Type
Text
Type
Dissertation
Advisor
Reuter, Matthew | Jiao, Xiangmin | Samulyak, Roman | Khairoutdinov, Marat | Zhang, Minghua | .
Date
2016-12-01
Keywords
Applied mathematics | geometric+algebraic multigrid, KKT systems, Krylov subspace methods, multigrid methods, null-space method, singular systems
Department
Department of Applied Mathematics and Statistics
Language
en_US
Source
This work is sponsored by the Stony Brook University Graduate School in compliance with the requirements for completion of degree.
Identifier
http://hdl.handle.net/11401/77232
Publisher
The Graduate School, Stony Brook University: Stony Brook, NY.
Format
application/pdf
Abstract
Iterative methods are some of the most important techniques in solving large scale linear systems. Compared to direct methods, iterative solvers have the advantage of lower storage cost and better scalability as the problem size increases. Among the methods, multigrid solvers and multigrid preconditioners are often the optimal choices for solving systems that arise from partial differential equations (PDEs). In this dissertation, we introduce several efficient algorithms for various scientific applications. Our first algorithm is a specialized geometric multigrid solver for ill-conditioned systems from Helmholtz equations. Such equations appear in climate models with pure Neumann boundary condition and small wave numbers. In numerical linear algebra, ill-conditioned and even singular systems are inherently hard to solve. Many standard methods are either slow or non-convergent. We demonstrate that our solver delivers accurate solutions with fast convergence. The second algorithm, HyGA, is a general hybrid geometric+algebraic multigrid framework for elliptic type PDEs. It leverages the rigor, accuracy and efficiency of geometric multigrid (GMG) for hierarchical unstructured meshes, with the flexibility of algebraic multigrid (AMG) at the coarsest level. We conduct numerical experiments using Poisson equations in both 2-D and 3-D, and demonstrate the advantage of HyGA over classical GMG and AMG. Besides the aforementioned algorithms, we introduce an orthogonally projected implicit null-space method (OPINS) for saddle point systems. The traditional null-space method is inefficient because it is expensive to find the null-space explicitly. Some alternatives, notably constraint-preconditioned or projected Krylov methods, are relatively efficient, but they can suffer from numerical instability. OPINS is equivalent to the null-space method with an orthogonal projector, without forming the orthogonal basis of the null space explicitly. Our results show that it is more stable than projected Krylov methods while achieving similar efficiency. | 116 pages
Recommended Citation
Lu, Cao, "Efficient Iterative and Multigrid Solvers with Applications" (2016). Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions). 3060.
https://commons.library.stonybrook.edu/stony-brook-theses-and-dissertations-collection/3060