The efficient solution of systems of nonlinear equations is an important tool for the modeling of physical phenomena. In this talk, we will discuss a powerful and low-cost method for accelerating the convergence of fixed-point iterations, known as Anderson acceleration (AA). First developed in 1965 in the context of integral equations, this method has become increasingly popular due to its efficacy on a wide range of problems, including optimization, machine learning and numerical PDE, including complex multiphysics simulations. AA requires the storage of a relatively small number of solution and update vectors, and the solution of an optimization problem that is generally posed as least-squares. On any given problem, how successful it is depends on the details of its implementation. We will introduce the algorithm and use standard tools and techniques from numerical linear algebra to develop an efficient filtering strategy to improve the numerical stability and performance of AA.
Filtered Anderson Acceleration for Numerical PDE
Sara Pollock, University of Florida
Authors: Sara Pollock and Leo Rebholz
2023 AWM Research Symposium
Recent Developments in Control, Optimization, and the Analysis of Partial Differential Equations [Organized by Lorena Bociu and Pelin Guven Geredeli]