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Paola F Antonietti - One of the best experts on this subject based on the ideXlab platform.
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an agglomeration based massively parallel non overlapping additive schwarz preconditioner for high order discontinuous galerkin methods on polytopic grids
arXiv: Numerical Analysis, 2019Co-Authors: Paola F Antonietti, Paul Houston, Giorgio Pennesi, Endre SuliAbstract:In this article we design and analyze a class of two-level non-overlapping additive Schwarz preconditioners for the solution of the linear System of equations stemming from discontinuous Galerkin discretizations of second-order elliptic partial differential equations on polytopic meshes. The preconditioner is based on a coarse space and a non-overlapping partition of the computational domain where local solvers are applied in parallel. In particular, the coarse space can potentially be chosen to be non-embedded with respect to the finer space; indeed it can be obtained from the fine grid by employing agglomeration and edge coarsening techniques. We investigate the dependence of the condition number of the Preconditioned System with respect to the diffusion coefficient and the discretization parameters, i.e., the mesh size and the polynomial degree of the fine and coarse spaces. Numerical examples are presented which confirm the theoretical bounds.
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schwarz domain decomposition preconditioners for discontinuous galerkin approximations of elliptic problems non overlapping case
Mathematical Modelling and Numerical Analysis, 2007Co-Authors: Paola F Antonietti, Blanca AyusoAbstract:We propose and study some new additive, two-level non-overlapping Schwarz preconditioners for the solution of the algebraic linear Systems arising from a wide class of discontinuous Galerkin approximations of elliptic problems that have been proposed up to now. In particular, two-level methods for both symmetric and non-symmetric schemes are introduced and some interesting features, which have no analog in the conforming case, are discussed. Both the construction and analysis of the proposed domain decomposition methods are presented in a unified framework. For symmetric schemes, it is shown that the condition number of the Preconditioned System is of order O(H/h) , where H and h are the mesh sizes of the coarse and fine grids respectively, which are assumed to be nested. For non-symmetric schemes, we show by numerical computations that the Eisenstat et al. [SIAM J. Numer. Anal. 20 (1983) 345–357] GMRES convergence theory, generally used in the analysis of Schwarz methods for non-symmetric problems, cannot be applied even if the numerical results show that the GMRES applied to the Preconditioned Systems converges in a finite number of steps and the proposed preconditioners seem to be scalable. Extensive numerical experiments to validate our theory and to illustrate the performance and robustness of the proposed two-level methods are presented.
Blanca Ayuso - One of the best experts on this subject based on the ideXlab platform.
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schwarz domain decomposition preconditioners for discontinuous galerkin approximations of elliptic problems non overlapping case
Mathematical Modelling and Numerical Analysis, 2007Co-Authors: Paola F Antonietti, Blanca AyusoAbstract:We propose and study some new additive, two-level non-overlapping Schwarz preconditioners for the solution of the algebraic linear Systems arising from a wide class of discontinuous Galerkin approximations of elliptic problems that have been proposed up to now. In particular, two-level methods for both symmetric and non-symmetric schemes are introduced and some interesting features, which have no analog in the conforming case, are discussed. Both the construction and analysis of the proposed domain decomposition methods are presented in a unified framework. For symmetric schemes, it is shown that the condition number of the Preconditioned System is of order O(H/h) , where H and h are the mesh sizes of the coarse and fine grids respectively, which are assumed to be nested. For non-symmetric schemes, we show by numerical computations that the Eisenstat et al. [SIAM J. Numer. Anal. 20 (1983) 345–357] GMRES convergence theory, generally used in the analysis of Schwarz methods for non-symmetric problems, cannot be applied even if the numerical results show that the GMRES applied to the Preconditioned Systems converges in a finite number of steps and the proposed preconditioners seem to be scalable. Extensive numerical experiments to validate our theory and to illustrate the performance and robustness of the proposed two-level methods are presented.
Cornelis Vuik - One of the best experts on this subject based on the ideXlab platform.
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a comparison of abstract versions of deflation balancing and additive coarse grid correction preconditioners
Numerical Linear Algebra With Applications, 2008Co-Authors: Reinhard Nabben, Cornelis VuikAbstract:SUMMARY In this paper we consider various preconditioners for the conjugate gradient (CG) method to solve large linear Systems of equations with symmetric positive definite System matrix. We continue the comparison between abstract versions of the deflation, balancing and additive coarse grid correction preconditioning techniques started in (SIAM J. Numer. Anal. 2004; 42:1631–1647; SIAM J. Sci. Comput. 2006; 27:1742–1759). There the deflation method is compared with the abstract additive coarse grid correction preconditioner and the abstract balancing preconditioner. Here, we close the triangle between these three methods. First of all, we show that a theoretical comparison of the condition numbers of the abstract additive coarse grid correction and the condition number of the System Preconditioned by the abstract balancing preconditioner is not possible. We present a counter example, for which the condition number of the abstract additive coarse grid correction Preconditioned System is below the condition number of the System Preconditioned with the abstract balancing preconditioner. However, if the CG method is Preconditioned by the abstract balancing preconditioner and is started with a special starting vector, the asymptotic convergence behavior of the CG method can be described by the so-called effective condition number with respect to the starting vector. We prove that this effective condition number of the System Preconditioned by the abstract balancing preconditioner is less than or equal to the condition number of the System Preconditioned by the abstract additive coarse grid correction method. We also provide a short proof of the relationship between the effective condition number and the convergence of CG. Moreover, we compare the A-norm of the errors of the iterates given by the different preconditioners and establish the orthogonal invariants of all three types of preconditioners. Copyright q 2008 John Wiley & Sons, Ltd.
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a comparison of deflation and the balancing preconditioner
SIAM Journal on Scientific Computing, 2005Co-Authors: Reinhard Nabben, Cornelis VuikAbstract:In this paper we compare various preconditioners for the numerical solution of partial differential equations. We compare the well-known balancing preconditioner used in domain decomposition methods with a so-called deflation preconditioner. We prove that the effective condition number of the deflated Preconditioned System is always, i.e., for all deflation vectors and all restrictions and prolongations, below the condition number of the System Preconditioned by the balancing preconditioner. Even more, we establish that both preconditioners lead to almost the same spectra. The zero eigenvalues of the deflation Preconditioned System are replaced by eigenvalues which are one if the balancing preconditioner is used. Moreover, we prove that the A-norm of the errors of the iterates built by the deflation preconditioner is always below the A-norm of the errors of the iterates built by the balancing preconditioner. Depending on the implementation of the balancing preconditioner the amount of work of one iteration of the deflation Preconditioned System is less than or equal to the amount of work of one iteration of the balancing Preconditioned System. If the amount of work is equal, both preconditioners are sensitive with respect to inexact computations. Finally, we establish that the deflation preconditioner and the balancing preconditioner produce the same iterates if one uses certain starting vectors. Numerical results for porous media flows emphasize the theoretical results.
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a comparison of deflation and coarse grid correction applied to porous media flow
SIAM Journal on Numerical Analysis, 2004Co-Authors: Reinhard Nabben, Cornelis VuikAbstract:In this paper we compare various preconditioners for the numerical solution of partial differential equations. We compare a coarse grid correction preconditioner used in domain decomposition methods with a so-called deflation preconditioner. We prove that the effective condition number of the deflated Preconditioned System is always, for all deflation vectors and all restrictions and prolongations, below the condition number of the System Preconditioned by the coarse grid correction. This implies that the conjugate gradient method applied to the deflated Preconditioned System is expected always to converge faster than the conjugate gradient method applied to the System Preconditioned by the coarse grid correction. Numerical results for porous media flows emphasize the theoretical results.
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a comparison of deflation and coarse grid correction applied to porous media flow
Reports of the Department of Applied Mathematical Analysis, 2003Co-Authors: Reinhard Nabben, Cornelis VuikAbstract:In this paper we compare various preconditioners for the numerical solution of partial dierential equations. We compare a coarse grid correction preconditioner used in domain decomposition methods with a so-called deflation preconditioner. We prove that the effective condition number of the de ated Preconditioned System is always, i.e. for all deflation vectors and all restrictions and prolongations, below the condition number of the System Preconditioned by the coarse grid correction. This implies that the Conjugate Gradient method applied to the de ated Preconditioned System converges always faster than the Conjugate Gradient method applied to the System Preconditioned by the coarse grid correction. Numerical results for porous media flows emphasize the theoretical results.
Brian Mackie-mason - One of the best experts on this subject based on the ideXlab platform.
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Domain Decomposition Preconditioning for Surface Integral Equations in Solving Challenging Electromagnetic Scattering Problems
IEEE Transactions on Antennas and Propagation, 2016Co-Authors: Zhen Peng, Ralf Hiptmair, Yang Shao, Brian Mackie-masonAbstract:We propose and study a nonoverlapping and nonconforming domain decomposition method for the integral-equation-based solution of large, complex electromagnetic (EM) scattering problems. The continuity of the electric surface current across the boundary between adjacent subdomains is enforced by a skew-symmetric interior penalty formulation. A nonoverlapping additive Schwarz preconditioner is designed and analyzed for the solution of the linear System of equations resulting from Galerkin boundary-element discretization. We show that the Preconditioned System exhibits a uniformly confined eigenspectrum with respect to changing problem and discretization parameters. Numerical examples are presented to demonstrate the fast convergence of iterative solvers and the superior accuracy of the solutions obtained by our method. The proposed work can be viewed as an effective preconditioning scheme that reduces the condition number of very large Systems of equations in challenging EM scattering problems. The strength and capability of the proposed method will be illustrated by means of several examples of practical interest.
Jianming Jin - One of the best experts on this subject based on the ideXlab platform.
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efie analysis of low frequency problems with loop star decomposition and calderon multiplicative preconditioner
IEEE Transactions on Antennas and Propagation, 2010Co-Authors: Su Yan, Jianming Jin, Zaiping NieAbstract:Low-frequency electromagnetic problems are analyzed using the electric field integral equation (EFIE) with loop-star basis functions to alleviate the low-frequency breakdown problem. By constructing the loop-star basis functions with the curvilinear RWG (CRWG) basis and the Buffa-Christiansen (BC) basis, respectively, the recently proposed Caldero?n multiplicative preconditioner (CMP) is improved to become applicable at low frequencies. The Gram matrix arisen from CRWG loop-star basis and BC loop-star basis is studied in detail. A direct solution approach is introduced to solve the Gram matrix equation. The proposed Calderon preconditioner improves the condition of the EFIE operator at low frequencies, which results in a fast convergence of the Preconditioned EFIE System. Several numerical examples demonstrate the fast and mesh-independent convergence of the Preconditioned System.
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a highly effective preconditioner for solving the finite element boundary integral matrix equation of 3 d scattering
IEEE Transactions on Antennas and Propagation, 2002Co-Authors: Jian Liu, Jianming JinAbstract:A highly effective preconditioner is presented for solving the System of equations obtained from the application of the hybrid finite element-boundary integral (FE-BI) method to three-dimensional (3-D) electromagnetic scattering problems. Different from widely used algebraic preconditioners, the proposed one is based on a physical approximation and is constructed from the finite element method (FEM) using an absorbing boundary condition (ABC) on the truncation boundary. It is shown that the large eigenvalues of the finite element (FE)-ABC System are similar to those of the FE-BI System. Hence, the Preconditioned System has a spectrum distribution clustered around 1 in the complex plane. Consequently, when a Krylov subspace based method is employed to solve the Preconditioned System, the convergence can be greatly accelerated. Numerical results show that the proposed preconditioner can improve the convergence of an iterative solution by approximately two orders of magnitude for large problems.