AndersonAcceleration

Anderson acceleration is a numerical method used to accelerate the convergence of iterative algorithms. It was introduced by David G. Anderson in 1965 and is particularly useful in the context of fixed-point iterations and nonlinear equations. The method is based on the idea of using a linear combination of previous iterates to form a new iterate, which can significantly improve the convergence rate compared to the original iterative scheme.

The Anderson acceleration method can be applied to a wide range of problems, including those arising in scientific computing, engineering, and machine learning. It is especially effective when the original iterative algorithm converges slowly or exhibits oscillatory behavior. The method involves storing a history of previous iterates and using them to construct a new iterate that is a better approximation to the fixed point.

One of the key advantages of Anderson acceleration is its simplicity and flexibility. It can be easily incorporated into existing iterative algorithms with minimal modifications. Additionally, the method does not require the computation of derivatives or the solution of linear systems, making it computationally efficient.

However, Anderson acceleration also has some limitations. It requires the storage of a history of iterates, which can be memory-intensive for large-scale problems. Furthermore, the method may not always lead to a significant improvement in convergence rate, depending on the specific problem and the original iterative algorithm.

In summary, Anderson acceleration is a powerful tool for accelerating the convergence of iterative algorithms. Its simplicity, flexibility, and effectiveness make it a valuable addition to the numerical analyst's toolbox.