A Case for a Biorthogonal Jacobi-Davidson Method: Restarting and Correction Equation

We propose a biorthogonal Jacobi-Davidson method (biJD), which can be viewed as an explicitly biorthogonalized, restarted Lanczos method, that uses the approximate solution of a correction equation to expand its basis. Through an elegant formulation, the algorithm allows for all the functionalities and features of the Jacobi-Davidson (JD), but it also includes some of the advantages of the nonsymmetric Lanczos. The motivation for this work stems mainly from a correction equation and a restarting scheme that are possible with biJD but not with JD. Speciically, a correction equation using the left approximate eigenvectors available in biJD yields cubic asymptotic convergence, as opposed to quadratic with the JD correction equation. In addition, a successful restarting scheme for the symmetric JD depends on the Lanczos three term recurrence, and thus can only apply to the biJD. We describe the algorithm, its features, and the possible functionalities. In addition, we develop an appropriate correction equation framework, and analyze the eeects of the new restarting scheme. Our numerical experiments connrm that biJD is a highly competitive method for a diicult problem. AMS Subject Classiication. 65F15 1. Introduction. The solution of the large eigenvalue problem A~ x = ~ ~ x for a few eigenvalues closest to a given value and their corresponding eigenvectors (together eigenpairs) is recognized as a harder problem than the solution of a linear system of equations with A. Because the eigenvalues are not known a priori, the system to be solved is non linear 38, 37]. Even if the eigenvalue were known, the resulting linear system would be indeenite for any eigenvalue that lies inside the spectrum. This usually implies slow convergence of the linear solver, and moreover it is hard to obtain good preconditioners 27, 1]. Non normality and ill conditioning exacerbate these problems further. Preconditioning is also not straightforward to apply on eigenvalue iterative solvers. Early attempts included variants of the Davidson's method 9] and shift and invert methods 23], but Jacobi-Davidson (JD) type methods have provided an appropriate preconditioning framework for eigensolvers 30]. Another important problem with eigensolvers is their high storage requirements. For linear systems storage is less of an issue, because three term recurrence methods, such as CG and BCG, are as eeective as full orthogonalization Arnoldi-type methods. In contrast, the three term recurrence Lanczos method for eigenproblems needs to store the basis vectors to recover eigenvector approximations. Moreover, most eigenvalue methods that use preconditioning …