  We present a model reduction approach that extends the original empirical
interpolation method to enable accurate and efficient reduced basis
approximation of parametrized nonlinear partial differential equations (PDEs).
In the presence of nonlinearity, the Galerkin reduced basis approximation
remains computationally expensive due to the high complexity of evaluating the
nonlinear terms, which depends on the dimension of the truth approximation. The
empirical interpolation method (EIM) was proposed as a nonlinear model
reduction technique to render the complexity of evaluating the nonlinear terms
independent of the dimension of the truth approximation. We introduce a
first-order empirical interpolation method (FOEIM) that makes use of the
partial derivative information to construct an inexpensive and stable
interpolation of the nonlinear terms. We propose two different FOEIM algorithms
to generate interpolation points and basis functions. We apply the FOEIM to
nonlinear elliptic PDEs and compare it to the Galerkin reduced basis
approximation and the EIM. Numerical results are presented to demonstrate the
performance of the three reduced basis approaches.
