On hybrid and resilient Monte Carlo methods for linear algebra problems

• Main Author: Straßburg, Janko
• Published: University of Reading 2014
• Subjects: 515.14
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.628535

This thesis aims to advance research in the area of Monte Carlo (MC) methods for linear algebra problems. It investigates the efficient application of Markov Chain Monte Carlo methods to matrix inversion, sparse approximate inverse preconditioning and solving of systems of linear algebraic equations. A Monte Carlo method for generating a rough approximation of a matrix inverse will be presented. Both serial and parallel algorithms are developed. An iterative refinement scheme to further improve the accuracy of the rough inverse is considered to build hybrid algorithms. Results are presented showing the performance of the implementations on a variety of test cases. Novel techniques for fault tolerance and resilience using an extension to the Message Passing Interface (MPI) standard are introduced and discussed. A main contribution is the development and implementation of a fault tolerant version of the MC algorithm that is based on the characteristics of the Monte Carlo methods. The behaviour of the algorithm is analysed at scale with the help of a high performance system simulator and findings concerning scalability and efficiency are presented. Results from the analysis directly impacted optimisation and enhancement of the program code. Improvements and advances that allow for application of the presented methods in wider areas are documented. Finally, considerations for a revised and restructured new implementation of the Monte Carlo algorithm that improves scalability and resilience characteristics are developed.