We present a new approach to treat nonlinear operators in reduced basis approximations of parametrized evolution equations. Our approach is based on empirical interpolation of nonlinear differential operators and their Fréchet derivatives. Efficient offline/online decomposition is obtained for discrete operators that allow an efficient evaluation for a certain set of interpolation functionals. An a posteriori error estimate for the resulting reduced basis method is derived and analyzed numerically. The approach is applied to nonlinear parabolic and hyperbolic equations based on explicit or implicit finite volume discretizations. We show that the resulting reduced scheme is able to capture the evolution of both smooth and discontinuous solutions. In case of symmetries of the problem, the approach realizes an automatic and intuitive space-compression or even space-dimensionality reduction. We perform empirical investigations of the error convergence and runtimes. In all cases we obtain a good runtime acceleration that grows with the problem's size.

@TechReport{DHO10,
  author       = {Drohmann, M. and Haasdonk, B. and Ohlberger, M.},
  title        = {Reduced Basis Approximation for Nonlinear Parametrized Evolution Equations based on Empirical Operator Interpolation},
  institution  = {FB 10, Universit\ät M\ünster},
  number       = {10/02 - N},
  month        = {october},
  year         = {2010},
  type         = {Preprint},
  keywords     = {model reduction, parametrized evolution equations, reduced basis methods, empirical interpolation, a posteriori error estimation},
  url          = \{/2010/DHO10},
}