Efficient and accurate nonlinear model reduction via first-order empirical interpolation

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. The main idea is to replace any nonlinear term with a reduced basis expansion expressed as a linear combination of pre-computed basis functions and parameter-dependent coefficients. The coefficients are determined efficiently by an inexpensive and stable interpolation procedure. In order to improve the approximation accuracy, we propose a first-order empirical interpolation method (FOEIM) that employs both the nonlinear function and its partial derivatives at selected parameter points to construct the reduced basis expansion of the nonlinear term. Our approach is applied to nonlinear elliptic PDEs and compared to the Galerkin reduced basis approximation and the EIM. Numerical results are presented to demonstrate the performance of the three reduced basis approaches.