Updating the inverse of a matrix
SIAM Review
Factorized sparse approximate inverse preconditionings I: theory
SIAM Journal on Matrix Analysis and Applications
Matrix computations (3rd ed.)
Parallel Preconditioning with Sparse Approximate Inverses
SIAM Journal on Scientific Computing
Experimental study of ILU preconditioners for indefinite matrices
Journal of Computational and Applied Mathematics
Approximate Inverse Techniques for Block-Partitioned Matrices
SIAM Journal on Scientific Computing
A Sparse Approximate Inverse Preconditioner for Nonsymmetric Linear Systems
SIAM Journal on Scientific Computing
Approximate Inverse Preconditioners via Sparse-Sparse Iterations
SIAM Journal on Scientific Computing
A comparative study of sparse approximate inverse preconditioners
IMACS'97 Proceedings on the on Iterative methods and preconditioners
Kalman Filtering and Neural Networks
Kalman Filtering and Neural Networks
Numerical Linear Algebra for High Performance Computers
Numerical Linear Algebra for High Performance Computers
Fast and Efficient Algorithms in Computational Electromagnetics
Fast and Efficient Algorithms in Computational Electromagnetics
A Note on Preconditioning Nonsymmetric Matrices
SIAM Journal on Scientific Computing
Preconditioning techniques for large linear systems: a survey
Journal of Computational Physics
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
Block LU Preconditioners for Symmetric and Nonsymmetric Saddle Point Problems
SIAM Journal on Scientific Computing
A Preconditioner for Generalized Saddle Point Problems
SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Scientific Computing
The effect of non-optimal bases on the convergence of Krylov subspace methods
Numerische Mathematik
SIAM Journal on Scientific Computing
Preconditioners for Generalized Saddle-Point Problems
SIAM Journal on Numerical Analysis
SIAM Journal on Scientific Computing
Pressure Schur Complement Preconditioners for the Discrete Oseen Problem
SIAM Journal on Scientific Computing
Photonic Crystals: Molding the Flow of Light
Photonic Crystals: Molding the Flow of Light
SIAM Journal on Scientific Computing
Iterative Near-Field Preconditioner for the Multilevel Fast Multipole Algorithm
SIAM Journal on Scientific Computing
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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.