A ‘best points’ interpolation method for efficient approximation of parametrized functions
A ‘best points’ interpolation method for efficient approximation of parametrized functions
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DOI：
10.1002/nme.2086
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发表时间：
2008-01
影响因子：
2.9
通讯作者：
N. Nguyen;A. Patera;J. Peraire
中科院分区：
文献类型：
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作者：
N. Nguyen;A. Patera;J. Peraire
提出了一种有效逼近参数化函数的插值方法。该方法识别并利用参数化函数的低维流形结构来提供良好的近似。基本成分包括一个特定的问题相关基集，定义参数化函数的低维表示，以及一组捕获参数化函数的空间参数变化的"最佳插值点"。最佳插值点被定义为最小二乘最小化问题的解，该最小二乘最小化问题可以使用标准优化算法有效地求解。然后通过一个廉价而稳定的插值过程，从基组和最佳插值点确定近似。此外，一个后验误差估计器被引入到量化的近似误差，并需要很少的额外费用。数值结果证明了该方法的准确性和有效性。版权所有© 2007约翰威利父子有限公司。
We present an interpolation method for efficient approximation of parametrized functions. The method recognizes and exploits the low‐dimensional manifold structure of the parametrized functions to provide good approximation. Basic ingredients include a specific problem‐dependent basis set defining a low‐dimensional representation of the parametrized functions, and a set of ‘best interpolation points’ capturing the spatial‐parameter variation of the parametrized functions. The best interpolation points are defined as solution of a least‐squares minimization problem which can be solved efficiently using standard optimization algorithms. The approximation is then determined from the basis set and the best interpolation points through an inexpensive and stable interpolation procedure. In addition, an a posteriori error estimator is introduced to quantify the approximation error and requires little additional cost. Numerical results are presented to demonstrate the accuracy and efficiency of the method. Copyright © 2007 John Wiley & Sons, Ltd.