New algorithms for convex interpolation
In various settings, from computer graphics to financial mathematics, it is necessary to smoothly interpolate a convex curve from a set of data points. Standard interpolation schemes do not respect convexity, and existing special purpose methods require arbitrary choices and/or give interpolants that are very flat between data points. We consider a broad set of spline-type schemes and show that convexity preservation requires the basic spline to be infinitely differentiable but nonanalytic at its endpoints. Using such a scheme - which essentially corresponds to building-in the possibility of "very flatness"
ab initio, rather than, say, enforcing it through extreme parameter choices - gives far more satisfactory numerical results.
Joint work with Eli Passov.