Course tutor: Dr Carola Schönlieb

Computational mathematics and scientific computing are central in translating rigorous mathematical analysis into a powerful took in applied mathematics and mathematical modelling. The purpose of this course is to familiarise the students both with a number of difference methodologies for three kinds of computational problems and also with the challenges implicit in their implementation.

The first two weeks are devoted to the *numerical solution of ordinary differential equations*. Four methods are introduced: an embedded Runge-Kutta pair, backward differentiation formulæ, symplectic Runge-Kutte and an explonential integrator. Subsequently, each team programs and implements one of the methods, applies it to a range of test problems and presents it to the entire forum. This part of the course concludes with a discussion of the merits of differential computational approaches.

The next four weeks are concerned with image processing, with an emphasis on image segmentation and inpainting with level-set methods and other methodologies based on partial differential equations. In the first part each team receives a reading assignment on a different aspect of image processing, which it then presents to the forum.

In the second part each team programs and implements a different image processing methodology. The segment, again, concludes with presentation and discussion. The final two weeks of the course are devoted to *approximation theory* of scattered deterministic and stochastic data. After brief introduction, each team programs a different algorithm for univariate data: B-splines, radial basis functions and wavelets, and applies it to a set of test problems. The segment concludes with presentation and discussion of computational results.