Preconditioned Techniques For Large Eigenvalue Problems

This research focuses on nding a large number of eigenvalues and eigenvectors of a sparse symmetric or Hermitian matrix, for example, nding 1000 eigenpairs of a 100,000 100,000 matrix. These eigenvalue problems are challenging because the matrix size is too large for traditional QR based algorithms and the number of desired eigenpairs is too large for most common sparse eigenvalue algorithms. In this thesis, we approach this problem in two steps. First, we identify a sound preconditioned eigenvalue procedure for computing multiple eigenpairs. Second, we improve the basic algorithm through new preconditioning schemes and spectrum transformations. Through careful analysis, we see that both the Arnoldi and Davidson methods have an appropriate structure for computing a large number of eigenpairs with preconditioning. We also study three variations of these two basic algorithms. Without preconditioning, these methods are mathematically equivalent but they di er in numerical stability and complexity. However, the Davidson method is much more successful when preconditioned. Despite its success, the preconditioning scheme in the Davidson method is seen as awed because the preconditioner becomes ill-conditioned near convergence. After comparison with other methods, we nd that the e ectiveness of the Davidson method is due to its preconditioning step being an inexact Newton method. We proceed to explore other Newton methods for eigenvalue problems to develop preconditioning schemes without the same aws. We found that the simplest and most e ective preconditioner is to use the Conjugate Gradient method to approximately solve equations generated by the Newton methods. Also, a di erent strategy of enhancing the performance of the Davidson method is to alternate between the regular Davidson iteration and a polynomial method for eigenvalue problems. To use these polynomials, the user must decide which intervals of the spectrum the polynomial should suppress. We studied di erent schemes of selecting these intervals, and found that these hybrid methods with polynomials can be e ective as well. Overall, the Davidson method with the CG preconditioner was the most successful method the eigenvalue problems we tested. Chapter 1 Electronic Structure Simulation Before entering into the main topic of this thesis, we use this chapter to introduces the context of our research which is also serve as motivation for our research. In this chapter, we focus on one application that is the main driving force behind this research in eigen-system solvers, the electronic structure simulation project at the University of Minnesota. We will give some background information about electronic structure simulation, and characteristics of the eigenvalue problems generated from the simulation. The eigenvalue algorithm developed in later chapters are designed to solve the eigenvalue problem of the same characteristics as these matrices. 1.