杂的几何形状是关键的未来的进展，在设备设计，成像和基础科学。 然而，在高频率下（系统中的许多波长），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。 研究人员试图建立在边界为基础的方法，已经独特的成功（比竞争对手快一千倍），在解决本征模问题的波长数百的大小与光谱精度在两个维度，并将其扩展到散射问题，更一般的介质和周期性的边界条件，并三维。 这些方法是通过特殊解基组的全局近似，以及Dirichlet本征模的标度方法。建议的扩展包括：1）基本解基组的使用，以及它们通过波场的解析延拓中奇点的作用的分析，2）利用线性梯度折射率材料中的基本解的鲜为人知的解析公式，使得能够在边界上求解非分段常数介质，3）通过域的Dirichlet-to-Neumann映射对缩放方法的重新表述进行误差分析，4）将这种方法应用于电介质光子晶体带结构的光谱精确解，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器件等的影响已经并将继续是深刻和深远的。 为了设计所有这些装置，必须计算它们将如何反射、引导和捕获波，这是一个耗时、困难和有时不可靠的计算，研究人员提出的计算机算法将使这种计算更快和更准确，特别是当物体较大或形状复杂时。 预计这将导致光学信号处理设备（依赖于光波长大小的微观周期性结构）的设计改进，有望成为下一代快速（后硅）计算机的候选者。 对量子混沌（被困在空腔中的波的行为，导致混乱的反弹）的更深入的理解将影响纳米级量子波器件，如量子点，超高速量子计算机，以及纯数学和物理理论领域。 该提案还提供研究生和本科生应用和计算数学方面的培训，以及“音乐和声音数学”课程，向非专业人员介绍波、模式和共振。

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.