Many important problems in mathematics and physics lead to (non-sparse) functions, vectors, or matrices in which the fraction of nonnegligible entries is vanishingly small compared the total number of entries as the size of the system tends to infinity. In other words, the nonnegligible entries tend to be localized, or concentrated, around a small region within the computational domain, with rapid decay away from this region (uniformly as the system size grows). When present, localization opens up the possibility of developing fast approximation algorithms, the complexity of which scales linearly in the size of the problem. While localization already plays an important role in various areas of quantum physics and chemistry, it has received until recently relatively little attention by researchers in numerical linear algebra. In this chapter we survey localization phenomena arising in various fields, and we provide unified theoretical explanations for such phenomena using general results on the decay behavior of matrix functions. We also discuss computational implications for a range of applications.

Localization in Matrix Computations: Theory and Applications

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.

/pdf/solutions-of-large-scale-electromagnetics-problems-using-an-4yiubr4mut.pdf

Solutions of large-scale electromagnetics problems using an iterative inner-outer scheme with ordinary and approximate multilevel fast multipole algorithms

The inherent difficulties of traditional preconditioning techniques are revealed by numerically analyzing characteristics of method of moments systems with respect to multiscale dynamic electromagnetic applications. Techniques are developed to solve these difficulties for the traditional sparse approximate inverse preconditioner. The proposed techniques include a skeleton-based filtering strategy aiming to overcome the awkward filtering strategy extensively employed in traditional preconditioners and the strategy to recover the block structure of the inverse matrix. The proposed techniques are investigated by numerical experiments on complex targets.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1002/2014RS005387

Sparse approximate inverse preconditioner for multiscale dynamic electromagnetic problems

A Tensor-Train accelerated solver for integral equations in complex geometries

We present fast and accurate solutions of electromagnet- ics problems involving realistic metamaterial structures using a low- frequency multilevel fast multipole algorithm (LF-MLFMA). Acceler- ating iterative solutions using robust preconditioning techniques may not be su-cient to reduce the overall processing time when the or- dinary high-frequency MLFMA is applied to metamaterial problems. The major bottleneck, i.e., the low-frequency breakdown, should be eliminated for e-cient solutions. We show that the combination of an LF-MLFMA implementation based on the multipole expansion with the sparse-approximate-inverse preconditioner enables e-cient and ac- curate analysis of realistic metamaterial structures. Using the robust LF-MLFMA implementation, we demonstrate how the transmission properties of metamaterial walls can be enhanced with randomly- oriented unit cells.

/pdf/efficient-solutions-of-metamaterial-problems-using-a-low-12q83uiros.pdf

Efficient Solutions of Metamaterial Problems Using a Low-Frequency Multilevel Fast Multipole Algorithm

With the help of the multilevel fast multipole algorithm, integral-equation methods can be used to solve real-life electromagnetics problems both accurately and efficiently. Increasing problem dimensions, on the other hand, necessitate effective parallel preconditioners with low setup costs. In this paper, we consider sparse approximate inverses generated from the sparse near-field part of the dense coefficient matrix. In particular, we analyze pattern selection strategies that can make efficient use of the block structure of the near-field matrix, and we propose a load-balancing method to obtain high scalability during the setup. We also present some implementation details, which reduce the computational cost of the setup phase. In conclusion, for the open-surface problems that are modeled by the electric-field integral equation, we have been able to solve ill-conditioned linear systems involving millions of unknowns with moderate computational requirements. For closed-surface problems that can be modeled by the combined-field integral equation, we reduce the solution times significantly compared to the commonly used block-diagonal preconditioner.

/pdf/accelerating-the-multilevel-fast-multipole-algorithm-with-1f02802gyi.pdf

Accelerating the Multilevel Fast Multipole Algorithm with the Sparse-Approximate-Inverse (SAI) Preconditioning

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 $2 \times 2$ 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.

/pdf/schur-complement-preconditioners-for-surface-integral-211o2kwssd.pdf

Schur Complement Preconditioners for Surface Integral-Equation Formulations of Dielectric Problems Solved with the Multilevel Fast Multipole Algorithm

For iterative solutions of large and difficult integral-equation problems in computational electromagnetics using the multilevel fast multipole algorithm (MLFMA), preconditioners are usually built from the available sparse near-field matrix. The exact solution of the near-field system for the preconditioning operation is infeasible because the LU factors lose their sparsity during the factorization. To prevent this, incomplete factors or approximate inverses can be generated so that the sparsity is preserved, but at the expense of losing some information stored in the near-field matrix. As an alternative strategy, the entire near-field matrix can be used in an iterative solver for preconditioning purposes. This can be accomplished with low cost and complexity since Krylov subspace solvers merely require matrix-vector multiplications and the near-field matrix is sparse. Therefore, the preconditioning solution can be obtained by another iterative process, nested in the outer solver, provided that the outer Krylov subspace solver is flexible. With this strategy, we propose using the iterative solution of the near-field system as a preconditioner for the original system, which is also solved iteratively. Furthermore, we use a fixed preconditioner obtained from the near-field matrix as a preconditioner to the inner iterative solver. MLFMA solutions of several model problems establish the effectiveness of the proposed nested iterative near-field preconditioner, allowing us to report the efficient solution of electric-field and combined-field integral-equation problems involving difficult geometries and millions of unknowns.

/pdf/iterative-near-field-preconditioner-for-the-multilevel-fast-4c8f88wufh.pdf

Iterative Near-Field Preconditioner for the Multilevel Fast Multipole Algorithm

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Papers

Accelerating the Multilevel Fast Multipole Algorithm with the Sparse-Approximate-Inverse (SAI) Preconditioning

Schur Complement Preconditioners for Surface Integral-Equation Formulations of Dielectric Problems Solved with the Multilevel Fast Multipole Algorithm

Iterative Near-Field Preconditioner for the Multilevel Fast Multipole Algorithm