Method of optimal variable-knot spline interpolation in the 2 discrete norm

Abstract

Interpolation to data by polynomial splines may be significantly improved if the knots of the splines are considered as free variables. When the appropriate problem of knot optimization is formulated (i.e. the problem of simultaneous interpolation and approximation with constraints) using the B-spline basis for interpolating splines, and separating the linear and non-linear aspects, that problem may be reduced to constrained non-linear least squares in the variable knots. The algorithmfor optimization of variable-knot interpolating Msplines in the 𝕝2 discrete norm is described. It is based ona logarithmic transformation of the knots being optimized which allows the original problem to be converted into the simpler and numerically better conditionedone. Thus, the desired knot distribution is obtained iteratively by the solution of two linear least squares problems which are strictly connected by a non-singular linear map. The practical usefulness of this approach arises from the fact that the optimally distributed knots can essentially reduce the amount of computer memory and work required to adequately represent data being interpolated. Some examples of cubic spline interpolation with such optimized knots are given.