The solution of large and sparse linear systems is often required by numerical simulations in many fields of science and engineering. Solving these linear systems is usually the major bottleneck for large-scale applications as it represents the most time-consuming part of the computations. For problems formulated in 2-D geometries, the state-of-the-art direct methods can efficiently solve fairly large sparse linear systems. On the other hand, for 3-D problems, the use of sparse direct methods has become prohibitive in terms of both the memory requirement and the computational complexity. For such problems, iterative methods have thus become a more attractive choice. Among these methods, Krylov subspace methods combined with incomplete LU (ILU) type preconditioners are among the most reliable general-purpose iterative solvers, which have been successfully employed in many applications. In spite of this, there are still two main drawbacks of the ILU-type preconditioners. In the first place is the robustness of these preconditioners. When the matrix is highly ill-conditioned or indefinite, ILU preconditioners are unlikely to work. Secondly, the construction and the application of these preconditioners represent a serial bottleneck, which leads to severe degradation of performance on modern parallel processors. This thesis proposes several preconditioning methods with the considerations of both the robustness for indefinite problems and the efficiency on modern parallel computing architectures. First, we discuss the acceleration techniques by the current many-core processors for several preconditioning approaches. Next, we present a class of new preconditioning techniques based on low-rank approximations by exploiting decay properties of eigenvalues. These preconditioning methods are proposed primarily as means to bypass the difficulties mentioned above that are encountered by standard ILU preconditioners. Implementations of these methods and the performance comparisons with standard preconditioners are also discussed.
University of Minnesota Ph.D. dissertation. June 2015. Major: Computer Science. Advisor: Yousef Saad. 1 computer file (PDF); xii, 188 pages.
Robust High Performance Preconditioning Techniques for Solving General Sparse Linear Systems.
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