2,150 research outputs found
A preconditioner for the Schur complement matrix
A preconditioner for iterative solution of the interface problem in Schur Complement Domain Decomposition Methods is presented. This preconditioner is based on solving a global problem in a narrow strip around the interface. It requires much less memory and computing time than classical Neumann–Neumann preconditioner and its variants, and handles correctly the flux splitting among subdomains that share the interface. The aim of this work is to present a theoretical basis (regarding the behavior of Schur complement matrix spectra) and some simple numerical experiments conducted in a sequential environment as a motivation for adopting the proposed preconditioner. Efficiency, scalability, and implementation details on a production parallel finite element code [Sonzogni V, Yommi A, Nigro N, Storti M. A parallel finite element program on a Beowulf cluster. Adv Eng Software 2002;33(7–10):427–43; Storti M, Nigro N, Paz R, Dalcín L. PETSc-FEM: a general purpose, parallel, multi-physics FEM program, 1999–2006] can be found in works [Paz R, Storti M. An interface strip preconditioner for domain decomposition methods: application to hydrology. Int J Numer Methods Eng 2005;62(13):1873–94; Paz R, Nigro N, Storti M. On the efficiency and quality of numerical solutions in cfd problems using the interface strip preconditioner for domain decomposition methods. Int J Numer Methods Fluids, in press].Fil: Storti, Mario Alberto. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Desarrollo Tecnológico para la Industria Química. Universidad Nacional del Litoral. Instituto de Desarrollo Tecnológico para la Industria Química; ArgentinaFil: Dalcin, Lisandro Daniel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Desarrollo Tecnológico para la Industria Química. Universidad Nacional del Litoral. Instituto de Desarrollo Tecnológico para la Industria Química; ArgentinaFil: Paz, Rodrigo Rafael. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Desarrollo Tecnológico para la Industria Química. Universidad Nacional del Litoral. Instituto de Desarrollo Tecnológico para la Industria Química; ArgentinaFil: Yommi, Alejandra Karina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Desarrollo Tecnológico para la Industria Química. Universidad Nacional del Litoral. Instituto de Desarrollo Tecnológico para la Industria Química; ArgentinaFil: Sonzogni, Victorio Enrique. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Desarrollo Tecnológico para la Industria Química. Universidad Nacional del Litoral. Instituto de Desarrollo Tecnológico para la Industria Química; ArgentinaFil: Nigro, Norberto Marcelo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Desarrollo Tecnológico para la Industria Química. Universidad Nacional del Litoral. Instituto de Desarrollo Tecnológico para la Industria Química; Argentin
On the efficient preconditioning of the Stokes equations in tight geometries
If the Stokes equations are properly discretized, it is well-known that the
Schur complement matrix is spectrally equivalent to the identity matrix.
Moreover, in the case of simple geometries, it is often observed that most of
its eigenvalues are equal to one. These facts form the basis for the famous
Uzawa and Krylov-Uzawa algorithms. However, in the case of complex geometries,
the Schur complement matrix can become arbitrarily ill-conditioned having a
significant portion of non-unit eigenvalues, which makes the established Uzawa
preconditioner inefficient. In this article, we study the Schur complement
formulation for the staggered finite-difference discretization of the Stokes
problem in 3D CT images and synthetic 2D geometries. We numerically investigate
the performance of the CG iterative method with the Uzawa and SIMPLE
preconditioners and draw several conclusions. First, we show that in the case
of low porosity, CG with the SIMPLE preconditioner converges faster to the
discrete pressure and provides a more accurate calculation of sample
permeability. Second, we show that an increase in the surface-to-volume ratio
leads to an increase in the condition number of the Schur complement matrix,
while the dependence is inverse for the Schur complement matrix preconditioned
with the SIMPLE. As an explanation, we conjecture that the no-slip boundary
conditions are the reason for non-unit eigenvalues of the Schur complement
A Parallel-in-Time Preconditioner for the Schur Complement of Parabolic Optimal Control Problems
For optimal control problems constrained by a initial-valued parabolic PDE,
we have to solve a large scale saddle point algebraic system consisting of
considering the discrete space and time points all together. A popular strategy
to handle such a system is the Krylov subspace method, for which an efficient
preconditioner plays a crucial role. The matching-Schur-complement
preconditioner has been extensively studied in literature and the
implementation of this preconditioner lies in solving the underlying PDEs
twice, sequentially in time. In this paper, we propose a new preconditioner for
the Schur complement, which can be used parallel-in-time (PinT) via the so
called diagonalization technique. We show that the eigenvalues of the
preconditioned matrix are low and upper bounded by positive constants
independent of matrix size and the regularization parameter. The uniform
boundedness of the eigenvalues leads to an optimal linear convergence rate of
conjugate gradient solver for the preconditioned Schur complement system. To
the best of our knowledge, it is the first time to have an optimal convergence
analysis for a PinT preconditioning technique of the optimal control problem.
Numerical results are reported to show that the performance of the proposed
preconditioner is robust with respect to the discretization step-sizes and the
regularization parameter
Hierarchical Schur complement preconditioner for the stochastic Galerkin finite element methods
Use of the stochastic Galerkin finite element methods leads to large systems
of linear equations obtained by the discretization of tensor product solution
spaces along their spatial and stochastic dimensions. These systems are
typically solved iteratively by a Krylov subspace method. We propose a
preconditioner which takes an advantage of the recursive hierarchy in the
structure of the global matrices. In particular, the matrices posses a
recursive hierarchical two-by-two structure, with one of the submatrices block
diagonal. Each one of the diagonal blocks in this submatrix is closely related
to the deterministic mean-value problem, and the action of its inverse is in
the implementation approximated by inner loops of Krylov iterations. Thus our
hierarchical Schur complement preconditioner combines, on each level in the
approximation of the hierarchical structure of the global matrix, the idea of
Schur complement with loops for a number of mutually independent inner Krylov
iterations, and several matrix-vector multiplications for the off-diagonal
blocks. Neither the global matrix, nor the matrix of the preconditioner need to
be formed explicitly. The ingredients include only the number of stiffness
matrices from the truncated Karhunen-Lo\`{e}ve expansion and a good
preconditioned for the mean-value deterministic problem. We provide a condition
number bound for a model elliptic problem and the performance of the method is
illustrated by numerical experiments.Comment: 15 pages, 2 figures, 9 tables, (updated numerical experiments
Sweeping Preconditioner for the Helmholtz Equation: Moving Perfectly Matched Layers
This paper introduces a new sweeping preconditioner for the iterative
solution of the variable coefficient Helmholtz equation in two and three
dimensions. The algorithms follow the general structure of constructing an
approximate factorization by eliminating the unknowns layer by layer
starting from an absorbing layer or boundary condition. The central idea of
this paper is to approximate the Schur complement matrices of the factorization
using moving perfectly matched layers (PMLs) introduced in the interior of the
domain. Applying each Schur complement matrix is equivalent to solving a
quasi-1D problem with a banded LU factorization in the 2D case and to solving a
quasi-2D problem with a multifrontal method in the 3D case. The resulting
preconditioner has linear application cost and the preconditioned iterative
solver converges in a number of iterations that is essentially indefinite of
the number of unknowns or the frequency. Numerical results are presented in
both two and three dimensions to demonstrate the efficiency of this new
preconditioner.Comment: 25 page
A class of nonsymmetric preconditioners for saddle point problems
For iterative solution of saddle point problems, a nonsymmetric preconditioning is studied which, with respect to the upper-left block of the system matrix, can be seen as a variant of SSOR. An idealized situation where the SSOR is taken with respect to the skew-symmetric part plus the diagonal part of the upper-left block is analyzed in detail. Since action of the preconditioner involves solution of a Schur complement system, an inexact form of the preconditioner can be of interest. This results in an inner-outer iterative process. Numerical experiments with solution of linearized Navier-Stokes equations demonstrate efficiency of the new preconditioner, especially when the left-upper block is far from symmetric
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