Subspace Methods for Spectral and Pseudo-Spectral Quantities

In this chapter, we will explain how subspace acceleration can be applied in the context of computing spectral and pseudo-spectral quantities. We will see techniques that allow us to reliably accelerate a possibly slowly convergent algorithm by storing important information in a subspace. Such ideas are not new and the most well-known examples are arguably Krylov subspace methods for solving linear systems and algebraic eigenvalue problems. Another family of methods were this occurs is in numerical optimization methods with momentum, although the idea of subspaces is there less obvious. What is atypical to the subspace approaches presented here is that they provably turn a linearly converging iteration into a superlinearly converging one. While this is similar to vector extrapolation techniques, it is important to note that subspace methods can in general not be seen as vector extrapolation instances.