Preconditioners for Krylov subspace methods: an overview

When simulating a mechanism from science or engineering, or an industrial process, one is frequently required to construct a mathematical model, and then resolve this model numerically. If accurate numerical solutions are necessary or desirable, this can involve solving large-scale systems of equations. One major class of solution methods is that of preconditioned iterative methods, involving preconditioners which are computationally cheap to apply while also capturing information contained in the linear system. In this article, we give a short survey of the field of preconditioning. We introduce a range of preconditioners for partial differential equations, followed by optimization problems, before discussing preconditioners constructed with less standard objectives in mind.