Anderson Accleration

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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.