Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Data Interpolation by Optimal Splines with Free Knots Using Linear Programming

Version 1 : Received: 19 April 2021 / Approved: 22 April 2021 / Online: 22 April 2021 (09:07:02 CEST)

A peer-reviewed article of this Preprint also exists.

Thakur, L.S.; Bragin, M.A. Data Interpolation by Near-Optimal Splines with Free Knots Using Linear Programming. Mathematics 2021, 9, 1099. Thakur, L.S.; Bragin, M.A. Data Interpolation by Near-Optimal Splines with Free Knots Using Linear Programming. Mathematics 2021, 9, 1099.

Journal reference: Mathematics 2021, 9, 1099
DOI: 10.3390/math9101099


Studies have shown that in many practical applications, data interpolation by splines leads to better approximation and higher computational efficiency as compared to data interpolation by a single polynomial. Data interpolation by splines can be significantly improved if knots are allowed to be free rather than at a priori fixed locations such as data points. In practical applications, the smallest possible curvature is often desired. Therefore, optimal splines are determined by minimizing a derivative of continuously differentiable functions comprising the spline of the required order. The problem of obtaining an optimal spline 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 analytically, interpolation by splines of higher orders or in higher dimensions is 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 substituted by their discrete counterparts. 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 appropriate 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 resulting splines provide a good approximation to the optimal splines.


Optimal Splines; Linear Programming; Data Interpolation; Splines with Free Knots



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