Modern simulation scenarios require real-time or many-query responses from a simulation model. This is the driving force for increased efforts in model order reduction for high-dimensional dynamical systems or partial differential equations. This demand for fast simulation models is even more critical for parameterized problems. Several snapshot-based methods for basis construction exist for parameterized model order reduction, for example, proper orthogonal decomposition or reduced basis methods. They require the careful choice of samples for generation of the reduced model. In this article we address two types of grid-based adaptivity that can be beneficial in such basis generation procedures. First, we describe an approach for training set adaptivity. Second, we introduce an approach for multiple bases on adaptive parameter domain partitions. Due to the modularity, both methods also can easily be combined. They result in efficient reduction schemes with accelerated training times, improved approximation properties and control on the reduced basis size. We demonstrate the applicability of the approaches for instationary partial differential equations and parameterized dynamical systems.

@Article{HDO11,
  author       = {Haasdonk, B. and Dihlmann, M. and Ohlberger, M.},
  title        = {A training set and multiple bases generation approach for parameterized model reduction based on adaptive grids in parameter space},
  journal      = {Mathematical and Computer Modelling of Dynamical Systems},
  number       = {4},
  volume       = {17},
  pages        = {423--442},
  year         = {2011},
  publisher    = {Taylor \& Francis},
  keywords     = {parameterized model order reduction, reduced basis methods, adaptive parameter grids,  snapshot and parameter selection},
  doi          = {10.1080/13873954.2011.547674},
  url          = \{/2011/HDO11},
}