Stochastically-inspired methods for solving systems of linear equations

工程和科学中的许多应用都需要求解线性代数方程组或偏微分方程（PDEs），而这些方程组是用线性方程求解器来求解的。通常，这些都是使用直接和迭代的方法来解决的。这个项目探索了第三种方法，通过使用随机方法求解线性方程，使用马尔可夫链上的随机行走。尽管这些方法的基础已经众所周知很多年了，但它们并没有得到广泛的应用，因为它们不被认为是可扩展的或不准确的。该项目开发了新技术，展示了这些方法如何成为传统方法的可行替代方案，并研究了这种方法，以扩大其理论和实践视野，包括与现有方法的杂交。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。这项工作有平行的研究重点，其中一个开发新的理论来提高这些方法的准确性和效率，而另一个将理论应用于特定的应用，使用特定问题的知识来进一步提高性能。预计这项研究将通过其对一系列科学和工程问题的适用性，以及在本项目所考虑的领域之外的更广泛应用的潜力，在其当前范围之外产生更广泛的影响。此外，这项工作的教育方面包括将这项工作的各个方面纳入课堂，并帮助培养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.