Augmented Krylov Subspace Research Articles

Computing interior eigenpairs in augmented Krylov subspace produced by Jacobi–Davidson correction equation

Computing interior eigenpairs in augmented Krylov subspace produced by Jacobi–Davidson correction equation

An augmented wavelet reconstructor for atmospheric tomography

Atmospheric tomography, i.e., the reconstruction of the turbulence profile in the atmosphere, is a challenging task for adaptive optics (AO) systems for the next generation of extremely large telescopes. Within the AO community, the solver of first choice is the so-called Matrix Vector Multiplication (MVM) method, which directly applies the (regularized) generalized inverse of the system operator to the data. For small telescopes this approach is feasible, however, for larger systems such as the European Extremely Large Telescope (ELT), the atmospheric tomography problem is considerably more complex, and the computational efficiency becomes an issue. Iterative methods such as the Finite Element Wavelet Hybrid Algorithm (FEWHA) are a promising alternative. FEWHA is a wavelet-based reconstructor that uses the well-known iterative preconditioned conjugate gradient (PCG) method as a solver. The number of floating point operations and the memory usage are decreased significantly by using a matrix-free representation of the forward operator. A crucial indicator for the real-time performance are the number of PCG iterations. In this paper, we propose an augmented version of FEWHA, where the number of iterations is decreased by 50% using an augmented Krylov subspace method. We demonstrate that a parallel implementation of augmented FEWHA allows the fulfilment of the real-time requirements of the ELT.

Computing eigenpairs of Hermitian matrices in augmented Krylov subspace produced by Rayleigh quotient iterations

In this paper, we first introduce a class of augmented Krylov subspace that is a union of the standard Krylov subspace and another low-dimension subspace, and then construct the corresponding augmented Krylov subspace method for computing some extreme eigenvalues and their associated eigenvectors of Hermitian matrices. In order to improve the overall performance of this method, we present some practical implementations with preconditioning, which then results in the preconditioned augmented Krylov subspace method. In addition, through convergence analysis it indicates that the preconditioned augmented Krylov subspace method attains cubic convergence rate locally. In the last, we use some numerical examples to demonstrate the convergence property and the superiority of this method.

Computing eigenpairs in augmented Krylov subspace produced by Jacobi–Davidson correction equation

In this paper, we present an augmented Krylov subspace method for computing some extreme eigenvalues and corresponding eigenvectors of Hermitian matrices. The augmented Krylov subspace, which is a union of the standard Krylov subspace and another low-dimension subspace used to extract the approximations to the desired eigenpairs, is essentially different from the projection subspace involved in the Jacobi–Davidson iteration method. The augmented Krylov subspace method converges globally and attains cubic convergence rate locally. Some numerical experiments are carried out to demonstrate the convergence property and the competitiveness of this method.

Deflated and Augmented Krylov Subspace Methods: A Framework for Deflated BiCG and Related Solvers

We present an extension of the framework of Gaul et al. [SIAM J. Matrix Anal. Appl., 34 (2013), pp. 495--518] for deflated and augmented Krylov subspace methods satisfying a Galerkin condition to more general Petrov--Galerkin conditions. The main goal is to apply the framework to the biconjugate gradient method (BiCG) and some of its generalizations, including BiCGStab and IDR$(s)$. For such applications the assumptions of Gaul et al. were too restrictive. Our abstract approach does not depend on particular recurrences and thus simplifies the derivation of theoretical results. It easily leads to a variety of realizations by specific algorithms. We do not go into algorithmic details, but we show that for every method there are two different approaches for extending it by augmentation and deflation: one that explicitly takes care of the augmentation space in every step, and one that applies the appropriate Krylov subspace method to a projected problem but requires a correction step at the end. The latter ap...

A Framework for Deflated and Augmented Krylov Subspace Methods

We consider deflation and augmentation techniques for accelerating the convergence of Krylov subspace methods for the solution of nonsingular linear algebraic systems. Despite some formal similarity, the two techniques are conceptually different from preconditioning. Deflation (in the sense the term is used here) "removes" certain parts from the operator making it singular, while augmentation adds a subspace to the Krylov subspace (often the one that is generated by the singular operator); in contrast, preconditioning changes the spectrum of the operator without making it singular. Deflation and augmentation have been used in a variety of methods and settings. Typically, deflation is combined with augmentation to compensate for the singularity of the operator, but both techniques can be applied separately. We introduce a framework of Krylov subspace methods that satisfy a Galerkin condition. It includes the families of orthogonal residual (OR) and minimal residual (MR) methods. We show that in this framework augmentation can be achieved either explicitly or, equivalently, implicitly by projecting the residuals appropriately and correcting the approximate solutions in a final step. We study conditions for a breakdown of the deflated methods, and we show several possibilities to avoid such breakdowns for the deflated MINRES method. Numerical experiments illustrate properties of different variants of deflated MINRES analyzed in this paper.

Krylov subspace method based on data preprocessing technology

The performance of adaptive beamforming techniques is limited by the nonhomogeneous clutter scenario. An augmented Krylov subspace method is proposed, which utilizes only a single snapshot of the data for adaptive processing. The novel algorithm puts together a data preprocessor and adaptive Krylov subspace algorithm, where the data preprocessor suppresses discrete interference and the adaptive Krylov subspace algorithm suppresses homogeneous clutter. The novel method uses a single snapshot of the data received by the array antenna to generate a cancellation matrix that does not contain the signal of interest (SOI) component, thus, it mitigates the problem of highly nonstationary clutter environment and it helps to operate in real-time. The benefit of not requiring the training data comes at the cost of a reduced degree of freedom (DOF) of the system. Simulation illustrates the effectiveness in clutter suppression and adaptive beamforming. The numeric results show good agreement with the proposed theorem.

A refined harmonic Rayleigh-Ritz procedure and an explicitly restarted refined harmonic Arnoldi algorithm

The work is fourfold. First, a refined harmonic Rayleigh-Ritz procedure is proposed, some relationships are established between the refined harmonic Ritz vector and the harmonic Ritz vector, an a priori error bound is derived for the refined harmonic Ritz vector, and some properties are established on Rayleigh quotients and residual norms. Second, a resulting refined harmonic Arnoldi method is discussed, and how to compute the refined harmonic Ritz vectors cheaply and accurately is considered. Third, an explicitly restarted refined harmonic Arnoldi algorithm is developed over an augmented Krylov subspace. Finally, numerical examples are reported that compare the new algorithm with the implicitly restarted harmonic Arnoldi algorithm (IRHA) and the implicitly restarted refined harmonic Arnoldi algorithm (IRRHA). Numerical results confirm efficiency of the new algorithm.

Convergence conditions for a restarted GMRES method augmented with eigenspaces

We consider the GMRES(m,k) method for the solution of linear systems Ax=b, i.e. the restarted GMRES with restart m where to the standard Krylov subspace of dimension m the other subspace of dimension k is added, resulting in an augmented Krylov subspace. This additional subspace approximates usually an A-invariant subspace. The eigenspaces associated with the eigenvalues closest to zero are commonly used, as those are thought to hinder convergence the most. The behaviour of residual bounds is described for various situations which can arise during the GMRES(m,k) process. The obtained estimates for the norm of the residual vector suggest sufficient conditions for convergence of GMRES(m,k) and illustrate that these augmentation techniques can remove stagnation of GMRES(m) in many cases. All estimates are independent of the choice of an initial approximation. Conclusions and remarks assessing numerically the quality of proposed bounds conclude the paper. Copyright © 2004 John Wiley & Sons, Ltd.

A Seed Method for Solving Nonsymmetric Linear Systems with Multiple Right-hand Sides

We present a seed method for solving large nonsymmetric linear systems with multiple right-hand sides. The method uses a single augmented Krylov subspace corresponding to a seed system as a generator of approximations to the nonseed systems. The residual evaluate of the method is shown, and a new strategy to form a seed system which could supply information shareable among the right-hand sides is given. Numerical experiments indicate that our seed selection strategy is more efficient than two existing strategies and our method has significant time saving compared with the block GMRES method and the GMRES method with a projection process.

Analysis of Augmented Krylov Subspace Methods

Residual norm estimates are derived for a general class of methods based on projection techniques on subspaces of the form $ K_m + {\cal W}$, where $K_m$ is the standard Krylov subspace associated with the original linear system and ${\cal W}$ is some other subspace. These "augmented Krylov subspace methods" include eigenvalue deflation techniques as well as block-Krylov methods. Residual bounds are established which suggest a convergence rate similar to one obtained by removing the components of the initial residual vector associated with the eigenvalues closest to zero. Both the symmetric and nonsymmetric cases are analyzed.

Deflated and Augmented Krylov Subspace Techniques

We present a general framework for a number of techniques based on projection methods on ‘augmented Krylov subspaces’. These methods include the deflated GMRES algorithm, an inner–outer FGMRES iteration algorithm, and the class of block Krylov methods. Augmented Krylov subspace methods often show a significant improvement in convergence rate when compared with their standard counterparts using the subspaces of the same dimension. The methods can all be implemented with a variant of the FGMRES algorithm. © 1997 by John Wiley & Sons, Ltd.

Deflated and Augmented Krylov Subspace Techniques

Deflated and Augmented Krylov Subspace Techniques