Development of Efficient Algorithms for Computational Electromagnetism and Computerized Tomography

本项目的第一部分是开发快速、高阶、精确的算法，用于求解某些描述电磁波在物体表面散射和光子晶体中波传播的椭圆偏微分方程的特征值和边值问题。这项工作是由作者最近提出的一套高效的数字技术发展而来的。布鲁诺，加州理工学院)的声学散射问题。这些方法允许计算百波长大小的物体的散射场，相对精度在0.0001到0.000001之间，但由于散射方程中超奇异核的结构不同，以及光子学特征问题的不同数学性质，这些方法不能直接适用于当前项目的问题。然而，这种非常有前途的通用方法的进一步发展承诺新的计算方法具有相同的前所未有的精度和效率。该项目的第二部分致力于设计解析（理论上精确）算法，并为单光子发射计算机断层扫描（SPECT）寻找新的解析反演公式。特别地，利用作者最近发现的显式反演公式，开发了一种有效的算法，用于从非平面方向进行的平行光束测量的3-DSPECT重建。此外，本文还对在180度角范围内采集的数据集进行二维SPECT重构的实际问题寻求了解析解。对高效计算方法的需求出现在几个广泛的应用领域。例如，非侵入性测试、遥感，以及最重要的雷达和雷达入侵技术，都需要非常精确地解决表面散射问题。研究者开发新的、快速的、非常精确的方法来计算这些问题的解决方案。新的计算技术在光子带隙结构（光子晶体）的理论和设计中得到了应用——光子晶体是一种人造材料，预计将成为下一代光通信网络设备的主要组成部分，以及未来的光学和量子计算技术。最后，本课题提出的新的SPECT分析方法和算法对于研究患者血流异常的功能医学断层扫描的理论和实践具有重要意义。

Kunyansky The first part of this project consists in developing fast,high-order accurate algorithms for solving eigenvalue andboundary value problems for certain elliptic partial differentialequations that describe electromagnetic wave scattering bysurfaces of objects and wave propagation in photonic crystals.This work grows from a set of highly efficient numericaltechniques recently proposed by the author (in collaboration withO. Bruno, Caltech) for problems of acoustic scattering. Thosemethods, which permit computing of scattering fields fromhundred-wavelength-sized objects, with relative accuracies in therange from 0.0001 to 0.000001, are not directly applicable to theproblems of the current project, due to a different structure ofhypersingular kernels in the scattering equations and a differentmathematical nature of the eigenproblems of photonics. However,further development of this very promising general approachpromises new computational methods of the same unprecedentedaccuracy and efficiency. The second part of the project isdevoted to the design of analytic (theoretically exact)algorithms and search for new analytic inversion formulas forSingle Photon Emission Computed Tomography (SPECT). Inparticular, explicit inversion formulas recently discovered bythe author are used to develop an efficient algorithm for 3-DSPECT reconstruction from parallel beam measurements made fromnon-planar directions. Also, an analytic solution is sought to apractically important problem of 2-D SPECT reconstruction from areduced (collected in the 180 degree angular range) set ofmeasurements. The need for efficient computational methods arises inseveral broad areas of application. For example, non-invasivetesting, remote sensing and, most importantly, radar and radarevasion technologies require very accurate solution of surfacescattering problems. The investigator develops new, fast, veryaccurate methods for computing solutions of such problems. Thenew computational techniques find applications in the theory anddesign of photonic bandgap structures (photonic crystals) --artificial man-made materials that are anticipated to become amain building block of devices for the next generation of opticalcommunication networks, and for optical and quantum computingtechnologies of tomorrow. Finally, the novel analytic methods andalgorithms of SPECT that result from this project are importantfor the theory and practice of functional medical tomography,studying abnormalities of blood flow in patients.