Preconditioning techniques for large linear systems: a survey
Journal of Computational Physics
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This thesis is concerned with numerical methods for the robust and efficient solution of fundamental problems in general relativity, fluid dynamics, chemical kinetics, electrical networks, and other areas of science and engineering. There is a critical need for mathematical software that solves such problems and, in certain cases, the governing set of equations can be treated as a set of differential-algebraic equations (DAEs). Fortunately, high-quality mathematical software for solving such DAEs exists (e.g., DASSL) and it can be used to solve a wide array of difficult problems. The main objective of this thesis, however, concerns how DAE codes such as DASSL can be enhanced in a way that enables scientists to efficiently solve a wide range of realistic, multidimensional DAE problems. The incorporation of robust, efficient preconditioned iterative methods within DASSL is a crucial component in extending the range of significant DAE problems that can be solved. For certain DAE problems, the unpreconditioned coefficient matrices are nonsymmetric and, possibly, indefinite. Moreover, they become increasingly ill-conditioned as DASSL decreases its stepsize. A key element of this thesis is the presentation of a preconditioner for a class of DAEs that transforms poorly conditioned, nonsymmetric (indefinite) matrices into reasonably conditioned matrices that approximate the identity matrix as the DAE solver decreases its stepsize. Another key point is that the robustness of Krylov methods is related to the ``departure from normality'''' of the preconditioned coefficient matrix. A study of nonnormality is central to Krylov method/DAE solver research since it affects the accuracy of eigenvalue estimation methods, the convergence behavior of Krylov methods, the relative effectiveness of left and right preconditioning, and the time-stepping behavior of the DAE solver. Our nonnormality results are also theoretically significant because they give new upper bounds to some classical problems in matrix perturbation theory. DASCHEBY is a newly developed and enhanced version of DASSL that uses the adaptive hybrid-Chebyshev algorithm as its linear equation solver. To improve the hybrid algorithm, we present a new power method (PM4) for stably estimating optimal iteration parameters. DASCHEBY also incorporates matrix-free methods and an inexact Newton method to minimize the work and storage requirements of integrating a large system of DAEs. Moreover, the theory, ideas and methods discussed in this thesis are a foundation for making further improvements to DASCHEBY for solving realistic, multidimensional DAE initial-value problems.