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.

翻译：我们提出了一种模型降阶方法，该方法将原始经验插值方法进行扩展，以实现对参数化非线性偏微分方程（PDE）的精确且高效的减基近似。在存在非线性的情况下，伽辽金减基近似由于评估非线性项的高复杂度（该复杂度依赖于真实近似的维度）而仍然计算成本高昂。经验插值方法（EIM）曾作为一种非线性模型降阶技术被提出，旨在使非线性项评估的复杂度独立于真实近似的维度。我们引入了一种一阶经验插值方法（FOEIM），该方法利用偏导数信息来构建低成本且稳定的非线性项插值。我们提出了两种不同的FOEIM算法来生成插值点和基函数。我们将FOEIM应用于非线性椭圆型偏微分方程，并将其与伽辽金减基近似及EIM进行比较。数值结果展示了这三种减基方法的性能。