Spatiospectral concentration on a sphere

We pose and solve the analogue of Slepian's time-frequency concentration problem on the surface of the unit sphere to determine an orthogonal family of strictly bandlimited functions that are optimally concentrated within a closed region of the sphere, or, alternatively, of strictly spacelimited functions that are optimally concentrated within the spherical harmonic domain. Such a basis of simultaneously spatially and spectrally concentrated functions should be a useful data analysis and representation tool in a variety of geophysical and planetary applications, as well as in medical imaging, computer science, cosmology and numerical analysis. The spherical Slepian functions can be found either by solving an algebraic eigenvalue problem in the spectral domain or by solving a Fredholm integral equation in the spatial domain. The associated eigenvalues are a measure of the spatiospectral concentration. When the concentration region is an axisymmetric polar cap the spatiospectral projection operator commutes with a Sturm-Liouville operator; this enables the eigenfunctions to be computed extremely accurately and efficiently, even when their area-bandwidth product, or Shannon number, is large. In the asymptotic limit of a small concentration region and a large spherical harmonic bandwidth the spherical concentration problem approaches its planar equivalent, which exhibits self-similarity when the Shannon number is kept invariant.Comment: 48 pages, 17 figures. Submitted to SIAM Review, August 24th, 200

Wavelet Radiosity

Radiosity methods have been shown to be an effective means to solve the global illumination problem in Lambertian diffuse environments. These methods approximate the radiosity integral equation by projecting the unknown radiosity function into a set of basis functions with limited support resulting in a set of n linear equations where n is the number of discrete elements in the scene. Classical radiosity methods required the evaluation of n2 interaction coefficients. Efforts to reduce the number of required coefficients without compromising error bounds have focused on raising the order of the basis functions, meshing, accounting for discontinuities, and on developing hierarchical approaches, which have been shown to reduce the required interactions to O(n). In this paper we show that the hierarchical radiosity formulation is an instance of a more general set of methods based on wavelet theory. This general framework offers a unified view of both higher order element approaches to radiosity and the hierarchical radiosity methods. After a discussion of the relevant theory, we discuss a new set of linear time hierarchical algorithms based on wavelets such as the multiwavelet family and a flatlet basis which we introduce. Initial results of experimentation with these basis sets are demonstrated and discussed.Engineering and Applied Science

Multi-solver schemes for electromagnetic modeling of large and complex objects

The work in this dissertation primarily focuses on the development of numerical algorithms for electromagnetic modeling of large and complex objects. First, a GPU-accelerated multilevel fast multipole algorithm (MLFMA) is presented to improve the efficiency of the traditional MLFMA by taking advantage of GPU hardware advancement. The proposed hierarchical parallelization strategy ensures a high computational throughput for the GPU calculation. The resulting OpenMP-based multi-GPU implementation is capable of solving real-life problems with over one million unknowns with a remarkable speedup. The radar cross sections (RCS) of a few benchmark objects are calculated to demonstrate the accuracy of the solution. The results are compared with those from the CPU-based MLFMA and measurements. The capability and efficiency of the presented method are analyzed through the examples of a sphere, an aircraft, and a missile-like object. Compared with the 8-threaded CPU-based MLFMA, the OpenMP-CUDA-MLFMA method can achieve from 5 to 20 times total speedup. Second, an efficient and accurate finite element--boundary integral (FE-BI) method is proposed for solving electromagnetic scattering and radiation problems. A mixed testing scheme, in which the Rao-Wilton-Glisson and the Buffa-Christiansen functions are both employed as the testing functions, is first presented to improve the accuracy of the FE-BI method. An efficient absorbing boundary condition (ABC)-based preconditioner is then proposed to accelerate the convergence of the iterative solution. To further improve the efficiency of the total computation, a GPU-accelerated MLFMA is applied to the iterative solution. The RCSs of several benchmark objects are calculated to demonstrate the numerical accuracy of the solution and also to show that the proposed method not only is free of interior resonance corruption, but also has a better convergence than the conventional FE-BI methods. The capability and efficiency of the proposed method are analyzed through several numerical examples, including a large dielectric coated sphere, a partial human body, and a coated missile-like object. Compared with the 8-threaded CPU-based algorithm, the GPU-accelerated FE-BI-MLFMA algorithm can achieve a total speedup up to 25.5 times. Third, a multi-solver (MS) scheme based on combined field integral equation (CFIE) is proposed. In this scheme, an object is decomposed into multiple bodies based on its material property and geometry. To model bodies with complicated materials, the FE-BI method is applied. To model bodies with homogeneous or conducting materials, the method of moments is employed. Specifically, three solvers are integrated in this multi-solver scheme: the FE-BI(CFIE) for inhomogeneous objects, the CFIE for dielectric objects, and the CFIE for conducting objects. A mixed testing scheme that utilizes both the Rao-Wilton-Glisson and the Buffa-Christiansen functions is adopted to obtain a good accuracy of the proposed multi-solver algorithm. In the iterative solution of the combined system, the MLFMA is applied to accelerate computation and reduce memory costs, and an ABC-based preconditioner is employed to speed up the convergence. In the numerical examples, the individual solvers are first demonstrated to be well conditioned and highly accurate. Then the validity of the proposed multi-solver scheme is demonstrated and its capability is shown by solving scattering problems of electrically large missile-like objects. Fourth, a MS scheme based on Robin transmission condition (RTC) is proposed. Different from the FE-BI method that applies BI equations to truncate the FE domain, this proposed multi-solver scheme employs both FE and BI equations to model an object along with its background. To be specific, the entire computational domain consisting of the object and its background is first decomposed into multiple non-overlapping subdomains with each modeled by either an FE or BI equation. The equations in the subdomains are then coupled into a multi-solver system by enforcing the RTC at the subdomain interfaces. Finally, the combined system is solved iteratively with the application of an extended ABC-based preconditioner and the MLFMA. To obtain an accurate solution, both the Rao-Wilton-Glisson and the Buffa-Christiansen functions are employed as the testing functions to discretize the BI equations. This scheme is applied to a variety of benchmark problems and the scattering from an aircraft with a launched missile to demonstrate its accuracy, versatility, and capability. The proposed scheme is compared with the MS-CFIE to illustrate the differences between the two schemes. Fifth, to further improve the modeling capability, an accelerated MS method is developed on distributed computing systems to simulate the scattering from very large and complex objects. The parallelization strategy is to parallelize different subdomains individually, which is different from the parallelized domain decomposition methods, where the subdomains are handled in parallel. The multilevel fast multipole algorithm is parallelized to enable computation on many processors. The modeling strategy using the MS-RTC method is also discussed so that one can easily follow the guideline to model large and complex objects. Numerical examples are given to show the parallel efficiency of the proposed strategy and the modeling capability of the proposed method. Finally, the specific absorption rate (SAR) in a human head at 5G frequencies is simulated by taking advantage of the MS-RTC method. Based on the strong skin effect, the human head model is first simplified to reduce the computation cost. Then the MS-RTC method is applied to model the human head. Numerical examples show that the MS method is very efficient in solving electromagnetic fields in the human head and the simplified human head model can be used in the SAR simulation with an acceptable accuracy