Stochastically-inspired methods for solving systems of linear equations

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Many applications in engineering and science require the solution of systems oflinear algebraic equations, or partial differential equations (PDEs) that are solved using linear equation solvers. Conventionally, these have been solved using direct and iterative approaches. This project explores a third way, through the use of stochastic methods for the solution of linear equations, using random walks on a Markov chain. Although the basis for these methods has been well known for many years, they have not found widespread application as they were not considered scalable or accurate. This project develops novel techniques that show how these methods can be viable alternatives to conventional methods, and researches this approach to expand its theoretical and practical horizons, including hybridizations with existing methods.The application of these techniques on a variety of practical engineering applications is under investigation, ranging from problems that require the solution of linear equations or PDEs, to a set of domain-specific applications, drawn from a range of fields. This effort has parallel research thrusts, of which one develops new theory to enhance the accuracy and efficiency of these methods, while the other applies the theory to specific applications, using problem-specific knowledge for further performance gains. The research is expected to have a broader impact beyond its immediate scope through its applicability to a range of problems in science and engineering, and its potential for wider application to fields beyond those considered in this project. In addition, the educational aspects of this work involve includingfacets of this work in the classroom, and in helping train the next generation of scientists and engineers.