Browsing by Subject "Preconditioners"

Now showing 1 - 11 of 11

Open Access
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.
Open Access
Approximate Schur preconditioners for efficient solutions of dielectric problems formulated with surface integral equations
(IEEE, 2009-07) Malas, Tahir; Gürel, Levent
We propose direct and iterative versions of approximate Schur preconditioners to increase robustness and efficiency of iterative solutions of dielectric problems formulated with surface integral equations. The performance of these preconditioners depends on the availability of fast and approximate solutions to reduced matrix systems. We show that sparse-approximate-inverse techniques provide a suitable mechanism for this purpose. The proposed preconditioners are demonstrated to significantly improve convergence rates of dielectric problems formulated with two different surface integral equations. ©2009 IEEE.
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, 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.
Open Access
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.
Open Access
Efficient solutions of metamaterial problems using a low-frequency multilevel fast multipole algorithm
(2010) Ergül, Özgür; Gürel, Levent
We present fast and accurate solutions of electromagnetics problems involving realistic metamaterial structures using a lowfrequency multilevel fast multipole algorithm (LF-MLFMA). Accelerating iterative solutions using robust preconditioning techniques may not be sufficient to reduce the overall processing time when the ordinary high-frequency MLFMA is applied to metamaterial problems. The major bottleneck, i.e., the low-frequency breakdown, should be eliminated for efficient solutions. We show that the combination of an LF-MLFMA implementation based on the multipole expansion with the sparse-approximate-inverse preconditioner enables efficient and accurate analysis of realistic metamaterial structures. Using the robust LF-MLFMA implementation, we demonstrate how the transmission properties of metamaterial walls can be enhanced with randomlyoriented unit cells.
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.
Open Access
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.
Open Access
Iterative solution of dielectric waveguide problems via schur complement preconditioners
(IEEE, 2010-07) Malas, Tahir; Gürel, Levent
Surface integral-equation methods accelerated with the multilevel fast multipole algorithm provide suitable mechanisms for the solution of dielectric problems. In particular, recently developed formulations increase the stability of the resulting matrix equations, hence they are more suitable for iterative solutions [1]. Among those formulations, we consider the combined tangential formulation (CTF), which produces more accurate results, and the electric and magnetic current combined-field integral equation (JMCFIE), which produces better-conditioned matrix systems than other formulations [1, 2]. ©2010 IEEE.
Open Access
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.
Open Access
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.
Open Access
Solutions of large-scale electromagnetics problems using an iterative inner-outer scheme with ordinary and approximate multilevel fast multipole algorithms
(2010) Ergül, A.; Malas, T.; Gürel, Levent
We present an iterative inner-outer scheme for the efficient solution of large-scale electromagnetics problems involving perfectly-conducting objects formulated with surface integral equations. Problems are solved by employing the multilevel fast multipole algorithm (MLFMA) on parallel computer systems. In order to construct a robust preconditioner, we develop an approximate MLFMA (AMLFMA) by systematically increasing the efficiency of the ordinary MLFMA. Using a flexible outer solver, iterative MLFMA solutions are accelerated via an inner iterative solver, employing AMLFMA and serving as a preconditioner to the outer solver. The resulting implementation is tested on various electromagnetics problems involving both open and closed conductors. We show that the processing time decreases significantly using the proposed method, compared to the solutions obtained with conventional preconditioners in the literature.