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，或使用线性方程解算器求解的偏微分方程组(PDE)。传统上，这些问题都是使用直接和迭代方法来解决的。这个项目探索了第三种方法，通过使用随机方法来解线性方程，使用马尔可夫链上的随机游动。虽然这些方法的基础已经众所周知很多年了，但由于它们被认为是不可扩展或不准确的，它们没有得到广泛的应用。该项目开发了新的技术，展示了这些方法如何成为传统方法的可行替代方法，并研究了这种方法以扩大其理论和实践视野，包括与现有方法的结合。这些技术在各种实际工程应用中的应用正在调查中，从需要求解线性方程或偏微分方程组的问题，到一系列来自一系列领域的特定领域的应用。这一努力具有平行的研究推动力，其中一个开发了新的理论来提高这些方法的准确性和效率，而另一个将该理论应用于具体应用，使用特定于问题的知识来进一步提高性能。预计这项研究将产生更广泛的影响，超出其直接范围，因为它适用于科学和工程中的一系列问题，并有可能更广泛地应用于本项目所考虑的领域之外的领域。此外，这项工作的教育方面包括在课堂上开展这项工作的方方面面，以及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.