Development of Efficient Algorithms for Computational Electromagnetism and Computerized Tomography

昆扬斯基 本项目的第一部分是发展快速、高精度的算法来求解某些椭圆型偏微分方程的特征值和边值问题，这些方程描述了电磁波在物体表面的散射和光子晶体中的传播。 Bruno，Caltech）用于声散射问题。 由于散射方程中的超奇异核的不同结构和光子学本征问题的不同数学性质，这些方法可以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。 然而，这个非常有前途的一般方法的进一步发展，保证了同样前所未有的精度和效率的新的计算方法。 第二部分致力于单光子发射计算机断层成像（SPECT）解析（理论上精确）算法的设计和新的解析反演公式的寻找。 特别地，作者最近发现的显式反演公式被用来发展一种有效的算法，从非平面方向的平行束测量的3-DSPECT重建。 本文还对从180度角范围内的测量数据集进行二维SPECT重建的一个重要问题寻求了一个解析解。 在几个广泛的应用领域中，需要有效的计算方法。 例如，非侵入性测试、遥感以及最重要的雷达和雷达防御技术都需要非常精确地解决表面散射问题。 研究人员开发新的，快速的，非常准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，有望成为下一代光通信网络设备的主要组成部分，也是未来光学和量子计算技术的主要组成部分。 最后，本课题所提出的SPECT新的分析方法和算法，对于功能医学断层成像、研究患者血流异常具有重要的理论和实践意义。

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.