Development of Efficient Algorithms for Computational Electromagnetism and Computerized Tomography

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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.