Reliable automatic model reduction combining a posteriori error analysis and a posteriori model reduction

Mathematical models are used for simulations and predictions of various phenomena and processes that can be translated into mathematical language. Nowadays, with easy access to powerful computers and specialised software, even complicated models can be handled. Model output can be generated even by users that may not have specialised mathematical training. A consequence of the increase in complexity of the models on the one hand, and the relative ease of numerical output generation on the other hand, can be a loss of insight into the mathematical model. One way to regain this is to reduce the original model to a simpler form that will nevertheless capture the key features of the model and elucidate the processes that are modelled, especially those that are of interest to the user. In this thesis I develop a model reduction algorithm that is based on a technique borrowed from finite element methods. An a posteriori error estimate, where the error is defined as the difference between the unknown solution and a FE approximation thereof, is used to ensure that the mesh on the computational domain is fine enough such that the discretised scheme approximates the differential operator sufficiently accurately. Model reduction on the other hand uses an a posteriori error estimate, where the error is defined as the difference between the solutions to the original model and a reduced model, to generate a differential operator that is a close enough approximation of the original operator. The main research chapter focuses on the mathematical analysis that this automated method entails. The model reduction technique is then applied to model problems to validate the numerical methods that are employed. An enzyme kinetic reaction model is discussed in detail as the application of the developed model reduction method satisfies the aim of this thesis to a great extent. In particular, I can show with that example that the reduced models which were automatically generated correspond to those obtained by asymptotic analysis. The same elucidation of the processes that are modelled could therefore be achieved with the automatic model reduction technique as with methods that usually need considerable mathematical expertise.