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Unformatted text preview: Generalized pointwise bias error bounds for response surface approximations Tushar Goel , Raphael T. Haftka , Melih Papila , Wei Shyy Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL 32611 Abstract Surrogate models such as response surface approximations are commonly employed for design optimization and sensitivity evaluations. With growing reliance on first-principle based, often expensive computations, to support the construction of response surfaces, the cost consideration can prevent one from generating enough data needed to ascertain their accuracy. Since response surfaces usually employ low-order polynomials bias (modeling) errors may be substantial. This paper proposes a generalized pointwise bias error bounds estimation method for polynomial based response surface approximations. The method is demonstrated with the help of an analytical example where the model is a quadratic polynomial while the true function is assumed to be cubic polynomial. A relaxation parameter is introduced to account for inconsistencies in the data and the assumed true model. The effect of noise in the data and variation in the relaxation parameter are studied. It is demonstrated that when bias errors dominate, the bias error bounds characterize the actual error field better than the prediction variance. The results also help identify regions in the design space where the accuracy of the response surface approximations is inadequate. Based on such information, local improvements in the maximum bias error bound predictions were accomplished with the aid of selectively generated new data. I. Nomenclature b Vector of estimated coefficients of basis functions A Alias matrix b j Estimated coefficients of basis functions e Vector of true errors at the data points e b ( x ) True bias error at the design point x e es ( x ) Estimated standard error at the design point x F ( x ), F (1) ( x ), F (2) (x) Vectors of basis functions at x f j ( x ) Basis function N Number of data points n 1 Number of basis functions in the regression model n 2 Number of missing basis functions in the regression model x Design point X, X (1) , X (2) Gramian (Design) matrices x 1 , x 2 , ,x n Design variables vectors x 1 (i) , x 2 (i) ,, x n (i) Design variables y Vector of observed responses ( x ) Prediction of the response surface approximation u Graduate Student, Student Member AIAA Distinguished Professor, Fellow AIAA Post Doctoral Research Associate, Member AIAA Distinguished Professor and Department Chair, Fellow AIAA 1 , (1) , (2) Vectors of basis function coefficients j , j (1) , j (2) Coefficients of basis functions ( x ) True mean response at x 2 Noise variance r Minimum relaxation Relaxation parameter k Degree of relaxation II. Introduction and Literature review Response surface approximations (RSAs) are widely accepted for solving optimization problems with high computational or experimental cost. RSAs offer a computationally less expensive way of evaluating designs. computational or experimental cost....
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This note was uploaded on 06/06/2011 for the course EAS 4240 taught by Professor Peterifju during the Spring '08 term at University of Florida.
- Spring '08