Summary
The problem of computing constrained spline functions, both for ideal data and noisy data, is considered. Two types of constriints are treated, namely convexity and convexity together with monotonity. A characterization result for constrained smoothing splines is derived. Based on this result a Newton-type algorithm is defined for computing the constrained spline function. Thereby it is possible to apply the constraints over a whole interval rather than at a discrete set of points. Results from numerical experiments are included.
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References
Andersson, L.-E., Elfving, T.: An Algorithm for Constrained Interpolation. SIAM J. Sci. Stat. Comput.8, 1012–1025 (1987)
Andersson, L.-E. Ivert, P.-A.: Constrained Interpolants with MinimalW k. p-Norm. J. Approximation Theory49, 283–288 (1987)
Dahlquist, G., Björck, Å.: Numerical Methods, Englewood Cliffs, NJ: Prentice Hall 1974
de Boor, C.: A Practical Guide to Splines. Berlin, Heidelberg, New York: Springer 1978
Craven, P., Wahba, G.: Smoothing Noisy Data with Spline Functions: Estimating the Correct Degree of Smoothing by the Method of Generalized Cross-Validation. Numer. Math.31, 377–403 (1979)
Eldén, L.: An Algorithm for the Regularization of Ill-Conditioned, Banded Least Squares Problems. SIAM J. Sci. Stat. Comput.5, 237–254 (1984)
Hornung, U.: Interpolation by Smooth Functions under Restrictions on the Derivatives. J. Approximation Theory28, 227–237 (1980)
Hu, C.L., Schumaker, L.L.: Complete Spline Smoothing. Numer. Math.49, 1–10 (1986)
Hutchinson, M.F., de Hoog, F.R.: Smoothing Noisy Data with Spline Functions. Numer. Math.47, 99–106 (1985)
Iliev, G.L., Pollul, W.: Convex Interpolation by Functions with MinimalL p-norm (1<p<∞) of thek:th derivative. Sonderforschungsbereich 72, Universität Bonn, preprint No. 665 (1984)
Irvine, L.D., Martin, S.P., Smith, P.W.: Constrained Interpolation and Smoothing. Constr. Approximation2, 129–151 (1986)
Miccheli, C.A., Smith, P.W., Swetits, J., Ward, J.D.: ConstrainedL p Approximation. Constr. Approximation1, 93–102 (1985)
Reinsch, C.H.: Smoothing by Spline Functions. Numer. Math.10, 177–183 (1967)
Schoenberg, I.J.: Spline Functions and the Problem of Graduation. Proc. Natl. Acad. Sci. USA52, 947–950 (1964)
Utreras, F.I.: Smoothing Noisy Data Under Monotonicity Constraints. Existence, characterization and convergence rates. Numer. Math.47, 611–626 (1985)
Wahba, G.: Smoothing Noisy Data with Spline Functions. Numer. Math.24, 383–392 (1975)
Wahba, G.: Constrained Regularization for Ill Posed Linear Operator Equations, with Applications in Meteorology and Medicine. In: Statistical Decision Theory and Related Topies Ill (S.S. Gupta, J.O. Berger, eds.), Vol. 2. New York: Academic Press 1982
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Elfving, T., Andersson, LE. An algorithm for computing constrained smoothing spline functions. Numer. Math. 52, 583–595 (1987). https://doi.org/10.1007/BF01400893
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DOI: https://doi.org/10.1007/BF01400893