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Recent computational developments in Krylov subspace methods for linear systems
 NUMER. LINEAR ALGEBRA APPL
, 2007
"... Many advances in the development of Krylov subspace methods for the iterative solution of linear systems during the last decade and a half are reviewed. These new developments include different versions of restarted, augmented, deflated, flexible, nested, and inexact methods. Also reviewed are metho ..."
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Cited by 85 (12 self)
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Many advances in the development of Krylov subspace methods for the iterative solution of linear systems during the last decade and a half are reviewed. These new developments include different versions of restarted, augmented, deflated, flexible, nested, and inexact methods. Also reviewed are methods specifically tailored to systems with special properties such as special forms of symmetry and those depending on one or more parameters.
Recycling Krylov Subspaces for Sequences of Linear Systems
 SIAM J. Sci. Comput
, 2004
"... Many problems in engineering and physics require the solution of a large sequence of linear systems. We can reduce the cost of solving subsequent systems in the sequence by recycling information from previous systems. We consider two dierent approaches. For several model problems, we demonstrate tha ..."
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Cited by 74 (6 self)
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Many problems in engineering and physics require the solution of a large sequence of linear systems. We can reduce the cost of solving subsequent systems in the sequence by recycling information from previous systems. We consider two dierent approaches. For several model problems, we demonstrate that we can reduce the iteration count required to solve a linear system by a factor of two. We consider both Hermitian and nonHermitian problems, and present numerical experiments to illustrate the eects of subspace recycling.
Deflated iterative methods for linear equations with multiple righthand sides
, 2004
"... Abstract. A new approach is discussed for solving large nonsymmetric systems of linear equations with multiple righthand sides. The first system is solved with a deflated GMRES method that generates eigenvector information at the same time that the linear equations are solved. Subsequent systems ar ..."
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Cited by 20 (7 self)
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Abstract. A new approach is discussed for solving large nonsymmetric systems of linear equations with multiple righthand sides. The first system is solved with a deflated GMRES method that generates eigenvector information at the same time that the linear equations are solved. Subsequent systems are solved by combining restarted GMRES with a projection over the previously determined eigenvectors. This approach offers an alternative to block methods, and it can also be combined with a block method. It is useful when there are a limited number of small eigenvalues that slow the convergence. An example is given showing significant improvement for a problem from quantum chromodynamics. The second and subsequent righthand sides are solved much quicker than without the deflation. This new approach is relatively simple to implement and is very efficient compared to other deflation methods.
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 ..."
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Cited by 20 (2 self)
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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.
Deflated GMRES for systems with multiple shifts and multiple righthand sides
, 2007
"... Abstract. We consider solution of multiply shifted systems of nonsymmetric linear equations, possibly also with multiple righthand sides. First, for a single righthand side, the matrix is shifted by several multiples of the identity. Such problems arise in a number of applications, including latti ..."
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Cited by 15 (1 self)
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Abstract. We consider solution of multiply shifted systems of nonsymmetric linear equations, possibly also with multiple righthand sides. First, for a single righthand side, the matrix is shifted by several multiples of the identity. Such problems arise in a number of applications, including lattice quantum chromodynamics where the matrices are complex and nonHermitian. Some Krylov iterative methods such as GMRES and BiCGStab have been used to solve multiply shifted systems for about the cost of solving just one system. Restarted GMRES can be improved by deflating eigenvalues for matrices that have a few small eigenvalues. We show that a particular deflated method, GMRESDR, can be applied to multiply shifted systems. In quantum chromodynamics, it is common to have multiple righthand sides with multiple shifts for each righthand side. We develop a method that efficiently solves the multiple righthand sides by using a deflated version of GMRES and yet keeps costs for all of the multiply shifted systems close to those for one shift. An example is given showing this can be extremely effective with a quantum chromodynamics matrix.
An efficient block variant of GMRES
 SIAM J. Sci. Comput
"... Abstract. We present an alternative to the standard restarted GMRES algorithm for solving a single righthand side linear system Ax = b based on solving the block linear system AX = B. Additional starting vectors and righthand sides are chosen to accelerate convergence. Algorithm performance, i.e. ..."
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Cited by 10 (2 self)
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Abstract. We present an alternative to the standard restarted GMRES algorithm for solving a single righthand side linear system Ax = b based on solving the block linear system AX = B. Additional starting vectors and righthand sides are chosen to accelerate convergence. Algorithm performance, i.e. time to solution, is improved by using the matrix A in operations on groups of vectors, or “multivectors, ” thereby reducing the movement of A through memory. The efficient implementation of our method depends on a fast matrixmultivector multiply routine. We present numerical results that show that the time to solution of the new method is up to two and half times faster than that of restarted GMRES on preconditioned problems. Furthermore, we demonstrate the impact of implementation choices on data movement and, as a result, algorithm performance. Key words. GMRES, block GMRES, iterative methods, Krylov subspace techniques, restart, nonsymmetric linear systems, memory access costs AMS subject classifications. 65F10
RECYCLING BICG WITH AN APPLICATION TO MODEL REDUCTION
, 2012
"... Science and engineering problems frequently require solving a sequence of dual linear systems. Besides having to store only a few Lanczos vectors, using the biconjugate gradient method (BiCG) to solve dual linear systems has advantages for specific applications. For example, using BiCG to solve the ..."
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Cited by 9 (4 self)
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Science and engineering problems frequently require solving a sequence of dual linear systems. Besides having to store only a few Lanczos vectors, using the biconjugate gradient method (BiCG) to solve dual linear systems has advantages for specific applications. For example, using BiCG to solve the dual linear systems arising in interpolatory model reduction provides a backward error formulation in the model reduction framework. Using BiCG to evaluate bilinear forms—for example, in quantum Monte Carlo (QMC) methods for electronic structure calculations—leads to a quadratic error bound. Since our focus is on sequences of dual linear systems, we introduce recycling BiCG, a BiCG method that recycles two Krylov subspaces from one pair of dual linear systems to the next pair. The derivation of recycling BiCG also builds the foundation for developing recycling variants of other biLanczos based methods, such as CGS, BiCGSTAB, QMR, and TFQMR. We develop an augmented biLanczos algorithm and a modified twoterm recurrence to include recycling in the iteration. The recycle spaces are approximate left and right invariant subspaces corresponding to the eigenvalues closest to the origin. These recycle spaces are found by solving a small generalized eigenvalue problem alongside the dual linear systems being solved in the sequence. We test our algorithm in two application areas. First, we solve a discretized partial differential equation (PDE) of convectiondiffusion type. Such a problem provides wellknown test cases that are easy to test and analyze further. Second, we use recycling BiCG in the iterative rational Krylov algorithm (IRKA) for interpolatory model reduction. IRKA requires solving sequences of slowly changing dual linear systems. We analyze the generated recycle spaces and show up to 70 % savings in iterations. For our model reduction test problem, we show that solving the problem without recycling leads to (about) a 50 % increase in runtime.
Low Frequency Tangential Filtering Decomposition
"... Abstract For unsymmetric blocktridiagonal systems of linear equations arising from the discretization of partial differential equations, a composite preconditioner is proposed and tested. It combines a classical ILU0 factorization for high frequencies with a tangential filtering preconditioner. Th ..."
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Cited by 8 (4 self)
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Abstract For unsymmetric blocktridiagonal systems of linear equations arising from the discretization of partial differential equations, a composite preconditioner is proposed and tested. It combines a classical ILU0 factorization for high frequencies with a tangential filtering preconditioner. The choice of the filtering vector is important: the testvector is the Ritz eigenvector corresponding to the lowest approximate eigenvalue, obtained after a limited number of iterations of a ILU0 preconditioned Krylov method. Numerical tests are carried out for this method.
On improving linear solver performance: A block variant of GMRes
 SIAM Journal on Scientific Computing
"... Abstract. The increasing gap between processor performance and memory access time warrants the reexamination of data movement in iterative linear solver algorithms. For this reason, we explore and establish the feasibility of modifying a standard iterative linear solver algorithm in a manner that ..."
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Cited by 8 (1 self)
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Abstract. The increasing gap between processor performance and memory access time warrants the reexamination of data movement in iterative linear solver algorithms. For this reason, we explore and establish the feasibility of modifying a standard iterative linear solver algorithm in a manner that reduces the movement of data through memory. In particular, we present an alternative to the restarted GMRES algorithm for solving a single righthand side linear system Ax = b based on solving the block linear system AX = B. Algorithm performance, i.e. time to solution, is improved by using the matrix A in operations on groups of vectors. Experimental results demonstrate the importance of implementation choices on data movement as well as the effectiveness of the new method on a variety of problems from different application areas.