Reduced basis modeling of hierarchical flow systems

Reduced basis methods are particularly attractive to use in order to reduce the number of degrees-of-freedom associated with a given numerical approximation to a set of partial differential equations; the computational complexity can be reduced to a level where potentially very complex systems can be simulated. The main idea is to construct basis functions with a large information content in order to reduce the number of basis coefficients needed to reach a certain level of accuracy in the outputs of inte rest. In this project, we propose to develop and analyze reduced basis methods for simulating hierarchical flow systems, which are of relevance for studying flow in a set of arteries or veins, flow in a respiratory system, or flow in a network of pipes. We prop ose to decompose the geometry into generic parts (e.g. pipes and bifurcations), and to contruct a reduced basis for these generic parts by considering representative geometrics snapshots. The global system will be constructed by glueing the individual basis solutions together via Lagrange multipliers. As far as we know, the use of reduced basis methods have previously been limited to finite-dimensional parameter spaces; defining the geomet ry as a generic "parameter" appears to be new.