Computational Electromagnetics Research Center (BİLCEM)

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Browsing Computational Electromagnetics Research Center (BİLCEM) by Author "Malas, Tahir"

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    Analysis of photonic-crystal problems with MLFMA and approximate Schur preconditioners
    (IEEE, 2009-07) Ergül, Özgür; Malas, Tahir; Kılınç, Seçil; Sarıtaş, Serkan; Gürel, Levent
    We consider fast and accurate solutions of electromagnetics problems involving three-dimensional photonic crystals (PhCs). Problems are formulated with the combined tangential formulation (CTF) and the electric and magnetic current combined-field integral equation (JMCFIE) discretized with the Rao-Wilton-Glisson functions. Matrix equations are solved iteratively by the multilevel fast multipole algorithm. Since PhC problems are difficult to solve iteratively, robust preconditioning techniques are required to accelerate iterative solutions. We show that novel approximate Schur preconditioners enable efficient solutions of PhC problems by reducing the number of iterations significantly for both CTF and JMCFIE. ©2009 IEEE.
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    Approximate MLFMA as an efficient preconditioner
    (IEEE, 2007) Malas, Tahir; Ergül, Özgür; Gürel, Levent
    In this work, we propose a preconditioner that approximates the dense system operator. For this purpose, we develop an approximate multilevel fast multipole algorithm (AMLFMA), which performs a much faster matrix-vector multiplication with some relative error compared to the original MLFMA. We use AMLFMA to solve a closely related system, which makes up the preconditioner. Then, this solution is embedded in the main solution that uses MLFMA. By taking into account the far-field elements wisely, this preconditioner proves to be much more effective compared to the near-field preconditioners.
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    Computational analysis of complicated metamaterial structures using MLFMA and nested preconditioners
    (IEEE, 2007-11) Ergül, Özgür; Malas, Tahir; Yavuz, Ç; Ünal, Alper; Gürel, Levent
    We 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.
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    An effective preconditioner based on schur complement reduction for integral-equation formulations of dielectric problems
    (IEEE, 2009) Malas, Tahir; Gürel, Levent
    The author consider effective preconditioning of recently proposed two integral-equation formulations for dielectrics; the combined tangential formulation (CTF) and the electric and magnetic current combined-field integral equation (JMCFIE). These two formulations are of utmost interest since CTF yields more accurate results and JMCFIE yields better-conditioned systems than other formulations.
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    Effective preconditioners for large integral-equation problems
    (IET, 2007-11) Malas, Tahir; Ergül, Özgür; Gürel, Levent
    We consider effective preconditioning schemes for the iterative solution of integral-equation methods. For parallel implementations, the sparse approximate inverse or the iterative solution of the near-field system enables fast convergence up to certain problem sizes. However, for very large problems, the near-field matrix itself becomes too crude approximation to the dense system matrix and preconditioners generated from the near-field interactions cannot be effective. Therefore, we propose an approximation strategy to the multilevel fast multipole algorithm (MLFMA) to be used as a preconditioner. Our numerical experiments reveal that this scheme significantly outperforms other preconditioners. With the combined effort of effective preconditioners and an efficiently parallelized MLFMA, we are able to solve targets with tens of millions of unknowns in a few hours.
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    Incomplete LU preconditioning for the electric-field integral equation
    (2006) Malas, Tahir; Gürel, Levent
    Linear systems resulting from the electric-field integral equation (EFIE) become ill-conditioned, particularly for large-scale problems. Hence, effective preconditioners should be used to obtain the iterative solution with the multilevel fast multipole algorithm in a reasonable time. In this paper, we show that a threshold-based incomplete LU (ILU) preconditioner, i.e., ILUT, can be used safely for such systems, provided that column pivoting is applied for the stability of the incomplete factors. It is observed that the resulting preconditioner ILUTP reduces the solution times by an order of magnitude, compared to simple Jacobi preconditioner. Moreover, we also use the iterative solution of the near-field system as a preconditioner, and use ILUTP as the preconditioner for the near-field system. This way, the effectiveness of the ILUTP is further improved.
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    Parallel preconditioners for solutions of dense linear systems with tens of millions of unknowns
    (2007-11) Malas, Tahir; Ergül, Özgür; Gürel, Levent
    We propose novel parallel preconditioning schemes for the iterative solution of integral equation methods. In particular, we try to improve convergence rate of the ill-conditioned linear systems formulated by the electric-field integral equation, which is the only integral-equation formulation for targets having open surfaces. For moderate-size problems, iterative solution of the near-field system enables much faster convergence compared to the widely used sparse approximate inverse preconditioner. For larger systems, we propose an approximation strategy to the multilevel fast multipole algorithm (MLFMA) to be used as a preconditioner. Our numerical experiments reveal that this scheme significantly outperforms other preconditioners. With the combined effort of effective preconditioners and an efficiently parallelized MLFMA, we are able to solve targets with tens of millions of unknowns, which are the largest problems ever reported in computational electromagnetics. ©2007 IEEE.
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    Preconditioning iterative MLFMA solutions of integral equations
    (IEEE, 2010) Gürel, Levent; Malas, Tahir; Ergül, Özgür
    The multilevel fast multipole algorithm (MLFMA) is a powerful method that enables iterative solutions of electromagnetics problems with low complexity. Iterative solvers, however, are not robust for three-dimensional complex reallife problems unless suitable preconditioners are used. In this paper, we present our efforts to devise effective preconditioners for MLFMA solutions of difficult electromagnetics problems involving both conductors and dielectrics. © 2010 IEEE.
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    Scalable parallelization of the sparse-approximate-inverse (SAl) preconditioner for the solution of large-scale integral-equation problems
    (IEEE, 2009-06) Malas, Tahir; Gürel, Levent
    In this paper, we consider efficient parallelization of the sparse approximate inverse (SAI) preconditioner in the context of the multilevel fast multipole algorithm (MLFMA). Then, we report the use of SAI in the solution of very large EFIE problems. The SAI preconditioner is important not only because it is a robust preconditioner that renders many difficult and large problems solvable, but also it can be utilized for the construction of more effective preconditioners.
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    Schur complement preconditioners for surface integral-equation formulations of dielectric problems solved with the multilevel fast multipole algorithm
    (Society for Industrial and Applied Mathematics, 2011-10-04) Malas, Tahir; Gürel, Levent
    Surface integral-equation methods accelerated with the multilevel fast multipole algorithm (MLFMA) provide a suitable mechanism for electromagnetic analysis of real-life dielectric problems. Unlike the perfect-electric-conductor case, discretizations of surface formulations of dielectric problems yield 2 × 2 partitioned linear systems. Among various surface formulations, the combined tangential formulation (CTF) is the closest to the category of first-kind integral equations, and hence it yields the most accurate results, particularly when the dielectric constant is high and/or the dielectric problem involves sharp edges and corners. However, matrix equations of CTF are highly ill-conditioned, and their iterative solutions require powerful preconditioners for convergence. Second-kind surface integral-equation formulations yield better conditioned systems, but their conditionings significantly degrade when real-life problems include high dielectric constants. In this paper, for the first time in the context of surface integral-equation methods of dielectric objects, we propose Schur complement preconditioners to increase their robustness and efficiency. First, we approximate the dense system matrix by a sparse near-field matrix, which is formed naturally by MLFMA. The Schur complement preconditioning requires approximate solutions of systems involving the (1,1) partition and the Schur complement. We approximate the inverse of the (1,1) partition with a sparse approximate inverse (SAI) based on the Frobenius norm minimization. For the Schur complement, we first approximate it via incomplete sparse matrix-matrix multiplications, and then we generate its approximate inverse with the same SAI technique. Numerical experiments on sphere, lens, and photonic crystal problems demonstrate the effectiveness of the proposed preconditioners. In particular, the results for the photonic crystal problem, which has both surface singularity and a high dielectric constant, shows that accurate CTF solutions for such problems can be obtained even faster than with second-kind integral equation formulations, with the acceleration provided by the proposed Schur complement preconditioners.
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    Sequential and parallel preconditioners for large-scale integral-equation problems
    (IEEE, 2007) Malas, Tahir; Ergül, Özgür; Gürel, Levent
    For efficient solutions of integral-equation methods via the multilevel fast multipole algorithm (MLFMA), effective preconditioners are required. In this paper we review appropriate preconditioners that have been used for sparse systems and developed specially in the context of MLFMA. First we review the ILU-type preconditioners that are suitable for sequential implementations. We can make these preconditioners robust and efficient for integral-equation methods by making appropriate selections and by employing pivoting to suppress the instability problem. For parallel implementations, the sparse approximate inverse or the iterative solution of the near-field system enables fast convergence up to certain problem sizes. However, for very large problems, the near-field matrix itself becomes insufficient to approximate the dense system matrix and preconditioners generated from the near-field interactions cannot be effective. Therefore, we propose an approximation strategy to MLFMA to be used as an effective preconditioner. Our numerical experiments reveal that this scheme significantly outperforms other preconditioners. With the combined effort of effective preconditioners and an efficiently parallelized MLFMA, we are able to solve problems with tens of millions of unknowns in a few hours. We report the solution of integral-equation problems that are among the largest in their classes. © 2007 IEEE.
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    Solution of extremely large integral-equation problems
    (IEEE, 2007) Ergül, Özgür; Malas, Tahir; Gürel, Levent
    We report the solution of extremely large integral-equation problems involving electromagnetic scattering from conducting bodies. By orchestrating diverse activities, such as the multilevel fast multipole algorithm, iterative methods, preconditioning techniques, and parallelization, we are able to solve scattering problems that are discretized with tens of millions of unknowns. Specifically, we report the solution of a closed geometry containing 42 million unknowns and an open geometry containing 20 million unknowns, which are the largest problems of their classes, to the best of our knowledge.
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    The solution of large EFIE problems via preconditioned multilevel fast multipole algorithm
    (Institution of Engineering and Technology, 2007) Malas, Tahir; Gürel, Levent
    We propose an effective preconditioning scheme for the iterative solution of the systems formulated by the electric- field integral equation (EFIE). EFIE is notorious for producing difficult-to-solve systems. Especially, if the target is complex and the utilized frequency is high, it becomes a challenge to solve these dense systems with even robust solvers such as full GMRES. For this purpose, we use an inner-outer solver scheme and use an approximate multilevel fast multipole algorithm for the inner solver to provide a very efficient approximation to the dense linear system matrix. We explore approximation level and inner-solver accuracy to optimize the efficiency of the inner-outer solution scheme. We report the solution of large EFIE systems of several targets to show the effectiveness of the proposed approach.
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    Solutions of large integral-equation problems with preconditioned MLFMA
    (IEEE, 2007) Ergül, Özgür; Malas, Tahir; Ünal, Alper; Gürel, Levent
    We report the solution of the largest integral-equation problems in computational electromagnetics. We consider matrix equations obtained from the discretization of the integral-equation formulations that are solved iteratively by employing parallel multilevel fast multipole algorithm (MLFMA). With the efficient parallelization of MLFMA, scattering and radiation problems with millions of unknowns are easily solved on relatively inexpensive computational platforms. For the iterative solutions of the matrix equations, we are able to obtain accelerated convergence even for ill-conditioned matrix equations using advanced preconditioning schemes, such as nested preconditioned based on an approximate MLFMA. By orchestrating these diverse activities, we have been able to solve a closed geometry formulated with the CFIE containing 33 millions of unknowns and an open geometry formulated with the EFIE containing 12 millions of unknowns, which are the largest problems of their classes, to the best of our knowledge.