Algebraic multigrid for complex symmetric systems

Lahaye, Domenico; Gersem, Herbert De; Vandewalle, Stefan; Hameyer, Kay

doi:10.1109/20.877730

Cited by 28 publications

(2 citation statements)

References 10 publications

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“…Helmholtz or frequency-domain Maxwell problems, the matrix is often indefinite, complex-valued, and non-Hermitian. In addition, the near null-space may no longer be slowly varying, which is S k = S t r e n g t h ( A k ) 6 C k = A g g r e g a t e ( S k ) 7 T k , B k+1 = I n j e c t C a n d i d a t e s ( C k , B k ) 8 P k = S m o o t h P r o l o n g a t o r ( A k , T k ) 9 i f A i s H e r m i t i a n 10 A k+1 = P * k A k P k …”

Section: Smoothed Aggregationmentioning

confidence: 99%

Smoothed aggregation for Helmholtz problems

Olson

Schroder

2010

Numerical Linear Algebra App

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SUMMARYWe outline a smoothed aggregation algebraic multigrid method for 1D and 2D scalar Helmholtz problems with exterior radiation boundary conditions. We consider standard 1D finite difference discretizations and 2D discontinuous Galerkin discretizations. The scalar Helmholtz problem is particularly difficult for algebraic multigrid solvers. Not only can the discrete operator be complex-valued, indefinite, and non-self-adjoint, but it also allows for oscillatory error components that yield relatively small residuals. These oscillatory error components are not effectively handled by either standard relaxation or standard coarsening procedures. We address these difficulties through modifications of SA and by providing the SA setup phase with appropriate wave-like near null-space candidates. Much is known a priori about the character of the near null-space, and our method uses this knowledge in an adaptive fashion to find appropriate candidate vectors. Our results for GMRES preconditioned with the proposed SA method exhibit consistent performance for fixed points-per-wavelength and decreasing mesh size.

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Section: Smoothed Aggregationmentioning

confidence: 99%

Smoothed aggregation for Helmholtz problems

Olson

Schroder

2010

Numerical Linear Algebra App

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“…For positive-definite and complex Hermitian problems, Geometric and/or Algebraic Multigrid are powerful solvers which provide solutions in linear time complexity by constructing coarser linear systems based on the information provided by the finegrid linear system. For complex Hermitian matrices, the coarse linear systems can be build using the Galerkin condition, which has been studied in [22,25]. If the use of a simple smoother by means of a stationary iterative method such as weighted-Jacobi or Gauss-Seidel is preferred, the coarse grid correction scheme should balance the simplicity of the smoother by accurately transferring smooth errors from the coarse grid to the fine grid [25].…”

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confidence: 99%

Scalable Convergence Using Two-Level Deflation Preconditioning for the Helmholtz Equation

Dwarka¹,

Vuik²

2020

SIAM J. Sci. Comput.

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Recent research efforts aimed at iteratively solving the Helmholtz equation have focused on incorporating deflation techniques for accelerating the convergence of Krylov subpsace methods. The requisite for these efforts lies in the fact that the widely used and well-acknowledged complex shifted Laplacian preconditioner (CSLP) shifts the eigenvalues of the preconditioned system towards the origin as the wave number increases. The two-level-deflation preconditioner combined with CSLP showed encouraging results in moderating the rate at which the eigenvalues approach the origin. However, for large wave numbers the initial problem resurfaces and the near-zero eigenvalues reappear. Our findings reveal that the reappearance of these near-zero eigenvalues occurs if the near-singular eigenmodes of the fine-grid operator and the coarse-grid operator are not properly aligned. This misalignment is caused by accumulating approximation errors during the inter-grid transfer operations. We propose the use of higher-order approximation schemes to construct the deflation vectors. The results from rigorous Fourier analysis and numerical experiments confirm that our newly proposed scheme outperforms any other deflation-based preconditioner for the Helmholtz problem. In particular, the spectrum of the adjusted preconditioned operator stays fixed near one. These results can be generalized to general shifted indefinite systems with random right-hand sides. For the first time, the convergence properties for very large wave numbers (k = 10 6 in one dimension and k = 10 3 in two dimensions) have been studied, and the convergence is close to wave number independence. Wave number independence for three dimensions has been obtained for wave numbers up to k = 75. The new scheme additionally shows very promising results for the more challenging Marmousi problem. Irrespective of the strongly varying wave number, we obtain a constant number of iterations and a reduction in computational time as the results remain robust without the use of the CSLP preconditioner.

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Application of a two-step preconditioning strategy to the finite element analysis for electromagnetic problems

Rui

Chen

Yung

et al. 2006

Microw. Opt. Technol. Lett.

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A two-step preconditioning strategy is presented for the conjugate gradient (CG) iterative method to solve a large system of linear equations resulting from the use of edge-based finite-element discretizations of Helmholtz equations. The key idea is to combine both the factorized sparse approximate inverse (FSAI) and the symmetric successive overrelaxation (SSOR) preconditioning techniques in two successive steps in order to obtain a better preconditioner for the original matrix equations. The newly constructed preconditioner combines the advantages of both the FSAI and SSOR preconditioners with less computational complexity without the breakdowns of incomplete factorization INTRODUCTIONThe finite-element method (FEM) is one of the most effective and versatile techniques which has been widely applied to electromagnetic-field problems. Its popularity in modeling microwave devices lies in its ability to deal with complex geometries and materials of any composition. Systematic description for the application of this method in electromagnetic can be found [1,2]. The application of the FEM to electromagnetic problems often yields a large, sparse, complex symmetric system of linear equations. These highly sparse linear equations can be solved using efficient solution techniques for sparse matrices based on either direct methods or iterative methods. Direct methods have the advantage that multiple right-hand sides can be treated efficiently. However, storing the factorized matrix is memory intensive for large matrices. Classical direct method includes Gaussian elimination method and the closely related LU decomposition with O(N 3 ) computational complexity. Moreover, these so-called "direct" methods bring "fill-in," that is, nonzero entries are created in certain positions where the coefficient matrix originally has zeros. Fill-in is undesirable because it increases both the computing time and the storage requirement. Direct solvers usually suffer from fill-in to an extent that these large problems cannot be solved at a reasonable cost, even on the state-of-the-art parallel machines. This necessitates the use of iterative algorithms instead of direct solvers to preserve the sparsity of the finite-element matrix. Especially attractive are iterative methods that involve the coefficient matrices only in terms of matrix-vector multiplication. The most powerful iterative algorithm of these types is the conjugate gradient algorithm for solving positive definite linear system [3]. The convergence rate of the CG method is mainly determined by the condition number of the coefficient matrix, which is closely related to the distribution of the eigenvalues of the coefficient matrix [4]. The condition number of a linear system of equations usually increases with the number of unknowns. Therefore, the solution of the linear system is the main expense of the whole problem solution, which consumes more than 95% of the CPU time for FEM application in electromagnetics. It is then desirable to precondition the coefficient matrix...

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Algebraic multigrid for complex symmetric systems

Cited by 28 publications

References 10 publications

Smoothed aggregation for Helmholtz problems

Smoothed aggregation for Helmholtz problems

Scalable Convergence Using Two-Level Deflation Preconditioning for the Helmholtz Equation

Application of a two-step preconditioning strategy to the finite element analysis for electromagnetic problems

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