Large Dense Complex Linear Systems Research Articles

Compressed representation of matrix decomposition algorithm-singular value decomposition for full-wave analysis microstrip problems

In order to efficiently solve large dense complex linear systems arising from electric field integral equations (EFIE) of electromagnetic problems, matrix decomposition algorithm-singular value decomposition (MDA-SVD) is used to accelerate the matrix-vector product (MVP) operations and decrease memory usage. Based on the symmetry of the impedance matrix resulting from the discretization of the EFIE, we introduce a compressed representation of MDA-SVD in this paper. We obtain a sparse representation of the far-field interaction parts of impedance matrix and perform a fast MVP operation. Numerical experiments demonstrate that the compressed representation of MDA-SVD can reduce both the MVP time and memory usage by around 50% with similar accuracy.

Shifted SSOR preconditioning technique for improved electric field integral equations

AbstractRecently, the improved electric field integral equation (IEFIE) is proposed to efficiently solve large dense complex linear system arising from three‐dimensional electromagnetic scattering problems. In this article, the shifted symmetric successive over‐relaxation preconditioner is constructed which combines the near‐field matrix of the EFIE and the principle value term of the magnetic field integral equation. The present method can achieve stably faster convergence of iterations than IEFIE with low cost for construction and implementation. Numerical results demonstrate the efficiency and validity of the present method. © 2012 Wiley Periodicals, Inc. Microwave Opt Technol Lett 55:304–308, 2012; View this article online at wileyonlinelibrary.com. DOI 10.1002/mop.27289

Efficient Analyzing EM Scattering of Objects Above a Lossy Half-Space by the Combined MLQR/MLSSM

A large dense complex linear system is obtained when solving an electromagnetic scattering of arbitrary shaped object located above a lossy half-space with the surface integral equation approach. To analyze the large dense complex linear system efficiently, the multilevel QR (MLQR) is used to accelerate the matrix-vector multiplication when the corresponding matrix equation is solved by a Krylov-subspace iterative method. Although the MLQR is more efficient than the direct solution, this paper presents a novel recompression technique to further reduce computation time and storage memory. The technique applies the multilevel simply sparse method (MLSSM) to the matrices of MLQR. Using the MLSSM, a sparser representation of the impedance matrix is obtained, and a more efficient matrix-vector multiplication is implemented. The combined MLQR/MLSSM is comparable to the MLQR and the adaptive cross-approximation/singular value decomposition (ACA-SVD) in terms of computation time and memory requirement. Remarkably, the new formulation can reduce the computational time and memory by about one order of magnitude, with excellent accuracy.

Modified compressed block decomposition preconditioner for electromagnetic problems

A modified compressed block decomposition (CBD) preconditioner combined with the multilevel fast multipole algorithm (MLFMA) is proposed for efficiently solving large dense complex linear systems that arise in electromagnetic problems.The modified preconditioner utilizes some improvements to improve the CBD preconditioner which is constructed from the available sparse near-field matrix. On the basis of the modified preconditioner, an efficient matrix–vector multiplication is implemented. Accordingly, the modified preconditioner is comparable with the unpreconditioned generalized minimal residual algorithm (GMRES) in terms of the number of iterations and the total solution time. Remarkably, the modified preconditioner provides a reduction close to one order of magnitude in both the number of iterations and the total solution time. © 2011 Wiley Periodicals, Inc. Microwave Opt Technol Lett, 2011; View this article online at wileyonlinelibrary.com. DOI 10.1002/mop.26149

A novel flexible GMRES with deflated restarted for efficient solution of electromagnetic problems

AbstractTo iterative solve large dense complex linear systems arising from electric field integral equations formulation of electromagnetic scattering problems, multilevel fast multipole algorithm (MLFMA) is built to speed up the matrix‐vector product (MVP) operations. This article presents a novel flexible generalized minimum residual with deflated restarted (FGMRES‐DR), which can greatly improve the convergence of those large dense complex linear systems. Based on physical mechanism of the scattering problems, in inner iteration of FGMRES‐DR, an iterative solution of the near‐field matrix in MLFMA is adopted. In outer iteration, deflation method which is purely algebraic is used to eliminate the influence on convergence because of the smallest eigenvalues. Numerical experiments indicate that FGMRES‐DR is very effective and can reduce the number of MVP compared with the FGMRES in large‐scale electromagnetic problems. © 2011 Wiley Periodicals, Inc. Microwave Opt Technol Lett 53:1360–1364, 2011; View this article online at wileyonlinelibrary.com. DOI 10.1002/mop.26019

Fast iterative solution of integral equation with matrix decomposition algorithm and multilevel simple sparse method

In order to efficiently solve large dense complex linear systems arising from the electric field integral equations formulation of electromagnetic problems, matrix decomposition algorithm–singular value decomposition (MDA–SVD) is used to accelerate the matrix–vector product (MVP) operation. Based on the symmetry of the impedance matrix, a modified multilevel simple sparse method (M-MLSSM) is proposed. Combining MDA and M-MLSSM, the authors obtain a sparse representation of the impedance matrix and construct a fast MVP operation. A perfectly electrically conducting sphere and the series-fed microstrip antenna array are simulated to demonstrate the validity of the proposed method. Moreover, numerical experiments demonstrate that combined MDA and M-MLSSM can accelerate the MVP operation and reduce memory requirements by at least half an order of magnitude compared to MDA–SVD.

Shifted SSOR preconditioning technique for electromagnetic wave scattering problems

AbstractTo efficiently solve large dense complex linear system arising from electric field integral equations (EFIE) formulation of electromagnetic scattering problems, the multilevel fast multipole method (MLFMM) is used to accelerate the matrix‐vector product operations. The symmetric successive over‐relaxation (SSOR) preconditioner is constructed based on the near‐field matrix of the EFIE and employed to speed up the convergence rate of iterative methods. This technique can be greatly improved by shifting the near‐field matrix of the EFIE with the principle value term of the magnetic field integral equation (MFIE) operator. Numerical results demonstrate that this method can reduce both the number of iterations and the computational time significantly with low cost for construction and implementation of preconditioners. © 2009 Wiley Periodicals, Inc. Microwave Opt Technol Lett 51: 1035–1039, 2009; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.24254

APPLICATION OF TWO-STEP SPECTRAL PRECONDITIONING TECHNIQUE FOR ELECTROMAGNETIC SCATTERING IN A HALF SPACE

To e-ciently solve large dense complex linear system arising from electric fleld integral equations (EFIE) formulation of half-space electromagnetic scattering problems, the multilevel fast multipole algorithm (MLFMA) is used to accelerate the matrix- vector product operations. The two-step spectral preconditioning is developed for the generalized minimal residual iterative method (GMRES). The two-step spectral preconditioner is constructed by combining the spectral preconditioner and sparse approximate inverse (SAI) preconditioner to speed up the convergence rate of iterative methods. Numerical experiments for scattering from conducting objects above or embeded in a lossy half-space are given to demonstrate the e-ciency of the proposed method.

SSOR PRECONDITIONED INNER-OUTER FLEXIBLE GMRES METHOD FOR MLFMM ANALYSIS OF SCATTERING OF OPEN OBJECTS

To efficiently solve large dense complex linear system arising from electric field integral equations (EFIE) formulation of electromagnetic scattering problems, the multilevel fast multipole method (MLFMM) is used to accelerate the matrix-vector product operations. The inner-outer flexible generalized minimum residual method (FGMRES) is combined with the symmetric successive over- relaxation (SSOR) preconditioner based on the near-part matrix of the EFIE in the inner iteration of FGMRES to speed up the convergence rate of iterative methods. Numerical experiments with a few electromagnetic scattering problems for open structures are given to demonstrate the efficiency of the proposed method.

Enhanced GMRES method combined with MLFMA for solving electromagnetic wave scattering problems

AbstractFor efficiently solving large dense complex linear systems that arise in the electric field integral equation (EFIE) formulation of electromagnetic wave scattering problems, the multilevel fast multipole algorithm (MLFMA) is used to speed up the matrix vector product operations, and the sparse approximate inverse (SAI) preconditioning technique is employed to accelerate the convergence rate of the generalized minimal residual (GMRES) iterative method. We show that the convergence rate can be greatly improved by augmenting to the GMRES method a few eigenvectors associated with the smallest eigenvalues of the preconditioned system. Numerical experiments indicate that this new variant GMRES method is very effective with the MLFMA and can reduce both the iteration number and the computational time significantly. © 2008 Wiley Periodicals, Inc. Microwave Opt Technol Lett 50: 1433–1439, 2008; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.23385

Multipreconditioned GMRES method for electromagnetic wave scattering problems

AbstractA new multipreconditioned generalized minimal residual method (GMRES) combined with the multilevel fast multipole method (MLFMM) is proposed for efficiently solving large dense complex linear systems that arise in electromagnetic wave scattering problems. The MLFMM is used to accelerate the matrix‐vector product operations, and the multipreconditioning technique is employed to speed up the convergence rate of the GMRES iterations. Unlike traditional preconditioning techniques, which usually work alone, the multipreconditioned GMRES algorithm has the capability to combine different preconditioning techniques all together. Numerical results show that the convergence rate of the GMRES method can be significantly improved by automatically taking advantages of all available preconditioners. © 2007 Wiley Periodicals, Inc. Microwave Opt Technol Lett 50: 150–152, 2008; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.23004

An efficient sparse approximate inverse preconditioning for FMM implementation

AbstractFor efficiently solving large dense complex linear systems that arise in electric field integral equations (EFIE) of electromagnetic scattering problems, the fast multipole method (FMM) is used to accelerate the matrix‐vector product operations, and the sparse approximate inverse (SAI) preconditioning technique is employed to speed up the convergence rate of the Krylov iterations. A good quality SAI preconditioner in the FMM context is constructed based on the near‐field matrix of the EFIE. The main purpose of this paper is to show that the quality of the SAI preconditioner can be greatly improved by use of information from FMM implementation. Numerical experiments indicate that the resulted SAI preconditioner is very effective with FMM and can reduce the overall simulation time significantly. © 2007 Wiley Periodicals, Inc. Microwave Opt Technol Lett 49: 1746–1750, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.22538

Combining Fast Multipole Techniques and an Approximate Inverse Preconditioner for Large Electromagnetism Calculations

The boundary element method has become a popular tool for the solution of Maxwell's equations in electromagnetism. From a linear algebra point of view, this leads to the solution of large dense complex linear systems, where the unknowns are associated with the edges of the mesh defined on the surface of the illuminated object. In this paper, we address the iterative solution of these linear systems via preconditioned Krylov solvers. Our primary focus is on the design of an efficient parallelizable preconditioner. In that respect, we consider an approximate inverse method based on the Frobenius-norm minimization. The preconditioner is constructed from a sparse approximation of the dense coefficient matrix, and the patterns both for the preconditioner and for the coefficient matrix are computed a priori using geometric information from the mesh. We describe how such a preconditioner can be naturally implemented in a parallel code that implements the multipole technique for the matrix-vector product calculation. We investigate the numerical scalability of our preconditioner on realistic industrial test problems and show that it exhibits some limitations on very large problems of size close to one million unknowns. To improve its robustness on those large problems we propose an embedded iterative scheme that combines nested GMRES solvers with different fast multipole computations. We show through extensive numerical experiments that this new scheme is extremely robust at affordable memory and CPU costs for the solution of very large and challenging problems.

Using spectral low rank preconditioners for large electromagnetic calculations

AbstractFor solving large dense complex linear systems that arise in electromagnetic calculations, we perform experiments using a general purpose spectral low rank update preconditioner in the context of the GMRES method preconditioned by an approximate inverse preconditioner. The goal of the spectral preconditioner is to improve the convergence properties by shifting by one the smallest eigenvalues of the original preconditioned system. Numerical experiments on parallel distributed memory computers are presented to illustrate the efficiency of this technique on large and challenging real‐life industrial problems. Copyright © 2004 John Wiley & Sons, Ltd.

Incomplete LU preconditioning for large scale dense complex linear systems from electromagnetic wave scattering problems

We consider the preconditioned iterative solution of large dense linear systems, where the coefficient matrix is a complex valued matrix arising from discretizing the integral equation of electromagnetic scattering. For some scattering structures this matrix can be poorly conditioned. The main purpose of this study is to evaluate the efficiency of a class of incomplete LU (ILU) factorization preconditioners for solving this type of matrices. We solve the electromagnetic wave equations using the BiCG method with an ILU preconditioner in the context of a multilevel fast multipole algorithm (MLFMA). The novelty of this work is that the ILU preconditioner is constructed using the near part block diagonal submatrices generated from the MLFMA. Experimental results show that the ILU preconditioner reduces the number of BiCG iterations substantially, compared to the block diagonal preconditioner. The preconditioned iteration scheme also maintains the computational complexity of the MLFMA, and consequently reduces the total CPU time.