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Incomplete factorization-based preconditionings for solving the Helmholtz equation

Preconditioning techniques based on incomplete factorization of matrices are investigated, to solve highly indefinite complex-symmetric linear systems. A novel preconditioning is introduced. The real part of the matrix is made positive definite, or less indefinite, by adding properly defined perturbations to the diagonal entries, while the imaginary part is unaltered. The resulting preconditioning matrix, which is obtained by applying standard methods to the perturbed complex matrix, turns out to perform significantly better than classical incomplete factorization schemes. For realistic values of the GMRES restart parameter, spectacular reduction of iteration counts is observed. A theoretical spectral analysis is provided, in which the spectrum of the preconditioner applied to indefinite matrix is related to the spectrum of the same preconditioner applied to a Stieltjes matrix extracted from the indefinite matrix. Results of numerical experiments are reported, which display the efficiency of the new preconditioning. Copyright

Parallel incomplete factorizations with pseudo­overlapped subdomains

Abstract We address the hard question of efficient use on parallel platforms, of incomplete factorization preconditioning techniques for solving large and sparse linear systems by Krylov subspace methods. A novel parallelization strategy based on pseudo-overlapped subdomains is explored. This results in efficient parallelizable preconditioners. Numerical results give evidence that high performance can be achieved.

Sparse symmetric preconditioners for dense linear systems in electromagnetism

We consider symmetric preconditioning strategies for the iterative solution of dense complex symmetric non-Hermitian systems arising in computational electromagnetics. In particular, we report on the numerical behaviour of the classical incomplete Cholesky factorization as well as some of its recent variants and consider also well-known factorized approximate inverses. We illustrate the difficulties that those techniques encounter on the linear systems under consideration and give some clues to explain their disappointing behaviour. We propose two symmetric preconditioned based on Frobenius-norm minimization that use a prescribed sparsity pattern. The numerical and computational efficiency of the proposed preconditioners are illustrated on a set of model problems arising both from academic and from industrial applications. Copyright

Preconditioning of discrete Helmholtz operators perturbed by a diagonal complex matrix

Incomplete factorizations are popular preconditioning techniques for solving large and sparse linear systems. In the case of highly indefinite complex–symmetric linear systems, the convergence of Krylov subspace methods sometimes degrades with increasing level of fill-in. The reasons for this disappointing behaviour are twofold. On the one hand, the eigenvalues of the preconditioned system tend to 1, but the ‘convergence’ is not monotonous. On the other hand, the eigenvalues with negative real part, on their move towards 1 have to cross the origin, whence the risk of clustering eigenvalues around 0 while ‘improving’ the preconditioner. This makes it risky to predict any gain when passing from a level to a higher one. We examine a remedy which consists in slightly moving the spectrum of the original system matrix along the imaginary axis. Theoretical analysis that motivates our approach and experimental results are presented, which displays the efficiency of the new preconditioning techniques. Copyright

A generalized domain decomposition paradigm for parallel incomplete LU factorization preconditionings

Abstract Incomplete LU (ILU) factorizations are popular preconditioning techniques for solving large linear systems that arise in scientific computations. We propose a (generalized) domain decomposition-based approach that leads to almost perfect speed-up with respect to standard ILU. Experimental results and theoretical spectral condition number are reported for two-dimensional problems.

Spectral analysis of parallel incomplete factorizations with implicit pseudo‐overlap

Two general parallel incomplete factorization strategies are investigated. The techniques may be interpreted as generalized domain decomposition methods. In contrast to classical domain decomposition methods, adjacent subdomains exchange data during the construction of the incomplete factorization matrix, as well as during each local forward elimination and each local backward elimination involved in the application of the preconditioner. Local renumberings of nodes are combined with suitable global fill-in strategy in an (successful) attempt to overcome the well-known trade-off between high parallelism (locality) and fast convergence (globality). From an algebraic viewpoint, our techniques may be implemented as global renumbering strategies. Theoretical spectral analysis is provided, which displays that the convergence rate weakly depends on the number of subdomains. Numerical results obtained on a 16-processor SGI Origin 2000 are reported, showing the efficiency of our parallel preconditionings. Copyright

ParIC: A Family of Parallel Incomplete Cholesky Preconditioners

A class of parallel incomplete factorization preconditionings for the solution of large linear systems is investigated. The approach may be regarded as a generalized domain decomposition method. Adjacent subdomains have to communicate during the setting up of the preconditioner, and during the application of the preconditioner. Overlap is not necessary to achieve high performance. Fill-in levels are considered in a global way. If necessary, the technique may be implemented as a global re-ordering of the unknowns. Experimental results are reported for two-dimensional problems.

Experimental comparison of three-dimensional point and line modified incomplete factorizations

We examine how the variations of the coefficients of 3-dimensional (3D) partial differential equations (PDEs) influence the convergence of the conjugate gradient method, preconditioned by standard pointwise and linewise modified incomplete factorizations. General analytical spectral bounds obtained previously are applied, which displays the conditions under which good performances could be expected. The arguments also reveal that, if the total number of unknowns is very large or the number of unknowns in one direction is much larger than in both other ones, or if there are strong jumps in the variation of the PDE coefficients or fewer Dirichlet boundary conditions, then linewise preconditionings could be significantly more efficient than the corresponding pointwise ones. We also discuss reasons to explain why in the case of constant PDE coefficients, the advantage of preferring linewise methods to pointwise ones is not as pronounced as in 2D problems. Results of numerical experiments are reported.

Performance of parallel incomplete LDLt factorizations for solving acoustic wave propagation problems from industry

Parallel incomplete LDLt (ParILDLt) factorizations are used to solve highly indefinite complex-symmetric linear systems that arise from finite element discretization of acoustic wave propagation problems. The parallelization strategy is a generalized domain decomposition type approach in which adjacent subdomains have to exchange data during the construction of the incomplete factorization preconditioning matrix, as well as during each local forward and backward substitution. Comparison with the SYSNOISE (LMS International NV) direct solver, and the finite element tearing and interconnecting method for the Helmholtz equation (FETI-H), is done in terms of execution time and memory usage. Challenging industrial problems are tested, showing that high performance is achieved with ParlLDLt. Copyright