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Faculty of Mathematics

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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.

An example of inpaintingAn example of inpainting

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.