TY - JOUR

T1 - Finding effective points by surrogate models with overcomplete bases

AU - Wang, Weichung

AU - Chen, Ray Bing

N1 - Funding Information:
The authors are grateful to Dianne P. O’Leary, Ying Nian Wu, John E. Dennis, Eric Jaehnig, and the anonymous referees for the helpful comments and suggestions. This work is partially supported by the National Science Council and the National Center for Theoretical Sciences in Taiwan.

PY - 2008/7/15

Y1 - 2008/7/15

N2 - We develop algorithms to find the so-called effective points (EPs) x ∈ X ⊂ Rn such that the corresponding responses f (x) ∈ R belong to a specific region of interest (ROI). Examples of an ROI include extreme values, bounded intervals, and positivity. We are especially interested in the problem defined by the following characteristics: (i) the definition of f (x) is either complicated or implicit, (ii) the response surface f (X) does not fit simple patterns, and (iii) computational costs of function evaluations is high. To solve this problem, we iteratively approximate the true yet unknown response surface with simplified surrogate models and then use the surrogate models to predict the possible EPs. Unlike interpolation schemes, the surrogate models are formed by linear combinations of a set of overcomplete bases and they are not obliged to fit the known response value. A numerical example that involves finding positive Lyapunov exponents of a dynamical system shows that the algorithm is efficient and practical.

AB - We develop algorithms to find the so-called effective points (EPs) x ∈ X ⊂ Rn such that the corresponding responses f (x) ∈ R belong to a specific region of interest (ROI). Examples of an ROI include extreme values, bounded intervals, and positivity. We are especially interested in the problem defined by the following characteristics: (i) the definition of f (x) is either complicated or implicit, (ii) the response surface f (X) does not fit simple patterns, and (iii) computational costs of function evaluations is high. To solve this problem, we iteratively approximate the true yet unknown response surface with simplified surrogate models and then use the surrogate models to predict the possible EPs. Unlike interpolation schemes, the surrogate models are formed by linear combinations of a set of overcomplete bases and they are not obliged to fit the known response value. A numerical example that involves finding positive Lyapunov exponents of a dynamical system shows that the algorithm is efficient and practical.

UR - http://www.scopus.com/inward/record.url?scp=43049085218&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=43049085218&partnerID=8YFLogxK

U2 - 10.1016/j.cam.2006.12.033

DO - 10.1016/j.cam.2006.12.033

M3 - Article

AN - SCOPUS:43049085218

VL - 217

SP - 110

EP - 122

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

IS - 1

ER -