Dr. Jose Carrillo Distinguished Lecture

Abstract

Functions appearing in many applications, for example as the solution of an integral or differential equation, are usually discretized and approximated in a basis. Each function in a certain space can be uniquely represented in that basis. Correspondingly, the discretization of the equation at hand leads to a square linear system of equations, that has a unique solution and that is hopefully well-conditioned. Unfortunately, that ideal scenario is often difficult to realize. Indeed, there are many situations in which it is not straightforward to come up with a suitable basis in the first place: perhaps the function is defined on a complicated domain, or it is non-smooth, it has corner singularities or certain oscillatory behaviour. We show that a lot of flexibility is gained by relaxing the uniqueness condition of a basis, and by allowing some redundancy in the discretisation. This rapidly leads to ill-conditioned problems, since there is no longer a unique solution. However, we show this is easily remedied and we identify a rather general setting where redundancy is accompanied by numerically stable and efficient algorithms. This allows for a lot of creativity in the discretization of problems.