A new method for interpolating in a convex subset of a Hilbert space
In this paper, interpolating curve or surface with linear inequality constraints is considered as a general convex optimization problem in a Reproducing Kernel Hilbert Space. The aim of the present paper is to propose an approximation method in a very general framework based on a discretized optimization problem in a finite-dimensional Hilbert space under the same set of constraints. We prove that the approximate solution converges uniformly to the optimal constrained interpolating function. Numerical examples are provided to illustrate this result in the case of boundedness and monotonicity constraints in one and two dimensions.
KeywordsOptimization RKHS Interpolation Inequality constraints
The authors would like to thank the Associate Editor and the two anonymous referees for their helpful suggestions.
- 12.Laurent, P.J.: An algorithm for the computation of spline functions with inequality constraints. Séminaire d’analyse numérique de Grenoble, No. 335. Grenoble (1980)Google Scholar
- 16.Rasmussen, C.E., Williams, C.K.I.: Gaussian Processes for Machine Learning (Adaptive Computation and Machine Learning). The MIT Press, Cambridge (2005)Google Scholar
- 20.Wolberg, G., Alfy, I.: Monotonic cubic spline interpolation. In: Proceedings of Computer Graphics International, pp. 188–195 (1999)Google Scholar