Abstract
In this paper, we study the problem of finding a real-valued function f on the interval [0, 1] with minimal L 2 norm of the second derivative that interpolates the points (t i, y i) and satisfies e(t) ≤ f(t) ≤ d(t) for t ∈ [0, 1]. The functions e and d are continuous in each interval (t i, t i+1) and at t 1 and t nbut may be discontinuous at t i. Based on an earlier paper by the first author [7] we characterize the solution in the case when e and d are linear in each interval (t i, t i+1). We present a method for the reduction of the problem to a convex finite-dimensional unconstrained minimization problem. When e and d are arbitrary continuous functions we approximate the problem by a sequence of finite-dimensional minimization problems and prove that the sequence of solutions to the approximating problems converges in the norm of W 2,2 to the solution of the original problem. Numerical examples are reported.
Similar content being viewed by others
References
L.-E.Andersson and T.Elfving, “An algorithm for constrained interpolation,” SIAM J. Sci. Stat. Comput., vol. 8, pp. 1012–025, 1987.
L.-E.Andersson and T.Elfving, “Best constrained approximation in Hilbert space and interpolation by cubic splines subject to obstacles,” SiAM J. Sci. Stat. Comput., vol. 16, pp. 1209–1232, 1995.
C.deBoor, “On “best” interpolation,” J. Approx. Theory, vol. 16, pp. 28–42, 1976.
J.M.Borwein and A.S.Lewis, “Partially finite convex programming,” Mathematical Programming, vol. 57, 1992, “Part I: Quasi relative interiors and duality theory,” pp. 15–48; “Part II: Explicit lattice models,” pp. 49–83.
C.K.Chui, F.Deutsch, and J.D.Ward, “Constrained best approximation in Hilbert space” Part I: Constr. Approx., vol. 6, pp. 35–64, 1990; Part II: J. Approx. Theory, vol. 71, pp. 213–238, 1992.
A.L.Dontchev, “Duality methods for constrained best interpolation,” Math. Balkanica, vol. 1, pp. 96–105, 1987.
A.L.Dontchev, “Best interpolation in a strip,” J. Approx. Theory, vol. 73, pp. 334–342, 1993.
A.L.Dontchev and B.D.Kalchev, “Duality and well-posedness in convex interpolation,” Numer. Func. Anal. Optim., vol. 10, pp. 637–687, 1989.
A.L.Dontchev and I.Kolmanovsky, “State constraints in the linear regulator problem: A case study,” J. Opt. Th. Appl., vol. 87, pp. 323–347, 1995.
B.Fischer, G.Opfer, and M.L.Puri, “A local algorithm for constructing non-negative cubic splines,” J. Approx. Theory, vol. 64, pp. 1–16, 1991.
J.C.Holladey, “A smoothest curvature approximation,” Math. Tables Aids Comput., vol. 11, pp. 233–243, 1957.
U.Hornung, “Interpolation by smooth functions under restrictions on the derivatives,” J. Approx. Theory, vol. 28, pp. 227–237, 1980.
L.D.Irvine, S.P.Marin, and P.W.Smith, “Constrained interpolation and smoothing,” Constructive Approx., vol. 1, pp. 129–151, 1986.
C.A.Micchelli, P.W.Smith, J.Swetits, and J.D.Ward, “Constrained L papproximation,” Constr. Approx., vol. 1, pp. 93–102, 1985.
C.A.Micchelli and F.I.Utreras, “Smoothing and interpolation in a convex subset of a Hilbert space,” SIAM J. Sci. Stat. Comput., vol. 9, pp. 728–746, 1988.
G.Opfer and H.J.Oberle, “The derivation of cubic splines with obstacles by methods of optimization and optimal control”, Numer. Math., vol. 52, pp. 17–31, 1988.
Author information
Authors and Affiliations
Additional information
The first author was supported by National Science Foundation Grant Number DMS 9404431. The second author was supported by a François-Xavier Bagnoud doctoral fellowship and by National Science Foundation Grant Number MSS 9114630.
Rights and permissions
About this article
Cite this article
Dontchev, A.L., Kolmanovsky, I. Best interpolation in a strip II: Reduction to unconstrained convex optimization. Comput Optim Applic 5, 233–251 (1996). https://doi.org/10.1007/BF00248266
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00248266