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.

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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.

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 fixed-point 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

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 Gauss-Seidel 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 Gauss-Seidel preconditioner used with the system permuted with the new algorithm can outperform the best ILUT preconditioners in a large set of experiments.

Abstract. We study an inexact implicitly restarted Arnoldi (IRA) method for computing a few eigenpairs of generalized non-Hermitian eigenvalue problems with spectral transformation, where in each Arnoldi step (outer iteration) the matrix-vector product involving the transformed operator is performed by iterative solution (inner iteration) of the corresponding linear system of equations. We provide new perspectives and analysis of two major strategies that help reduce the inner iteration cost: a special type of preconditioner with “tuning”, and gradually relaxed tolerances for the solution of the linear systems. We study a new tuning strategy constructed from vectors in both previous and the current IRA cycles, and we show how tuning is used in a new two-phase algorithm to greatly reduce inner iteration counts. We give an upper bound of the allowable tolerances of the linear systems and propose an alternative estimate of the tolerances. In addition, the inner iteration cost can be further reduced through the use of subspace recycling with iterative linear solvers. The effectiveness of these strategies is demonstrated by numerical experiments.

It is unknown how to include stochastic process variation into fast-multipole-method (FMM) for a full chip capacitance extraction. This paper presents a parallel FMM extraction using stochastic polynomial expanded geometrical moments. It utilizes multiprocessors to evaluate in parallel for the stochastic potential interaction and its matrix-vector product (MVP) with charge. Moreover, a generalized minimal residual (GMRES) method with deflation is modified to incrementally consider the nominal value and the variance. The overall extraction flow is called piCAP. Experiments show that the parallel MVP in piCAP is up to 3X faster than the serial MVP, and the incremental GMRES in pi-CAP is up to 15X faster than non-incremental GMRES methods.

Abstract. We study inexact Rayleigh quotient iteration (IRQI) for computing a simple interior eigenpair of the generalized eigenvalue problem Av λBv, providing new insights into a special type of preconditioners with “tuning ” for the efficient iterative solution of the shifted linear systems that arise in this algorithm. We first give a new convergence analysis of IRQI, showing that locally cubic and quadratic convergence can be achieved for Hermitian and non-Hermitian problems, respectively, if the shifted linear systems are solved by a generic Krylov subspace method with a tuned preconditioner to a reasonably small fixed tolerance. We then refine the study by Freitag and Spence [Linear Algebra Appl., 428 (2008), pp. 2049–2060] on the equivalence of the inner solves of IRQI and single-vector Jacobi–Davidson method where a full orthogonalization method with a tuned preconditioner is used as the inner solver. We also provide some new perspectives on the tuning strategy, showing that tuning is essentially needed only in the first inner iteration in the non-Hermitian case. Based on this observation, we propose a flexible GMRES algorithm with a special configuration in the first inner step, and show that this method is as efficient as GMRES with the tuned preconditioner.

Solution of linear systems from an optimal control problem