Speaker
Description
The Reduced Basis Method (RBM) is a well-established model reduction technique to realize multi-query and/or realtime applications of parameterized partial differential equations (PPDEs). The RBM relies on a well-posed variational formulation of the PPDE under consideration. Since the RBM is a linear approximation method, the best possible rate of convergence is given by the Kolmogorov N-width. It is well-known that the decay of the Kolmogorov N-width is exponentially fast for suitable elliptic and parabolic problems, but is poor for transport- or wave-type problems. This motivates our goal of developing a well-posed variational formulation for the wave equation, which also allows for a nonlinear model reduction in order to overcome the limitations of a possibly poor Kolmogorov N -width. To this end, starting from a strandard weak formulation of the wave equation, we construct a extended formulation with a parameter dependent ansatz space. This new formulation is well posed and optimally stable in the sens, that inf-sup and continuity constant are both equal to one. Due to the parameter dependent ansatz space, this formulation naturally leads to a nonlinear model order reduction. Further, we present numerical experiments regarding the performance of the RBM. Thereby we use an unconditionally stable space-time Petrov-Galerkin discretization for the wave equation based upon a modified Hilbert type transformation, leading to a parameter-independent discretization of the ansatz space with a parameter-dependent topology. Following the idea of a nonlinear model order reduction, a fully parameter-dependent discretization, not only in the topology, is a work in progress.
