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• W2126890944 abstract "In this dissertation, we focus on three areas of research related to preconditioning for linear systems and eigenvalue problems. We analyze the behaviour of the conjugate gradient method for linear systems with variable preconditioning. We consider the case where the preconditioner is “widely” variable, i.e., preconditioners used on different iterations are essentially unrelated. We assume that the preconditioner on every step is symmetric and positive definite and that the condition number of the preconditioned system matrix is bounded above by a constant independent of the step number. Our main result is of a negative nature: we show that the conjugate gradient method with variable preconditioning may not yield any improvement relative to the preconditioned steepest descent method. We describe a Matlab implementation of the element-based algebraic multigrid (AMG) methods that target linear systems of equations coming from finite element discretizations of elliptic partial differential equations. The individual element information (element matrices and element topology) is the main input to construct the AMG hierarchy. We analyze a number of variants of the spectral (based on solving a large number of local eigenvalue problems) agglomerate element based AMG method. The core of the algorithms relies on element agglomeration utilizing the element topology built recursively from fine to coarse levels. We investigate strategies for adaptive AMG as well as multigrid cycles that are more expensive than the V–cycle utilizing simple interpolation matrices and nested conjugate gradient based recursive calls between the levels. We perform an extensive set of numerical experiments. We describe our software package Block Locally Optimal Preconditioned Eigenvalue Xolvers (BLOPEX), which includes the Locally Optimal Block Preconditioned Conjugate Gradient method for symmetric eigenvalue problems. We demonstrate numerical scalability of BLOPEX by testing it on a number of distributed and shared memory parallel systems, including a Beowulf system, SUN Fire 880, an AMD dual-core Opteron workstation, and IBM BlueGene/L supercomputer, using PETSc (“Portable, Extensible Toolkit for Scientific Computation” by Argonne National Laboratory) domain decomposition and hypre (“High Performance Preconditioners” by Lawrence Livermore National Laboratory) multigrid preconditioning. This abstract accurately represents the content of the candidate’s thesis. I recommend its publication. Signed Andrew Knyazev" @default.
• W2126890944 created "2016-06-24" @default.
• W2126890944 creator A5011362871 @default.
• W2126890944 date "2007-01-01" @default.
• W2126890944 modified "2023-09-24" @default.
• W2126890944 title "On preconditioning for linear equations and eigenvalue problems" @default.
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