Stochastically-inspired methods for solving systems of linear equations

工程和科学中的许多应用需要求解线性代数方程组或使用线性方程求解器求解的偏微分方程 (PDE)。 传统上，这些问题是使用直接和迭代方法来解决的。 该项目探索了第三种方法，即使用马尔可夫链上的随机游走的随机方法来求解线性方程。尽管这些方法的基础多年来已广为人知，但由于不被认为可扩展或不准确，因此它们尚未得到广泛应用。 该项目开发了新技术，展示这些方法如何成为传统方法的可行替代品，并研究这种方法以扩展其理论和实践视野，包括与现有方法的混合。这些技术在各种实际工程应用中的应用正在研究中，范围从需要求解线性方程或偏微分方程的问题到一系列领域特定的应用。 这项工作有并行的研究重点，其中一个研究方向是开发新的理论来提高这些方法的准确性和效率，而另一个方向是将理论应用于特定的应用，使用特定于问题的知识来进一步提高性能。 该研究预计将通过其对科学和工程中一系列问题的适用性以及更广泛应用到本项目考虑范围之外的领域的潜力，产生超出其直接范围的更广泛影响。 此外，这项工作的教育方面涉及将这项工作的各个方面纳入课堂，并帮助培训下一代科学家和工程师。

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.