Anderson Accleration

非线性方程在科学和工程中无处不在。该项目旨在理解和改进计算其解决方案的方法。安德森加速度是一种重要的化学和物理计算方法。这个项目的主要重点是在这些应用程序的上下文中分析该算法。首席研究员将分析结果应用于太阳能、核工程等领域，其中设计更高效的光敏分子使用安德森加速来计算能量，核工程中几种不同物理模拟之间的耦合导致了困难的非线性方程，以及新材料的大规模模拟，其中材料能量状态的计算即使是最大的超级计算机也难以计算。一名研究生正在从事该项目的研究。该项目解决了数值分析理论和实现中尚未解决的算法问题，重点关注非线性方程的算法，这些算法在科学和工程代码中无处不在。对Anderson加速度的研究延续了主要研究者对一类无雅可比矩阵解的研究，这类解在雅可比矩阵或雅可比向量积不可用时是非常重要的。安德森加速被广泛应用于电子结构计算和多物理场仿真中，当一个或多个物理模拟器是一个“黑盒”代码时。研究者研究了在实践中被证明是有效的算法的变化，但没有理论可用，以了解它在某些应用中比预期的性能更好。他还研究了特征值计算的无矩阵方法对函数空间中潜在问题的离散化的依赖性。第三条研究路线是函数、雅可比矩阵和雅可比向量积评估中的误差对非线性求解方法的影响。这部分项目的重点是半光滑非线性方程和延拓方法。一名研究生正在从事该项目的研究。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Nonlinear equations are ubiquitous in science and engineering. This project is directed at understanding and improving methods for computing their solutions. Anderson acceleration is an important computational method for chemistry and physics applications. The main focus of this project is analysis of that algorithm in the context of those applications. The principal investigator deploys results of that analysis in applications such as solar power, where the design of more efficient photosensitive molecules uses Anderson acceleration to compute energy, nuclear engineering, where the coupling between several different physics simulations leads to difficult nonlinear equations, and large-scale simulations of novel materials, where the computation of material energy states taxes even the largest supercomputers. A graduate student is engaged in the research of the project.This project addresses unresolved algorithmic questions in numerical analysis theory and implementation, focusing on algorithms for nonlinear equations, which are ubiquitous in science and engineering codes. The research on Anderson acceleration continues the principal investigator's study of a class of Jacobian-free solvers, which are very important when Jacobians or Jacobian-vector products are unavailable. Anderson acceleration is widely used in electronic structure computations and in multi-physics simulations when one or more of the physics simulators is a "black-box" code. The investigator studies variations of the algorithm that have proved effective in practice, but for which no theory is available, to understand its better-than-predicted performance on some applications. He also examines the dependence of matrix-free methods for eigenvalue computations on discretizations of an underlying problem in a function space. A third line of study is the effects of errors in function, Jacobian, and Jacobian-vector product evaluations on nonlinear solver methods. The focus of this part of the project is semismooth nonlinear equations and continuation methods. A graduate student is engaged in the research of the project.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.