Interactive Simulations Using Localized Reduced Basis Methods

An interactive simulation tool should allow its user to change the geometry of the simulation and present an updated solution within a very short time span. To achieve this, the Reduced Basis Method can be used. For problems described by parametrized partial differential equations, it allows for very fast recomputation of the solution after parameter changes. In many cases, changes in the geometry can be accounted for by parametrization. However, this approach has two drawbacks: First, not all geometric variations can be described efficiently by parametrization. Second, the parametrization and thereby the type of changes possible has to be specified before the setup phase. The user is then restricted to these. To overcome these limitations, we propose to localize the basis generation in the Reduced Basis Method. Using basis functions having support only on a small subset of the domain, one can react to arbitrary local geometry modifications by recreating only the basis functions in an environment of the modification.