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48
An efficient preconditioned CG method for the solution of a class of layered problems with extreme contrasts in the coefficients
 J. Comput. Phys
, 1999
"... Knowledge of fluid pressure is important to predict the presence of oil and gas in reservoirs. A mathematical model for the prediction of fluid pressures is given by a timedependent diffusion equation. Application of the finite element method leads to a system of linear equations. A complication is ..."
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Cited by 49 (23 self)
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Knowledge of fluid pressure is important to predict the presence of oil and gas in reservoirs. A mathematical model for the prediction of fluid pressures is given by a timedependent diffusion equation. Application of the finite element method leads to a system of linear equations. A complication is that the underground consists of layers with very large differences in permeability. This implies that the symmetric and positive definite coefficient matrix has a very large condition number. Bad convergence behavior of the CG method has been observed; moreover, a classical termination criterion is not valid in this problem. After diagonal scaling of the matrix the number of extreme eigenvalues is reduced and it is proved to be equal to the number of layers with a high permeability. For the IC preconditioner the same behavior is observed. To annihilate the effect of the extreme eigenvalues a deflated CG method is used. The convergence rate improves considerably and the termination criterion becomes again reliable. Finally a cheap approximation of the eigenvectors is proposed. c ○ 1999 Academic Press Key Words: porous media; preconditioned conjugate gradients; deflation; Poisson equation; discontinuous coefficients across layers; eigenvectors; finite element method. 1.
The Idea Behind Krylov Methods
 American Mathematical Monthly
, 1997
"... We explain why Krylov methods make sense, and why it is natural to represent a solution to a linear system as a member of a Krylov space. In particular we show that the solution to a nonsingular linear system Ax = b lies in a Krylov space whose dimension is the degree of the minimal polynomial of A. ..."
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Cited by 29 (2 self)
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We explain why Krylov methods make sense, and why it is natural to represent a solution to a linear system as a member of a Krylov space. In particular we show that the solution to a nonsingular linear system Ax = b lies in a Krylov space whose dimension is the degree of the minimal polynomial of A. Therefore, if the minimal polynomial of A has low degree then the space in which a Krylov method searches for the solution is small. In this case a Krylov method has the opportunity to converge fast. When the matrix is singular, however, Krylov methods can fail. Even if the linear system does have a solution, it may not lie in a Krylov space. In this case we describe the class of righthand sides for which a solution lies in a Krylov space. As it happens, there is only a single solution that lies in a Krylov space, and it can be obtained from the Drazin inverse. Center for Research In Scientific Computation, Department of Mathematics, North Carolina State University, P. O. Box 8205, Rale...
Expressions And Bounds For The GMRES Residual
 BIT
, 1999
"... . Expressions and bounds are derived for the residual norm in GMRES. It is shown that the minimal residual norm is large as long as the Krylov basis is wellconditioned.For scaled Jordan blocks the minimal residual norm is expressed in terms of eigenvalues and departure from normality.For normal mat ..."
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Cited by 25 (0 self)
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. Expressions and bounds are derived for the residual norm in GMRES. It is shown that the minimal residual norm is large as long as the Krylov basis is wellconditioned.For scaled Jordan blocks the minimal residual norm is expressed in terms of eigenvalues and departure from normality.For normal matrices the minimal residual norm is expressed in terms of products of relative eigenvalue di#erences. Key words. linear system, Krylov methods, GMRES, MINRES, Vandermonde matrix, eigenvalues, departure from normality AMS subject classi#cation. 15A03, 15A06, 15A09, 15A12, 15A18, 15A60, 65F10, 65F15, 65F20, 65F35. 1. Introduction.. The generalised minimal residual method #GMRES# #31, 36# #and MINRES for Hermitian matrices #30## is an iterative method for solving systems of linear equations Ax = b. The approximate solution in iteration i minimises the twonorm of the residual b , Az over the Krylov space spanfb;Ab;:::;A i,1 bg. The goal of this paper is to express this minimal residual norm...
Solving Some Large Scale Semidefinite Programs Via the Conjugate Residual Method
, 2000
"... Most current implementations of interiorpoint methods for semidefinite programming use a direct method to solve the Schur complement equation (SCE) M y = h in computing the search direction. When the number of constraints is large, the problem of having insufficient memory to store M can be avoided ..."
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Cited by 25 (10 self)
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Most current implementations of interiorpoint methods for semidefinite programming use a direct method to solve the Schur complement equation (SCE) M y = h in computing the search direction. When the number of constraints is large, the problem of having insufficient memory to store M can be avoided if an iterative method is used instead. Numerical experiments have shown that the conjugate residual (CR) method typically takes a huge number of steps to generate a high accuracy solution. On the other hand, it is difficult to incorporate traditional preconditioners into the SCE, except for block diagonal preconditioners. We decompose the SCE into a 2 &times; 2 block system by decomposing y (similarly for h) into two orthogonal components with one lying in a certain subspace that is determined from the structure of M . Numerical experiments on semidefinite programming problems arising from Lovász function of graphs and MAXCUT problems show that high accuracy solutions can be obtained with moderate n...
An immersed interface method for viscous incompressible flows involving rigid and flexible boundaries
 J. Comp. Phys
, 2006
"... We present an immersed interface method for the incompressible NavierStokes equations capable of handling rigid immersed boundaries. The immersed boundary is represented by a set of Lagrangian control points. In order to guarantee that the noslip condition on the boundary is satisfied, singular fo ..."
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Cited by 24 (3 self)
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We present an immersed interface method for the incompressible NavierStokes equations capable of handling rigid immersed boundaries. The immersed boundary is represented by a set of Lagrangian control points. In order to guarantee that the noslip condition on the boundary is satisfied, singular forces are applied on the fluid. The forces are related to the jumps in pressure and the jumps in the derivatives of both pressure and velocity, and are interpolated using cubic splines. The strength of the singular forces is determined by solving a small system of equations iteratively at each time step. The NavierStokes equations are discretized on a staggered Cartesian grid by a second order accurate projection method for pressure and velocity. Keywords: Immersed interface method, NavierStokes equations, Cartesian grid method, finite difference, fast Poisson solvers, irregular domains.
Breakdownfree GMRES for singular systems
 SIAM J. Matrix Anal. Appl
"... Abstract. GMRES is a popular iterative method for the solution of large linear systems of equations with a square nonsingular matrix. When the matrix is singular, GMRES may break down before an acceptable approximate solution has been determined. This paper discusses properties of GMRES solutions at ..."
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Cited by 19 (10 self)
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Abstract. GMRES is a popular iterative method for the solution of large linear systems of equations with a square nonsingular matrix. When the matrix is singular, GMRES may break down before an acceptable approximate solution has been determined. This paper discusses properties of GMRES solutions at breakdown and presents a modification of GMRES to overcome the breakdown.
Algorithms for range restricted iterative methods for linear discrete illposed problems
 Numer. Algorithms
"... Abstract. This paper is concerned with iterative solution methods for large linear systems of equations with a matrix of illdetermined rank and an errorcontaminated righthand side. The numerical solution is delicate, because the matrix is very illconditioned and may be singular. It is natural to ..."
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Cited by 12 (7 self)
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Abstract. This paper is concerned with iterative solution methods for large linear systems of equations with a matrix of illdetermined rank and an errorcontaminated righthand side. The numerical solution is delicate, because the matrix is very illconditioned and may be singular. It is natural to require that the computed iterates live in the range of the matrix when the latter is symmetric, because then the iterates are orthogonal to the null space. Computational experience indicates that it can be beneficial to require that the iterates live in the range of the matrix also when the latter is nonsymmetric. We discuss the design and implementation of iterative methods that determine iterates with this property. New implementations that are particularly well suited for use with the discrepancy principle are described.
MODIFIED GRAM–SCHMIDT (MGS), LEAST SQUARES, AND BACKWARD STABILITY OF MGSGMRES
, 2006
"... The generalized minimum residual method (GMRES) [Y. Saad and M. Schultz, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 856–869] for solving linear systems Ax = b is implemented as a sequence of least squares problems involving Krylov subspaces of increasing dimensions. The most usual implementation ..."
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Cited by 11 (1 self)
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The generalized minimum residual method (GMRES) [Y. Saad and M. Schultz, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 856–869] for solving linear systems Ax = b is implemented as a sequence of least squares problems involving Krylov subspaces of increasing dimensions. The most usual implementation is modified Gram–Schmidt GMRES (MGSGMRES). Here we show that MGSGMRES is backward stable. The result depends on a more general result on the backward stability of a variant of the MGS algorithm applied to solving a linear least squares problem, and uses other new results on MGS and its loss of orthogonality, together with an important but neglected condition number, and a relation between residual norms and certain singular values.
Accurate Solution of Weighted Least Squares by Iterative Methods
 Tech. Rep. ANL/MCSP6440297, Mathematics and Computer Science Division, Argonne National Laboratory
, 1997
"... . We consider the weighted leastsquares (WLS) problem with a very illconditioned weight matrix. Weighted leastsquares problems arise in many applications including linear programming, electrical networks, boundary value problems, and structures. Because of roundoff errors, standard iterative meth ..."
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Cited by 10 (2 self)
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. We consider the weighted leastsquares (WLS) problem with a very illconditioned weight matrix. Weighted leastsquares problems arise in many applications including linear programming, electrical networks, boundary value problems, and structures. Because of roundoff errors, standard iterative methods for solving a WLS problem with illconditioned weights may not give the correct answer. Indeed, the difference between the true and computed solution (forward error) may be large. We propose an iterative algorithm, called MINRESL, for solving WLS problems. The MINRESL method is the application of MINRES, a Krylovspace method due to Paige and Saunders, to a certain layered linear system. Using a simplified model of the effects of roundoff error, we prove that MINRESL gives answers with small forward error. We present computational experiments for some applications. This work has been supported in part by an NSF Presidential Young Investigator grant, with matching funds received fr...
An algebraic multigrid preconditioner for a class of singular Mmatrices
 SIAMJ. Sci. Comput
, 1982
"... Abstract. We apply algebraic multigrid (AMG) as a preconditioner for solving large singular linear systems of the type (I − T T)x = 0 with GMRES. Here, T is assumed to be the transition matrix of a Markov process. Although AMG and GMRES were originally designed for the solution of regular systems, w ..."
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Cited by 9 (0 self)
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Abstract. We apply algebraic multigrid (AMG) as a preconditioner for solving large singular linear systems of the type (I − T T)x = 0 with GMRES. Here, T is assumed to be the transition matrix of a Markov process. Although AMG and GMRES were originally designed for the solution of regular systems, with adequate adaptation their applicability can be extended to problems as described above.