After following this course, the student:
- is able to derive numerical algorithms for analytical problems, investigate and prove theoretical properties and implement these algorithms in a suitable language such as MATLAB, Python or C;
- can interpret results from a numerical algorithm and quantify reliability using error analysis;
- is able to formulate mathematical models for basic (physical) systems and investigate their mathematical structure;
- can apply the acquired knowledge to perform numerical quadrature and data fitting and obtain accurate numerical solutions to initial and boundary value problems.
In this course, we introduce basic discretisation and (iterative) numerical solution methods for mathematical and physical problems. We theoretically derive and study algorithms to solve these problems. Since every numerical method will be accompanied by numerical errors due to round off and approximation, estimating numerical error and minimising computational cost are at the heart of the course. Error estimation for smooth problems will be approached by developing reliable extrapolation and efficient implementation.|
The course addresses six pivotal numerical mathematics themes. Each is composed of an oral lecture introducing the theme, a proof-lab in which theoretical aspects of the theme are elucidated, ahead of a practical in which specific implementations, testing, validation and numerical simulation are key. In detail, each theme includes
A challenging advanced question completes every practical. The practical component will add further intuition regarding performance limitations of numerical approaches.
- a theoretical component in which basic concepts are presented, algorithms are derived, and proofs of convergence are discussed together with theoretical error bounds. Each oral lecture is accompanied by an exercise class in which students work on a number of exercises to improve understanding of the new concepts and algorithms
- a practical component in which students work in pairs implementing, testing and applying basic algorithms and answering a number of questions linked to the numerical mathematics theme of that week. Next to questions pertaining to the performance of the algorithms, also questions on the theoretical basis are included.