Why Should I Care About Preconditioning?

Iterative solvers for large and sparse linear systems are often restricted in their effectiveness due to ill conditioning, an unfavorable spectral distribution of the matrix, or other difficulties. Preconditioning is a central tool in accelerating convergence. The design of effective preconditioners is challenging and is often tied to the underlying continuous problem and to various numerical properties of the matrix. In this talk I will provide an overview of preconditioning techniques and their qualities. We will briefly discuss incomplete factorizations, multigrid, and block preconditioners for saddle-point systems, among other examples of widely used preconditioners.