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

精确、快速地数值求解复杂几何中的Helmholtz方程和相关偏微分方程组，是未来器件设计、成像和基础科学发展的关键。然而，在高频(整个系统的许多波长)，由于问题的多尺度性质，使用直接离散化变得令人望而却步。研究人员试图建立基于边界的方法，这些方法在解决具有二维光谱精度的数百个波长的本征模式问题方面取得了独特的成功(比竞争对手快了1000倍)，并将它们扩展到散射问题，更一般的介质和周期边界条件，以及三维。这些方法是特解基集的整体逼近和Dirichlet本征模的标度方法。建议的扩展包括：1)使用基本解基集，并通过奇点在波场的解析延拓中的作用来分析它们，2)利用线性梯度折射率材料中鲜为人知的基本解的解析公式，使非分段常数介质能够在边界上求解，3)通过区域的Dirichlet-to-Neumann映射对标度方法的重新公式进行误差分析，4)将这种方法应用于介电光子晶体能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、医学超声成像和微型量子设备，已经并将继续产生深远的影响。要设计所有这些设备，必须计算它们将如何反射、引导和捕获波，这是一项耗时、困难、有时不可靠的计算。研究人员提出的计算机算法将使此类计算更快、更准确，特别是在物体较大或形状复杂的情况下。例如，这有望导致光学信号处理设备(依赖于光波大小的微观周期结构)设计的改进，有望成为下一代快速(后硅)计算机的候选者。对量子混沌(囚禁在空穴中的波的行为，导致射线的混沌反弹)的更深入理解将影响到纳米级的量子波设备，如量子点、超快量子计算机，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了应用数学和计算数学的培训，并开设了一门课程，向非主修学生介绍波、调子和共鸣的音乐与声音数学。

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.