Development of Efficient Algorithms for Computational Electromagnetism and Computerized Tomography

这个项目的第一部分是开发快速、高精度的算法来求解某些描述物体表面的电磁波散射和光子晶体中的波传播的椭圆型偏微分方程组的特征值和边值问题。这项工作源于作者最近提出的一套高效的数值技术(与作者合作)。加州理工大学布鲁诺)，以解决声散射问题。由于散射方程中超奇异核的不同结构和光子学本征问题的不同数学性质，这些方法允许计算数百波长大小物体的散射场，相对精度在0.0001到0.000001的范围内，并不直接适用于本项目的问题。然而，这种非常有希望的通用方法的进一步发展保证了新的计算方法具有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(理论上精确的)算法和寻找新的单光子发射计算机层析成像(SPECT)的解析反演公式。特别是，作者最近发现的显式反演公式被用来发展一种由非平面方向的平行束测量数据重建三维SPECT的有效算法。此外，本文还求出了由减少的(在180度角度范围内收集的)测量数据集重建二维SPECT的实际重要问题的解析解。在几个广泛的应用领域中出现了对高效计算方法的需求。例如，非侵入性测试、遥感以及最重要的雷达和雷达扩散技术需要非常精确地解决表面散射问题。研究人员开发了新的、快速的、非常准确的方法来计算这类问题的解。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。光子带隙结构是一种人造材料，有望成为下一代光通信网络以及明天的光学和量子计算技术的主要部件。最后，本项目所产生的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.