Browsing by Subject "Matrix vector multiplication"
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Item Open Access Advanced partitioning and communication strategies for the efficient parallelization of the multilevel fast multipole algorithm(IEEE, 2010) Ergül O.; Gürel, LeventLarge-scale electromagnetics problems can be solved efficiently with the multilevel fast multipole algorithm (MLFMA) [1], which reduces the complexity of matrix-vector multiplications required by iterative solvers from O(N 2) to O(N logN). Parallelization of MLFMA on distributed-memory architectures enables fast and accurate solutions of radiation and scattering problems discretized with millions of unknowns using limited computational resources. Recently, we developed a hierarchical partitioning strategy [2], which provides an efficient parallelization of MLFMA, allowing for the solution of very large problems involving hundreds of millions of unknowns. In this strategy, both clusters (sub-domains) of the multilevel tree structure and their samples are partitioned among processors, which leads to improved load-balancing. We also show that communications between processors are reduced and the communication time is shortened, compared to previous parallelization strategies in the literature. On the other hand, improved partitioning of the tree structure complicates the arrangement of communications between processors. In this paper, we discuss communications in detail when MLFMA is parallelized using the hierarchical partitioning strategy. We present well-organized arrangements of communications in order to maximize the efficiency offered by the improved partitioning. We demonstrate the effectiveness of the resulting parallel implementation on a very large scattering problem involving a conducting sphere discretized with 375 million unknowns. ©2010 IEEE.Item Open Access Computational analysis of complicated metamaterial structures using MLFMA and nested preconditioners(IEEE, 2007-11) Ergül, Özgür; Malas, Tahir; Yavuz, Ç; Ünal, Alper; Gürel, LeventWe consider accurate solution of scattering problems involving complicated metamaterial (MM) structures consisting of thin wires and split-ring resonators. The scattering problems are formulated by the electric-field integral equation (EFIE) discretized with the Rao-Wilton- Glisson basis functions defined on planar triangles. The resulting dense matrix equations are solved iteratively, where the matrix-vector multiplications that are required by the iterative solvers are accelerated with the multilevel fast multipole algorithm (MLFMA). Since EFIE usually produces matrix equations that are ill-conditioned and difficult to solve iteratively, we employ nested preconditioners to achieve rapid convergence of the iterative solutions. To further accelerate the simulations, we parallelize our algorithm and perform the solutions on a cluster of personal computers. This way, we are able to solve problems of MMs involving thousands of unit cells.Item Open Access Fast and accurate analysis of large metamaterial structures using the multilevel fast multipole algorithm(2009) Gürel, Levent; Ergül, Özgür; Ünal, A.; Malas, T.We report fast and accurate simulations of metamaterial structures constructed with large numbers of unit cells containing split-ring resonators and thin wires. Scattering problems involving various metamaterial walls are formulated rigorously using the electric-field integral equation, discretized with the Rao-Wilton-Glisson basis functions. Resulting dense matrix equations are solved iteratively, where the matrix-vector multiplications are performed efficiently with the multilevel fast multipole algorithm. For rapid solutions at resonance frequencies, convergence of the iterations is accelerated by using robust preconditioning techniques, such as the sparse-approximate-inverse preconditioner. Without resorting to homogenization approximations and periodicity assumptions, we are able to obtain accurate solutions of realistic metamaterial problems discretized with millions of unknowns.Item Open Access ON two-dimensional sparse matrix partitioning: models, methods, and a recipe(Society for Industrial and Applied Mathematics, 2010) Çatalyürek, U. V.; Aykanat, Cevdet; Uçar, A.We consider two-dimensional partitioning of general sparse matrices for parallel sparse matrix-vector multiply operation. We present three hypergraph-partitioning-based methods, each having unique advantages. The first one treats the nonzeros of the matrix individually and hence produces fine-grain partitions. The other two produce coarser partitions, where one of them imposes a limit on the number of messages sent and received by a single processor, and the other trades that limit for a lower communication volume. We also present a thorough experimental evaluation of the proposed two-dimensional partitioning methods together with the hypergraph-based one-dimensional partitioning methods, using an extensive set of public domain matrices. Furthermore, for the users of these partitioning methods, we present a partitioning recipe that chooses one of the partitioning methods according to some matrix characteristics. © 2010 Society for Industrial and Applied Mathematics.Item Open Access Solution of low-frequency electromagnetics problems using hierarchical matrices(IEEE, 2013) Kazempour, Mahdi; Gürel, LeventFast and accurate solutions of low-frequency electromagnetics problems are obtained with an iterative solver based on hierarchical matrices. Iterative solvers require matrix-vector multiplications (MVMs). The results show significant reductions both in CPU time and memory consumption compared to the O(N2) complexity of ordinary matrix filling and MVM.