Solution of Sparse Indefinite Systems of Linear Equations

Paige, Christopher C.; Saunders, Michael A.

doi:10.1137/0712047

Cited by 1,402 publications

(1,034 citation statements)

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“…Today there is a large community of researchers dedicated to the task of solving Cx = b and the spectral properties of C, appropriate iterative solvers and preconditioners have been well studied, [6]. The appearance of the zero matrix in the (2,2) block and the fact that A is positive definite mean that C falls into a relatively easy class of saddle point matrices, for which the minimal residual method (minres, [25]) is an optimal iterative solver. Convergence can be accelerated using symmetric and positive definite preconditioners, of which there are two well-known types.…”

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confidence: 99%

Preconditioning Stochastic Galerkin Saddle Point Systems

Powell¹,

Ullmann²

2010

SIAM J. Matrix Anal. & Appl.

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Abstract. Mixed finite element discretizations of deterministic second-order elliptic partial differential equations (PDEs) lead to saddle point systems for which the study of iterative solvers and preconditioners is mature. Galerkin approximation of solutions of stochastic second-order elliptic PDEs, which couple standard mixed finite element discretizations in physical space with global polynomial approximation on a probability space, also give rise to linear systems with familiar saddle point structure. For stochastically nonlinear problems, the solution of such systems presents a serious computational challenge. The blocks are sums of Kronecker products of pairs of matrices associated with two distinct discretizations and the systems are large, reflecting the curse of dimensionality inherent in most stochastic approximation schemes. Moreover, for the problems considered herein, the leading blocks of the saddle point matrices are block-dense and the cost of a matrix vector product is non-trivial.We implement a stochastic Galerkin discretization for the steady-state diffusion problem written as a mixed firstorder system. The diffusion coefficient is assumed to be a lognormal random field, approximated via a nonlinear function of a finite number of unbounded random parameters. We study the resulting saddle point systems and investigate the efficiency of block-diagonal preconditioners of Schur complement and augmented type, for use with minres. By introducing so-called Kronecker product preconditioners we improve the robustness of cheap, mean-based preconditioners with respect to the statistical properties of the stochastically nonlinear diffusion coefficients.

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Preconditioning Stochastic Galerkin Saddle Point Systems

Powell¹,

Ullmann²

2010

SIAM J. Matrix Anal. & Appl.

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“…This is the case when we use a Krylov subspace method starting with a zero vector. For example, the conjugate gradient (CG) method on the normal equation leads to the min-length solution (see Paige and Saunders [20]). In practice, CGLS [16] or LSQR [21] are preferable because they are equivalent to applying CG to the normal equation in exact arithmetic but they are numerically more stable.…”

Section: Least Squares Solversmentioning

confidence: 99%

LSRN: A Parallel Iterative Solver for Strongly Over- or Underdetermined Systems

Meng

Saunders

Mahoney

2014

SIAM J. Sci. Comput.

111

127

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We describe a parallel iterative least squares solver named that is based on random normal projection. computes the min-length solution to minx∈ℝn ‖Ax − b‖2, where A ∈ ℝm × n with m ≫ n or m ≪ n, and where A may be rank-deficient. Tikhonov regularization may also be included. Since A is involved only in matrix-matrix and matrix-vector multiplications, it can be a dense or sparse matrix or a linear operator, and automatically speeds up when A is sparse or a fast linear operator. The preconditioning phase consists of a random normal projection, which is embarrassingly parallel, and a singular value decomposition of size ⌈γ min(m, n)⌉ × min(m, n), where γ is moderately larger than 1, e.g., γ = 2. We prove that the preconditioned system is well-conditioned, with a strong concentration result on the extreme singular values, and hence that the number of iterations is fully predictable when we apply LSQR or the Chebyshev semi-iterative method. As we demonstrate, the Chebyshev method is particularly efficient for solving large problems on clusters with high communication cost. Numerical results show that on a shared-memory machine, is very competitive with LAPACK’s DGELSD and a fast randomized least squares solver called Blendenpik on large dense problems, and it outperforms the least squares solver from SuiteSparseQR on sparse problems without sparsity patterns that can be exploited to reduce fill-in. Further experiments show that scales well on an Amazon Elastic Compute Cloud cluster.

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“…Finally, after it is decided that the estimate of the residual norm is small enough, the final factorization of Hm will be used to fully solve the system (3.4). The Gaussian elimination with partial pivoting gives satisfactory results in general, but one might as well use a more stable decomposition, as the LQ decomposition in [14], [15], although at a high cost.…”

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“…When the matrix A is symmetric, then, by taking/» = 2, we obtain a version of the conjugate gradient method which is known to be equivalent to the Lanczos algorithm; see [14]. In that case the vectors vx, .…”

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“…Let us give a few indications about the problem of computing the estimation \e"y(m\ since it will appear in several parts along the paper. Parlett [15] suggests utilizing a recurrence relation proposed by Paige and Saunders [14], which is based upon the LQ factorization of //". Another interesting possibility is to perform the more economical factorization provided by the Gaussian elimination with partial pivoting on the matrix Hj.…”

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Krylov subspace methods for solving large unsymmetric linear systems

Saad¹

1981

Math. Comp.

381

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Abstract. Some algorithms based upon a projection process onto the Krylov subspace Km = Span(r0, Ar^, .. . ,Am~>r¿) are developed, generalizing the method of conjugate gradients to unsymmetric systems. These methods are extensions of Arnoldi's algorithm for solving eigenvalue problems. The convergence is analyzed in terms of the distance of the solution to the subspace Km and some error bounds are established showing, in particular, a similarity with the conjugate gradient method (for symmetric matrices) when the eigenvalues are real. Several numerical experiments are described and discussed.

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Solution of Sparse Indefinite Systems of Linear Equations

Cited by 1,402 publications

References 11 publications

Preconditioning Stochastic Galerkin Saddle Point Systems

Preconditioning Stochastic Galerkin Saddle Point Systems

LSRN: A Parallel Iterative Solver for Strongly Over- or Underdetermined Systems

Krylov subspace methods for solving large unsymmetric linear systems

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