Abstract
Numerical and theoretical questions related to constrained interpolation and smoothing are treated. The prototype problem is that of finding the smoothest convex interpolant to given univariate data. Recent results have shown that this convex programming problem with infinite constraints can be recast as a finite parametric nonlinear system whose solution is closely related to the second derivative of the desired interpolating function. This paper focuses on the analysis of numerical techniques for solving the nonlinear system and on the theoretical issues that arise when certain extensions of the problem are considered.
In particular, we show that two standard iteration techniques, the Jacobi and Gauss-Seidel methods, are globally convergent when applied to this problem. In addition we use the problem structure to develop an efficient implementation of Newton's method and observe consistent quadratic convergence. We also develop a theory for the existence, uniqueness, and representation of solutions to the convex interpolation problem with nonzero lower bounds on the second derivative (strict convexity). Finally, a smoothing spline analogue to the convex interpolation problem is studied with reference to the computation of convex approximations to noisy data.
Similar content being viewed by others
References
C. de Boor (1978): A Practical Guide to Splines. New York: Springer-Verlag.
G. Iliev, W. Pollul (1984):Convex interpolation by functions with minimal Lp-Norm (1<p<∞) of the kth derivative. In: Proceedings of the Thirteenth Spring Conference of the Union of Bulgarian Mathematicians, Sunny Beach, April 6–9, 1984, pp. 31–42.
D. G. Luenberger (1973): Introduction to Linear and Nonlinear Programming. Reading, Massachusetts: Addison-Wesley.
C. A. Micchelli, P. W. Smith, J. Swetits, J. D. Ward (1985):Constrained L p approximation. Constr. Approx.,1:93–102.
T. A. Porsching (1969):Jacobi and Gauss-Seidel methods for nonlinear network problems. SIAM J. Numer. Anal.,6:437–449.
W. C. Rheinboldt (1970):On M-functions and their application to nonlinear Gauss-Seidel iterations and to network flows. J. Math. Anal. Appl.,32:274–307.
Author information
Authors and Affiliations
Additional information
Communicated by Larry Schumaker.
Rights and permissions
About this article
Cite this article
Irvine, L.D., Marin, S.P. & Smith, P.W. Constrained interpolation and smoothing. Constr. Approx 2, 129–151 (1986). https://doi.org/10.1007/BF01893421
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01893421