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

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.