Results 1  10
of
43
Computing and deflating eigenvalues while solving multiple right hand side linear systems with an application to quantum chromodynamics
, 2008
"... Abstract. We present a new algorithm that computes eigenvalues and eigenvectors of a Hermitian positive definite matrix while solving a linear system of equations with Conjugate Gradient (CG). Traditionally, all the CG iteration vectors could be saved and recombined through the eigenvectors of the t ..."
Abstract

Cited by 11 (2 self)
 Add to MetaCart
Abstract. We present a new algorithm that computes eigenvalues and eigenvectors of a Hermitian positive definite matrix while solving a linear system of equations with Conjugate Gradient (CG). Traditionally, all the CG iteration vectors could be saved and recombined through the eigenvectors of the tridiagonal projection matrix, which is equivalent theoretically to unrestarted Lanczos. Our algorithm capitalizes on the iteration vectors produced by CG to update only a small window of about ten vectors that approximate the eigenvectors. While this window is restarted in a locally optimal way, the CG algorithm for the linear system is unaffected. Yet, in all our experiments, this small window converges to the required eigenvectors at a rate identical to unrestarted Lanczos. After the solution of the linear system, eigenvectors that have not accurately converged can be improved in an incremental fashion by solving additional linear systems. In this case, eigenvectors identified in earlier systems can be used to deflate, and thus accelerate, the convergence of subsequent systems. We have used this algorithm with excellent results in lattice QCD applications, where hundreds of right hand sides may be needed. Specifically, about 70 eigenvectors are obtained to full accuracy after solving 24 right hand sides. Deflating these from the large number of subsequent right hand sides removes the dreaded critical slowdown, where the conditioning of the matrix increases as the quark mass reaches a critical value. Our experiments show almost a constant number of iterations for our method, regardless of quark mass, and speedups of 8 over original CG for light quark masses.
Block preconditioning of realvalued iterative algorithms for complex linear systems
, 2008
"... ..."
Analysis of iterative methods for the forcebased quasicontinuum method. manuscript
, 2009
"... Abstract. Forcebased atomisticcontinuum hybrid methods are the only known pointwise consistent methods for coupling a general atomistic model to a finite element continuum model. For this reason, and due to their algorithmic simplicity, forcebased coupling methods have become a popular class of a ..."
Abstract

Cited by 8 (8 self)
 Add to MetaCart
Abstract. Forcebased atomisticcontinuum hybrid methods are the only known pointwise consistent methods for coupling a general atomistic model to a finite element continuum model. For this reason, and due to their algorithmic simplicity, forcebased coupling methods have become a popular class of atomisticcontinuum hybrid models as well as other types of multiphysics models. However, the recently discovered unusual stability properties of the linearized forcebased quasicontinuum (QCF) approximation, especially its indefiniteness, present a challenge to the development of efficient and reliable iterative methods. We present analytic and computational results for the generalized minimal residual (GMRES) solution of the linearized QCF equilibrium equations. We show that the GMRES method accurately reproduces the stability of the forcebased approximation and conclude that an appropriately preconditioned GMRES method results in a reliable and efficient solution method. 1.
Block triangular preconditioners for Mmatrices and Markov chains
, 2006
"... We consider preconditioned Krylov subspace methods for solving large sparse linear systems under the assumption that the coefficient matrix is a (possibly singular)matrix. The matrices are partitioned into 2 x 2 block form using graph partitioning. Approximations to the Schur complement are use ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
We consider preconditioned Krylov subspace methods for solving large sparse linear systems under the assumption that the coefficient matrix is a (possibly singular)matrix. The matrices are partitioned into 2 x 2 block form using graph partitioning. Approximations to the Schur complement are used to produce various preconditioners of block triangular and block diagonal type. A few properties of the preconditioners are established, and extensive numerical experiments are used to illustrate the performance of the various preconditioners on singular linear systems arising from Markov modeling.
INTERPRETING IDR AS A PETROVGALERKIN METHOD
, 2009
"... The IDR method of Sonneveld and van Gijzen [SIAM J. Sci. Comput., 31:1035–1062, 2008] is shown to be a PetrovGalerkin (projection) method with a particular choice of left Krylov subspaces; these left subspaces are rational Krylov spaces. Consequently, other methods, such as BiCGStab and ML(s)BiCG ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
The IDR method of Sonneveld and van Gijzen [SIAM J. Sci. Comput., 31:1035–1062, 2008] is shown to be a PetrovGalerkin (projection) method with a particular choice of left Krylov subspaces; these left subspaces are rational Krylov spaces. Consequently, other methods, such as BiCGStab and ML(s)BiCG, which are mathematically equivalent to some versions of IDR, can also be interpreted as PetrovGalerkin methods. The connection with rational Krylov spaces inspired a new version of IDR, called RitzIDR, where the poles of the rational function are chosen as certain Ritz values. Experiments are presented illustrating the effectiveness of this new version.
EXTENSIONS OF CERTAIN GRAPHBASED ALGORITHMS FOR PRECONDITIONING
, 2007
"... The original TPABLO algorithms are a collection of algorithms which compute a symmetric permutation of a linear system such that the permuted system has a relatively full block diagonal with relatively large nonzero entries. This block diagonal can then be used as a preconditioner. We propose and an ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
The original TPABLO algorithms are a collection of algorithms which compute a symmetric permutation of a linear system such that the permuted system has a relatively full block diagonal with relatively large nonzero entries. This block diagonal can then be used as a preconditioner. We propose and analyze three extensions of this approach: we incorporate a nonsymmetric permutation to obtain a large diagonal, we use a more general parametrization for TPABLO, and we use a block GaussSeidel preconditioner which can be implemented to have the same execution time as the corresponding block Jacobi preconditioner. Experiments are presented showing that for certain classes of matrices, the block GaussSeidel preconditioner used with the system permuted with the new algorithm can outperform the best ILUT preconditioners in a large set of experiments.
A Dimensional Split Preconditioner for Stokes and Linearized Navier–Stokes Equations
, 2010
"... In this paper we introduce a new preconditioner for linear systems of saddle point type arising from the numerical solution of the Navier– Stokes equations. Our approach is based on a dimensional splitting of the problem along the components of the velocity field, resulting in a convergent fixedpoi ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
In this paper we introduce a new preconditioner for linear systems of saddle point type arising from the numerical solution of the Navier– Stokes equations. Our approach is based on a dimensional splitting of the problem along the components of the velocity field, resulting in a convergent fixedpoint iteration. The basic iteration is accelerated by a Krylov subspace method like restarted GMRES. The corresponding preconditioner requires at each iteration the solution of a set of discrete scalar elliptic equations, one for each component of the velocity field. Numerical experiments illustrating the convergence behavior for different finite element discretizations of Stokes and Oseen problems are included. Key words. saddle point problems, matrix splittings, iterative methods, preconditioning, Stokes problem, Oseen problem, stretched grids