Abstract
Reduced basis methods (RBM) are a class of numerical methods developed for settings which require a large number of solutions to a parameterized problem. In this work, we consider the case where this problem is a discretization scheme for a parameterized steady-state nonlinear partial differential equation. By utilizing an offline-online procedure and exploiting the fact that the parameter-induced solution manifolds can be well approximated by low-dimensional spaces, RBMs can improve efficiency by several orders of magnitude while maintaining accuracy through a rigorous a posteriori error estimate. The reduced collocation method (RCM), a variant of RBM for the collocation setting, uses greedy selection to choose both the basis and a complimentary set of collocation points. We adapt the RCM to the nonlinear case and produce an online-efficient scheme. This is achieved by using a simultaneous empirical interpolation and reduced basis approach, referred to as the SER method. This approach constructs an EIM approximation of the nonlinear terms in tandem with the greedy selection of the reduced basis and a judicious choice of the reduced set of collocation points. One EIM basis and one collocation point are chosen for every reduced basis selection, creating an approximation which improves as the surrogate approximation improves. This results in an online-efficient scheme, using this EIM approximation to efficiently calculate the residuals for the error estimator. We call this the SERCM, the simultaneous empirical reduced collocation method..