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Many physical, chemical or biological processes can be described with the help of partial differential equations. Since an analytical solution of the equations is rarely possible, numerical discretization methods must be applied in order to be able to analyze and predict the behavior of these processes. However, despite the many technical advances in high-performance computing, for many applications even a numerical solution can only be obtained with considerable effort. It is therefore of great interest to develop model reduction methods which, starting from a given high-dimensional discrete model can provide efficient surrogate models of low dimension that deliver fast and accurate predictions for varying model parameters.
In this lecture we will primarily consider so-called reduced basis methods for parameterized partial differential equations. These methods are based on a (Petrov-)Galerkin projection of the solution onto a suitably chosen low-dimensional subspace of the discrete solution space. The resulting reduced order models can then be studied using well-known techniques from numerics of partial differential equations (Céa lemma, residual-based error estimators, etc.). Using results from approximation theory, appropriate algorithms for constructing the reduced solution space can be devised that guarantee the quality of the resulting reduced order model. In the integrated practical exercises of this course, we will gain hands-on experience with the application of reduced-basis methods to concrete examples using the model-reduction library pyMOR. |
eG