The main theme of this work is effectiveness and efficiency of Krylov subspace methods and Krylov subspace recycling. While solving long, slowly changing sequences of large linear systems, such as the ones that arise in engineering, there are many issues we need to consider if we want to make the process reliable (converging to a correct solution) and as fast as possible. This thesis is built on three main components. At first, we target bilinear and quadratic form estimation. Bilinear form $c^TA^{-1}b$ is often associated with long sequences of linear systems, especially in optimization problems. Thus, we devise algorithms that adapt cheap bilinear and quadratic form estimates for Krylov subspace recycling. In the second part, we develop a hybrid recycling method that is inspired by a complex CFD application. We aim to make the method robust and cheap at the same time. In the third part of the thesis, we optimize the implementation of Krylov subspace methods on Graphic Processing Units (GPUs). Since preconditioners based on incomplete matrix factorization (ILU, Cholesky) are very slow on the GPUs, we develop a preconditioner that is effective but well suited for GPU implementation. / Ph. D. / In many applications we encounter the repeated solution of a large number of slowly changing large linear systems. The cost of solving these systems typically dominates the computation. This is often the case in medical imaging, or more generally inverse problems, and optimization of designs. Because of the size of the matrices, Gaussian elimination is infeasible. Instead, we find a sufficiently accurate solution using iterative methods, so-called Krylov subspace methods, that improve the solution with every iteration computing a sequence of approximations spanning a Krylov subspace. However, these methods often take many iterations to construct a good solution, and these iterations can be expensive. Hence, we consider methods to reduce the number of iterations while keeping the iterations cheap. One such approach is Krylov subspace recycling, in which we recycle judiciously selected subspaces from previous linear solves to improve the rate of convergence and get a good initial guess.
In this thesis, we focus on improving efficiency (runtimes) and effectiveness (number of iterations) of Krylov subspace methods. The thesis has three parts. In the first part, we focus on efficiently estimating sequences of bilinear forms, c<sup>T</sup>A⁻¹b. We approximate the bilinear forms using the properties of Krylov subspaces and Krylov subspace solvers. We devise an algorithm that allows us to use Krylov subspace recycling methods to efficiently estimate bilinear forms, and we test our approach on three applications: topology optimization for the optimal design of structures, diffuse optical tomography, and error estimation and grid adaptation in computational fluid dynamics. In the second part, we focus on finding the best strategy for Krylov subspace recycling for two large computational fluid dynamics problems. We also present a new approach, which lets us reduce the computational cost of Krylov subspace recycling. In the third part, we investigate Krylov subspace methods on Graphics Processing Units. We use a lid driven cavity problem from computational fluid dynamics to perform a thorough analysis of how the choice of the Krylov subspace solver and preconditioner influences runtimes. We propose a new preconditioner, which is designed to work well on Graphics Processing Units.
| Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/78695 |
| Date | 10 August 2017 |
| Creators | Swirydowicz, Katarzyna |
| Contributors | Mathematics, de Sturler, Eric, Roy, Christopher J., Zietsman, Lizette, Embree, Mark P. |
| Publisher | Virginia Tech |
| Source Sets | Virginia Tech Theses and Dissertation |
| Detected Language | English |
| Type | Dissertation |
| Format | ETD, application/pdf |
| Rights | In Copyright, http://rightsstatements.org/vocab/InC/1.0/ |
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