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A ‘best points’ interpolation method for efficient approximation of parametrized functions
Author(s) -
Nguyen N. C.,
Patera A. T.,
Peraire J.
Publication year - 2007
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.2086
Subject(s) - interpolation (computer graphics) , mathematics , estimator , trilinear interpolation , mathematical optimization , basis function , multivariate interpolation , a priori and a posteriori , approximation error , stairstep interpolation , representation (politics) , nearest neighbor interpolation , algorithm , set (abstract data type) , bilinear interpolation , computer science , mathematical analysis , frame (networking) , telecommunications , philosophy , statistics , epistemology , politics , law , political science , programming language
Abstract 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.