A C∞ numerical method for the basic problem in the calculus of variations - Gregory, John; Yang, George (Utilitas Mathematica, 1999)
In previous work Gregory, along with Lin and Wang, gave accurate numerical algorithms to efficiently solve general (constrained) variational problems in the calculus of variations/optimal control theory. Of special note was that the authors (usually) used spline hat functions, getting an a priori, pointwise maximal error of O(h2). The drawback of these methods was that our solution was only piecewise C1 and it was not obvious how to extend these results to smooth solutions when they exist. We now propose to obtain numerical algorithms which retain the efficiency, accuracy and generality of this earlier work in using the same intervals of support but whose solutions are C∞. In this case our a priori pointwise maximal error will remain O(h2), a very surprising result. In addition, we will immediately be able to extend our results to free and constrained calculus of variations/optimal control theory problems, obtaining smooth solutions when they exist.