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

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Abstract
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 © 2008 John Wiley & Sons, Ltd.

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