Asgeir Birkisson

My research focuses on the solution of nonlinear boundary-value problems (BVPs) of ordinary differential equations (ODEs). It is based on the Chebfun system, www.chebfun.org. Chebfun extends familiar methods and techniques of scientific computing, normally associated with vectors of matrices, to the functional setting where vectors become functions, and matrices become linear operators.

Working in such a functional setting, nonlinear BVPs are solved via Newton iteration in function space. Such an iteration scheme requires Fréchet derivatives of the nonlinear differential operators, which we compute using the techniques of automatic differentiation. I first explored these ideas while writing my M.Sc. thesis for the course Mathematical Modelling and Scientific Computing in Oxford, working with Nick Trefethen and Toby Driscoll (Univ. of Delaware).

These ideas, along with object-oriented programming lead to the introduction of chebops, a class in Chebfun which allows users to setup and solve BVPs with few lines of Matlab code. In order to further improve user-friendliness of the BVP solving, I along with Nick Hale, wrote the chebgui, a graphical interface to Chebfun.

Current topics I'm paying particular interest to at the moment are techniques for automatic linearity detection, automatic computation of multiple solution of nonlinear BVPs, and how to solve problems of calculus of variations in an automated way.

Further to my mathematical research, I'm a keen developer in the Chebfun project, focusing on parts of Chebfun related to solving differential equations.