Abstract. We present a deflated version of the conjugate gradient algorithm for solving linear systems. The new algorithm can be useful in cases when a small number of eigenvalues of the iteration matrix are very close to the origin. It can also be useful when solving linear systems with multiple right-hand sides, since the eigenvalue information gathered from solving one linear system can be recycled for solving the next systems and then updated.Key words. conjugate gradient, deflation, multiple right-hand sides, Lanczos algorithm AMS subject classifications. 65F10, 65F15PII. S10648295983397611. Introduction. A number of recent articles have established the benefits of using eigenvalue deflation when solving nonsymmetric linear systems with Krylov subspace methods. It has been observed that significant improvements in convergence rates can be achieved from Krylov subspace methods by adding to these subspaces a few approximate eigenvectors associated with the eigenvalues closest to zero [2,4,7,8,13,14]. In practice, approximations to the eigenvectors closest to zero are obtained from the use of a certain Krylov subspace; then these approximations are dynamically updated using the new Krylov subspace. Results of experiments obtained from these variations indicate that the improvement in convergence over standard Krylov subspaces of the same dimension can sometimes be substantial, especially when the convergence of the original scheme is hampered by a small number of eigenvalues near zero; see, e.g., [2,8].In this paper we consider extensions of this idea to the conjugate gradient (CG) algorithm for the symmetric case. Our starting point is an algorithm recently proposed by Erhel and Guyomarc'h [5]. This is an augmented subspace CG method aimed at linear systems with several right-hand sides. Erhel and Guyomarc'h [5] propose an algorithm which adds one specific vector obtained from a subspace related to a previous right-hand side. We first extend this algorithm to one which handles an arbitrary block W of vectors. We note that introducing an arbitrary W into the Krylov subspace of CG has already been considered by Nicolaides in [9]. The algorithm introduced in this paper is mathematically equivalent to the one in [9]. Nicolaides's algorithm is directly derived from a deflated Lanczos procedure and uses the 3-term recurrence version of the conjugate gradient algorithm. The algorithm in this paper exploits the link between the Lanczos algorithm and the standard CG algorithm. The LDL T factorization of the projected system obtained from the same

This paper presents a new preconditioning technique for solving linear systems. It is based on an invariant subspace approximation for the restarted GMRES algorithm. It uses the exible GMRES scheme by d esigning a new preconditioning after each restart. Numerical examples show that this approach m a y c o n verge almost as fast as full-GMRES at a, possibly, m uch l o wer cost.

[1] We determine the asymptotic dispersion coefficients in 2D exponentially correlated lognormally distributed permeability fields by using parallel computing. Fluid flow is computed by solving the flow equation discretized on a regular grid and transport triggered by advection and diffusion is simulated by a particle tracker. To obtain a welldefined asymptotic regime under ergodic conditions (initial plume size much larger than the correlation length of the permeability field), the characteristic dimension of the simulated computational domains was of the order of 10 3 correlation lengths with a resolution of ten cells by correlation length. We determine numerically the asymptotic effective longitudinal and transverse dispersion coefficients over 100 simulations for a broad range of heterogeneities s 2 2 [0, 9], where s 2 is the lognormal permeability variance. For purely advective transport, the asymptotic longitudinal dispersion coefficient depends linearly on s 2 for s 2 < 1 and quadratically on s 2 for s 2 > 1 and the asymptotic transverse dispersion coefficient is zero. Addition of homogeneous isotropic diffusion induces an increase of transverse dispersion and a decrease of longitudinal dispersion.

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