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Parameter optimization in multiquadric response surface approximations

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Abstract

Multiquadric (MQ) response surface approximation uses the function ∑Ci|X-Xi|2+h1/2 to interpolate a given set of data. The performance of MQ approximation depends on the shift parameter h. Efficient methods of computing the optimal shift parameter based on the leave-one-out cross validation technique are presented in this paper. We also proved that the condition number of the MQ coefficient matrix is an increasing function of the shift parameter h. Two numerical examples are included to illustrate the proposed formulation.

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References

  1. Carlson, R.E.; Foley, T.A. 1991: The parameter R2 in multiquadric interpolation. Comput. Math. Appl. 21, 29–42

    Google Scholar 

  2. Dyn, N.; Levin, D.; Rippa, S. 1986: Numerical procedures for surface fitting of scattered data by radial basis functions. SIAM J. Sci. Stat. Comput. 7, 639–659

  3. Franke, R. 1982: Scattered data interpolation: tests of some methods. Math. Comput. 48, 181–200

    Google Scholar 

  4. Hardy, R.L. 1971: Multiquadratic equations of topography and other irregular surfaces. J. Geophys. Res. 76, 1905–1915

    Google Scholar 

  5. Jin, R.; Chen, W.; Simpson, T. 2000: Comparative studies of metamodeling techniques under multiple modeling criteria. 8 th AIAA/NASA/USAF/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Long beach, CA, 6–8 September 2000. Also see: Struct. Multidisc. Optim. 22, 5, 1–13

  6. Kansa, E.J. 1990: Multiquadrics – a scattered data approximation scheme with applications to computational fluid dynamics. I. Comput. Math. Appl. 19, 127–145

    Google Scholar 

  7. Kleijnen, J.P.C. 1987: Statistical tools for simulation practitioners. New York: Marcel Dekker

  8. McDonald, D.B.; Grantham, W.J.; Tabor, W.L. 2000: Response surface model development for global/local optimization using radial basis functions. 8 th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Long Beach, CA, AIAA 2000-4776

  9. Micchelli, C.A. 1986: Interpolation of scattered data: distance matrices and conditionally positive definite functions. Constr. Approx. 2, 11–22

    Google Scholar 

  10. Myers, R.H.; Montgomery, D.C. 1995: Response Surface methodology: Process and Product Optimization Using Designed Experiments. New York: Wiley

  11. Simpson, T.W.; Peplinski, J.; Koch, P.N.; Allen, J.K. 1997: On the use of statistics in design and implications for deterministic computer experiments. Design Theory and Methodology – DTM’97, Sacramento, CA, ASME, 14–17 September 1997. Paper No. DETC97/DTM-3881

  12. Yang, R.J.; Gu, L.; Liaw, L.; Gearhart, C.; Tho, C.H.; Liu, X.; Wang, B.P. 2001: Approximations for safety optimization of large systems. Proceeding of DETC ’00 and Computers and Information in Engineering Conference, Baltimore, MD, 10–13 September 2000. DETC2000/DAC-14245

  13. Yang, R.J.; Wang, N.; Tho, C.H.; Bobineau, J.P.; Wang, B.P. 2001: Metamodelling development for vehicle frontal impact simulation. Proceeding of DETC ’00 and Computers and Information in Engineering Conference, Baltimore, MD, 9–12 September 2001. DETC2000/DAC-21012

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  1. Department of Mechanical and Aerospace Engineering, University of Texas at Arlington, Box 19023, 76019, Arlington, TX, USA

    B.P. Wang

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  1. B.P. Wang
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Wang , B. Parameter optimization in multiquadric response surface approximations. Struct Multidisc Optim 26, 219–223 (2004). https://doi.org/10.1007/s00158-003-0341-4

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  • DOI: https://doi.org/10.1007/s00158-003-0341-4

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