high performance computing on graphics processing units: hgpu.org

This dissertation focuses on block cyclic reduction (BCR) type fast direct solvers, graphics processing unit (GPU) computation, and image denoising. The fast direct solvers are specialized methods for solving certain types of linear systems. They take into account specific characteristics of the system and are therefore able to solve the system much more efficiently than less specialized methods. In particular, this dissertation focuses on symmetric block tridiagonal linear systems that can be presented in a separable form using the so-called Kronecker matrix tensor product. Modern GPUs can provide significantly more floating point processing power than traditional central processing units (CPUs) and could therefore potentially improve the efficiency of fast direct solvers. Image denoising is a process in which a given noisy image is cleared of excess noise. Recently, higher order models that utilize mean curvature information in their regularization term have received a lot of attention. These models are expensive to solve, but fast direct solvers and GPUs could be a solution to this problem. A total of five articles are included in this article-style dissertation. The first three articles deal with the BCR methods and two present GPU implementations of different variants and compare the implementations against similar CPU implementations. The two remaining articles focus on a so-called L1-mean curvature image denoising model. The fourth article introduces a new augmented Lagrangian-based solution algorithm and the fifth article describes an efficient GPU implementation of the algorithm. The included articles show that GPUs can provide significant performance benefits in the context of the BCR type fast direct solvers and higher order image denoising models.