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.

翻译：本文提出了一种模型降阶方法，将原始经验插值方法扩展，以实现对参数化非线性偏微分方程（PDE）的精确且高效的约化基逼近。在存在非线性的情况下，伽辽金约化基逼近仍面临计算成本高昂的问题，这是因为非线性项的计算复杂度依赖于真实逼近的维数。经验插值方法（EIM）作为一种非线性模型降阶技术被提出，旨在使非线性项的计算复杂度独立于真实逼近的维数。其主要思想是用预设基函数和参数依赖系数的线性组合构成的约化基展开来替代任何非线性项，而系数则通过一种廉价且稳定的插值过程高效确定。为了提高逼近精度，我们提出了一种一阶经验插值方法（FOEIM），该方法在选定参数点处同时利用非线性函数及其偏导数来构造非线性项的约化基展开。我们将该方法应用于非线性椭圆型偏微分方程，并与伽辽金约化基逼近及EIM进行了比较。通过数值结果展示了三种约化基方法的性能。