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Data Interpolation by Near-Optimal Splines with Free Knots Using Linear Programming

Author

Listed:
  • Lakshman S. Thakur

    (Department of Operations and Information Management, School of Business, University of Connecticut, Storrs, CT 06269, USA)

  • Mikhail A. Bragin

    (Department of Electrical and Computer Engineering, School of Engineering, University of Connecticut, Storrs, CT 06269, USA)

Abstract

The problem of obtaining an optimal spline with free knots is tantamount to minimizing derivatives of a nonlinear differentiable function over a Banach space on a compact set. While the problem of data interpolation by quadratic splines has been accomplished, interpolation by splines of higher orders is far more challenging. In this paper, to overcome difficulties associated with the complexity of the interpolation problem, the interval over which data points are defined is discretized and continuous derivatives are replaced by their discrete counterparts. The l ∞ -norm used for maximum r th order curvature (a derivative of order r ) is then linearized, and the problem to obtain a near-optimal spline becomes a linear programming (LP) problem, which is solved in polynomial time by using LP methods, e.g., by using the Simplex method implemented in modern software such as CPLEX. It is shown that, as the mesh of the discretization approaches zero, a resulting near-optimal spline approaches an optimal spline. Splines with the desired accuracy can be obtained by choosing an appropriately fine mesh of the discretization. By using cubic splines as an example, numerical results demonstrate that the linear programming (LP) formulation, resulting from the discretization of the interpolation problem, can be solved by linear solvers with high computational efficiency and the resulting spline provides a good approximation to the sought-for optimal spline.

Suggested Citation

  • Lakshman S. Thakur & Mikhail A. Bragin, 2021. "Data Interpolation by Near-Optimal Splines with Free Knots Using Linear Programming," Mathematics, MDPI, vol. 9(10), pages 1-12, May.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:10:p:1099-:d:553756
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    References listed on IDEAS

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    2. Konstantinos Demertzis & Dimitrios Tsiotas & Lykourgos Magafas, 2020. "Modeling and Forecasting the COVID-19 Temporal Spread in Greece: An Exploratory Approach Based on Complex Network Defined Splines," IJERPH, MDPI, vol. 17(13), pages 1-17, June.
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    4. D. P. de Farias & B. Van Roy, 2003. "The Linear Programming Approach to Approximate Dynamic Programming," Operations Research, INFORMS, vol. 51(6), pages 850-865, December.
    5. Daniel Gervini, 2006. "Free‐knot spline smoothing for functional data," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(4), pages 671-687, September.
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