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Sparse grid interpolation and engineering applications

Introduction

The interpolation problem considered with sparse grid interpolation is an optimal recovery problem (i.e. the selection of points such that a smooth multivariate function can be approximated with a suitable interpolation formula). Depending on the characteristics of the function to interpolate (degree of smoothness, periodicity), various interpolation techniques based on sparse grids exist. All of them employ Smolyak's construction, which forms the basis of all sparse grid methods. The asymptotic error decay of full grid interpolation is preserved up to a logarithmic factor with increasing grid resolution. An additional benefit of the method is its hierarchical structure, which can be used to obtain an estimate of the current approximation error. Thus, one can easily develop an interpolation algorithm that aborts automatically when a desired accuracy is reached.

Sparse Grid Interpolation Toolbox for Matlab

A Matlab toolbox has been developed that includes hierarchical sparse grid interpolation algorithms based on both piecewise multilinear and polynomial basis functions. Special emphasis is placed on an efficient implementation that performs well even for very large dimensions d > 10. There are many ways to customize the behavior of the interpolation routines. Furthermore, additional tasks involving the interpolants can be performed, such as computing derivatives or performing an optimization or integration.

The following list gives an overview of the options that are available:

Link: Sparse Grid Interplation Toolbox Pfeil

Examples for Dimension-Adaptive Sparse Grid Interpolation

With piecewise linear basis functions, f(x,y) = exp(-5x2) + exp(-5y2):

spanim.gif

With polynomial basis functions, Branin's function:

spanim.gif

Publications

Publications of the group on sparse grids -including many applications to engineering problems- are listed below.

Author(s) Title Appeared In Year Type Download
Corinna Hager, Stefan Hüeber, Barbara Wohlmuth Numerical techniques for the valuation of basket options and its Greeks J. Comput. Fin. 13(4), 1-31 2010 Article in Journal Abstract
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Andreas Klimke, Christopher J. Pye Sparse grid meta-models for model updating Proceedings of the IMAC XXVII Conference, Orlando, Fl 2009 Article in Conference Proceedings Abstract
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Andreas Klimke Sparse grid surrogate functions for nonlinear systems with parameter uncertainty Proceedings of the 1st International Conference on Uncertainty in Structural Dynamics, 159-168 2007 Article in Conference Proceedings Abstract
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Andreas Klimke Uncertainty Modeling using Fuzzy Arithmetic and Sparse Grids PhD thesis, Universität Stuttgart 2006 PhD Thesis Abstract
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Andreas Klimke Sparse Grid Interpolation Toolbox User's Guide IANS Documentation 2006/001 2006 Technical Manual Abstract
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Andreas Klimke, Barbara Wohlmuth Constructing dimension-adaptive sparse grid interpolants using parallel function evaluations Parallel Process. Lett. 16, 407-418 2006 Article in Journal Abstract
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Andreas Klimke Construction of Hierarchical Polynomial Sparse Grid Interpolants using the Fast Discrete Cosine Transform IANS Preprint 2006/007 2006 Preprint Abstract
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Andreas Klimke, Ronaldo Nunes, Barbara Wohlmuth Fuzzy arithmetic based on dimension-adaptive sparse grids: a case study of a large-scale finite element model under uncertain parameters Internat. J. Uncertain. Fuzziness Knowledge-Based Systems 14, 561-577 2006 Article in Journal Abstract
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Andreas Klimke, Barbara Wohlmuth Computing expensive multivariate functions of fuzzy numbers using sparse grids Fuzzy Sets and Systems 154, 432-453 2005 Article in Journal Abstract
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Andreas Klimke, Barbara Wohlmuth Piecewise multilinear hierarchical sparse grid interpolation in Matlab ACM Trans. Math. Software. 31, 561-579 2005 Article in Journal Abstract
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Andreas Klimke Uncertainty modeling using fuzzy arithmetic based on dimension-adaptive sparse grids Proceedings of CANCAM 2005, Montreal, pp. 596-597 2005 Article in Conference Proceedings Abstract

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Andreas Klimke, Kai Willner, Barbara Wohlmuth Uncertainty modeling using fuzzy arithmetic based on sparse grids: applications to dynamic systems Internat. J. Uncertain. Fuzziness Knowledge-Based Systems 12, 745-759 2004 Article in Journal Abstract
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Andreas Klimke, Barbara Wohlmuth Efficient fuzzy arithmetic for nonlinear functions of modest dimension using sparse grids Proceedings of FUZZ-IEEE 2004, Budapest, Hungary 2004 Article in Conference Proceedings Abstract
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Andreas Klimke Piecewise multilinear sparse grid interpolation in Matlab IANS Technical Report 2003/019 2003 Technical Report Abstract
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