An efficient reduced-order modeling approach for non-linear parametrized partial differential equations

For general non-linear parametrized partial differential equations (PDEs), the standard Galerkin projection is no longer efficient to generate reduced-order models. This is because the evaluation of the integrals involving the non-linear terms has a high computational complexity and cannot be pre-computed. This situation also occurs for linear equations when the parametric dependence is nonaffine. In this paper, we propose an efficient approach to generate reduced-order models for large-scale systems derived from PDEs, which may involve non-linear terms and nonaffine parametric dependence. The main idea is to replace the non-linear and nonaffine terms with a coefficient-function approximation consisting of a linear combination of pre-computed basis functions with parameter-dependent coefficients. The coefficients are determined efficiently by an inexpensive and stable interpolation at some pre-computed points. The efficiency and accuracy of this method are demonstrated on several test cases, which show significant computational savings relative to the standard Galerkin projection reduced-order approach. Copyright q 2008 John Wiley & Sons, Ltd.