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Multiscale Reduced Basis Methods

Principal investigators
Bernard Haasdonk
Staff
Sven Kaulmann
Begin 5/2/11
End 1/31/12

Reduced basis (RB) methods are a well-established technique for model reduction of parametrized partial differential equations. These methods generate efficient, globally approximating, reduced models, which allow a fast parameter variation by an offline/online decomposition of the compu- tational procedure. However, RB-methods are in practice mostly rather static and only applicable for decent parameter dimensions, parameter ranges and computational domains. In the current project, we will make reduced basis methods suitable for complex parameter influences, i.e. lar- ge parameter domains, high parameter dimensions, nonlocal and local parameter dependencies.

The key for this will be making use of different types of redundancy via adaptivity. The approaches will consider both the offline phase (dictionary construction, h-adaptivity for snapshot computation) and the online phase (basis selection, dimensionality adaptation). Mathematical challenges are the development of methodology and analysis, in particular convergence and approximation error analysis of the adaptive schemes. Implementational challenges are the focus on using existing ad- aptive high-performance-computing simulation packages for the model generation in combination with online-capable platforms. These methodological extensions will turn the RB-scheme into an adaptive multiscale approach applicable to geophysical applications.