Tahir Malas

Incomplete LU Preconditioning with the Multilevel Fast Multipole Algorithm for Electromagnetic Scattering

Iterative solution of large-scale scattering problems in computational electromagnetics with the multilevel fast multipole algorithm (MLFMA) requires strong preconditioners, especially for the electric-field integral equation (EFIE) formulation. Incomplete LU (ILU) preconditioners are widely used and available in several solver packages. However, they lack robustness due to potential instability problems. In this study, we consider various ILU-class preconditioners and investigate the parameters that render them safely applicable to common surface integral formulations without increasing the

SOLUTIONS OF LARGE-SCALE ELECTROMAGNETICS PROBLEMS USING AN ITERATIVE INNER-OUTER SCHEME WITH ORDINARY AND APPROXIMATE MULTILEVEL FAST MULTIPOLE ALGORITHMS

{\cal O}(n\log n)

Fast and Accurate Analysis of Large Metamaterial Structures Using the Multilevel Fast Multipole Algorithm

complexity of MLFMA. We conclude that the no-fill ILU(0) preconditioner is an optimal choice for the combined-field integral equation (CFIE). For EFIE, we establish the need to resort to methods depending on drop tolerance and apply pivoting for problems with high condition estimate. We propose a strategy for the selection of the parameters so that the preconditioner can be used as a black-box method. Robustness and efficiency of the employed preconditioners are demonstrated over several test problems.

Approximate MLFMA as an efficient preconditioner

We present an iterative inner-outer scheme for the e-cient 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 e-ciency of the ordinary MLFMA. Using a ∞exible 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 signiflcantly using the proposed method, compared to the solutions obtained with conventional preconditioners in the literature.

Sequential and parallel preconditioners for large-scale integral-equation problems

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-fleld integral equation, discretized with the Rao-Wilton- Glisson basis functions. Resulting dense matrix equations are solved iteratively, where the matrix-vector multiplications are performed e-ciently 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.

Incomplete LU Preconditioning Strategies for MLFMA

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.

Parallel image restoration using surrogate constraint methods

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 if 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.

Solution of Extremely Large Integral-Equation Problems

This paper shows that the ILU preconditioners can be used in the iterative solutions of scattering problems. It is deduced that ILU(0) can be safely applied to CFIE, yielding very close performance to the exact solution of the near-field matrix. For EFIE, establishing condest to be a strong indicator for the quality of the resulting ILU preconditioner, the following strategy is proposed. Before the iterations begin, compute condest for ILUT. If the condition estimate is not very high, (e.g., less than 104), use ILUT as the preconditioner. Otherwise, switch to ILUTP5. With this strategy, robust and effective preconditioners are obtained for the test problems

An effective preconditioner based on schur complement reduction for integral-equation formulations of dielectric problems

When formulated as a system of linear inequalities, the image restoration problem yields huge, unstructured, sparse matrices even for images of small size. To solve the image restoration problem, we use the surrogate constraint methods that can work efficiently for large problems. Among variants of the surrogate constraint method, we consider a basic method performing a single block projection in each step and a coarse-grain parallel version making simultaneous block projections. Using several state-of-the-art partitioning strategies and adopting different communication models, we develop competing parallel implementations of the two methods. The implementations are evaluated based on the per iteration performance and on the overall performance. The experimental results on a PC cluster reveal that the proposed parallelization schemes are quite beneficial.

Scalable parallelization of the sparse-approximate-inverse (SAI) preconditioner for the solution of large-scale integral-equation problems

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.