A Relation Between Anderson Acceleration and GMRES

Sammanfattning: A very common type of problem within mathematics and numerical analysis are fixed-point problems, which can arise as sub-problems of optimization methods, differential equations solvers and much more. The most basic iterative approach for fixed-point problems is fixed-point iteration, special cases of which actually date back as far as the Babylonians, where it was used to to find the square roots of positive numbers. An issue with fixed-point iteration is that it can be very slow, as a consequence, acceleration methods have been developed, which are, not surprisingly, methods for speeding up fixed-point iteration. One of these methods are called Anderson acceleration, which has a very strong relationship with GMRES, an algorithm for solving systems of linear equations, which at first glance, seems completely unrelated. The purpose of this thesis is to investigate the theory behind this relationship and to test it numerically.