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April  2009, 5(2): 351-361. doi: 10.3934/jimo.2009.5.351

Robust univariate cubic $L_2$ splines: Interpolating data with uncertain positions of measurements

Igor Averbakh 1, , Shu-Cherng Fang 2, and  Yun-Bin Zhao 3,

1. 

Division of Management, University of Toronto at Scarborough, Scarborough, Ontario M1C 1A4, Canada
2. 

Industrial Engineering and Operations Research, North Carolina State University, NC 27695-7906, USA, and Departments of Mathematical Sciences and Industrial Engineering, Tsinghua University, Beijing, China
3. 

School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, United Kingdom

Received  April 2008 Revised  September 2008 Published  April 2009

Traditional univariate cubic spline models assume that the position and function value of each knot are given precisely. It has been observed that errors in data could result in significant fluctuations of the resulting spline. To handle situations that involve uncertainty only in measurements of function values, the concept of a robust spline has been developed in the literature. We propose a more general concept of a PH-robust cubic spline that takes into account also uncertainty in positions of measurements (knots or boundary points) using the paradigm of robust optimization. This bridges the robustness concepts developed in the interpolation/approximation and the optimization communities. Our model handles the case of "coordinated" variations of positions of measurements. It is formulated as a semi-infinite convex optimization problem. We develop a reformulation of the model as a finite explicit convex optimization problem, which makes it possible to use standard convex optimization algorithms for computation.
Keywords: spline function, robust optimization., Approximation, piecewise polynomial interpolation.
Mathematics Subject Classification: Primary: 65D07, 65D10; Secondary: 90C47, 90C3.
Citation: Igor Averbakh, Shu-Cherng Fang, Yun-Bin Zhao. Robust univariate cubic $L_2$ splines: Interpolating data with uncertain positions of measurements. Journal of Industrial and Management Optimization, 2009, 5 (2) : 351-361. doi: 10.3934/jimo.2009.5.351
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