Two-frequency decomposition

SUMMARYLarge linear systems arising from the discretization of partial di erential equations with ÿnite di erences or ÿnite elements on structured grids in dimension d (d¿3) require e cient preconditioners. For a symmetric and positive deÿnite (SPD) matrix, we propose a SPD block LDL T preconditioner whose factorized form requires a smaller amount of memory than the original matrix. Moreover, the computing time for the preconditioner solves is linear with respect to the number of unknowns. The preconditioner is built in d stages: in a ÿrst stage, we use the tangential ÿltering decomposition of Wittum et al. and obtain a preconditioner which remains rather di cult to factorize. Then, in a second stage, we apply tangential ÿltering decomposition recursively to the diagonal blocks of this ÿrst preconditioner. The ÿnal stage consists of factorizing exactly the blocks corresponding to one dimensional problems. Such preconditioners can also be computed adaptively and combined in a multiplicative way. A generic programming implementation is discussed and numerical tests are presented, in particular for problems with highly heterogeneous media.

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“…and D n + L n−1 are symmetric and positive deÿnite, it is proved in Reference [9] that the T i are also positive deÿnite.…”

Section: Other ÿLtering Methodsmentioning

confidence: 94%

“…In [5,10,9], the authors carry out a somewhat similar analysis, but they minimize the rate of convergence of the Richardson algorithm instead of the condition number.…”

Section: Choosing An Optimal Test-vectormentioning

confidence: 97%

An iterated tangential filtering decomposition

Achdou

Nataf

2003

Numerical Linear Algebra App

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“…The optimal condition numbers derived here and in [1] have the same order. In Reference [13], the condition number of optimized two-frequency filtering decomposition is of same order. However, it is not tangential filtering decomposition.…”

Section: Lemmamentioning

confidence: 99%

“…Similar to some popular preconditioning techniques discussed in [1,3,8,12,13,19,23,[40][41][42]44], TFFD preconditioner can be used either as a preconditioner or as a smoother for multigrid methods. The preconditioner satisfies a filtering property (M − A) f = 0 for a vector f , whereM is the TFFD preconditioner.…”

Section: Introductionmentioning

confidence: 99%

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Modified tangential frequency filtering decomposition and its fourier analysis

et al. 2010

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In this paper, a modified tangential frequency filtering decomposition (MTFFD) preconditioner is proposed. The optimal order of the modification and the optimal relaxation parameter is determined by Fourier analysis. With the choice of optimal order of modification, the Fourier results show that the condition number of the preconditioned matrix is O(h − 2 3 ), and the spectrum distribution of the preconditioned matrix can be predicted by the Fourier results. The performance of MTFFD preconditioner is compared with tangential frequency filtering (TFFD) preconditioner on a variety of large sparse matrices arising from the discretization of PDEs with discontinuous coefficients. The numerical results show that the MTFFD preconditioner is much more efficient than the TFFD preconditioner.

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IBLU decompositions based on Padé approximants

Buzdin

Logashenko

Wittum

2008

Numerical Linear Algebra App

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In this paper, we introduce a new class of frequency-filtering IBLU decompositions that use continuedfraction approximation for the diagonal blocks. This technique allows us to construct efficient frequencyfiltering preconditioners for discretizations of elliptic partial differential equations on domains with non-trivial geometries. We prove theoretically for a class of model problems that the application of the proposed preconditioners leads to a convergence rate of up to 1− O(h 1/4 ) of the CG iteration. are high-oscillating grid functions, the linear iteration preconditioned with such a decomposition is a good smoother for geometric multi-grid methods. Using smooth test vectors, we make W an efficient preconditioner for the conjugate gradient method. Furthermore, filtering decompositions with different test vectors can be used successively in the linear iteration. Then the logarithmic growth of the computational complexity of solving the linear systems on structured rectangular grids can be achieved (cf. [4, 7]). A deep convergence theory of the frequency-filtering decompositions was worked out by Wittum and his successors (cf. [4, 7, 10, 11, 13, 14]) for a defined class of model problems.However, an essential problem of these decompositions is the strong restrictions on the structure of the matrix blocks of A. For this reason, only their simplest subclass (the so-called FF decomposition) has been applied to problems with complicated domain geometries (cf. [15]). For other IBLU decompositions satisfying (6), e.g. those proposed in [8], the thorough quantitative convergence analysis could not be carried out, not even for model problems.In this paper, we use the same theoretical approach as for the frequency-filtering decompositions but weaken these restrictions. The proposed approximation of T k is based on rational functions. This allows us to carry out the quantitative convergence analysis for a class of model problems and generalize the technique for non-trivial discretization grids. The proposed methods are introduced in Section 2 for a general matrix (2). Section 2 also explains the main idea of these methods for a model problem. The theoretical analysis of the convergence properties is presented in Sections 3 and 4. In Section 5, we consider the application of the proposed decompositions to the general block-tridiagonal matrices in detail. Section 6 presents results of numerical experiments. We summarize our main conclusions and give an outlook in Section 7. We use the following notation: For two symmetric matrices A and B, A

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Two-frequency decomposition

Cited by 9 publications

References 8 publications

An iterated tangential filtering decomposition

An iterated tangential filtering decomposition

Modified tangential frequency filtering decomposition and its fourier analysis

IBLU decompositions based on Padé approximants

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