An unconstrained dual program for computing convexC 1-spline approximants

Abstract

--ZusammenfassungAn Unconstrained Dual Program for Computing Convex C1-Spline Approximants. In the present paper the problem of approximating given data sets by convex cubic CX-splines is considered. To this programming problem a dual program is constructed which is unconstrained. Therefore an efficient computational treatment is possible. AMS (MOS)SubjectClassifications: 65D 10, 41A15, 90C20. Key words: Specia~y structured optimization problem, Fenchel dualization, return-formula.. Eine dnale Optimierongsaufgabe ohne Nebenbedingungen zum Problem der konvexen Approximationdutch C1-Splines. Es wird die Aufgabe betrachtet, eine vorgegebene Datenmenge dutch konvexe kubische C~-Splines zu approximierert. Zu diesem Problem wird eine zugeh6rige duale Optimierungsaufgabe konstruiert, welche den Vorteil hat, dab keine Nebenbedingungen auftreten, und welche sich daher effektiver ats ~tas Ausgangsproblem numerisch behandeln l~iBt.

Convex interval interpolation with cubic splines, II

“…The functional (4.3) is widely in use for fitting given data (xo, Wo), (xl, wl) ..... (x,, w,); see [13], [4], [9] and several further papers. Now, tc is seen more dearly to be a parameter by means of which the compromise between curvature minimization and interpolation is controlled.…”

Finite termination of a dual Newton method for convex best C 1 interpolation and smoothing

“…Then we show that the finite termination of the Newton method follows when the iterate p k is sufficiently close to a solution p * and it lies in the same piece U i as p * does. The finite termination property explains the numerically observed behavior of the Newton method that it terminates after a small number of steps when starting from a sufficiently good point [26,2,23,25].…”

“…Based on those theoretical results, it is suggested that the (ordinary) Newton method is applied to (10) or equivalently to (11). Numerical experiments [2,[23][24][25] show that the Newton method terminates in a small number of steps (averaging 3−5 steps) if a sufficiently good starting point is used, which is often provided by the steepest descent method. The finite termination is even observed when starting from rough points.…”

“…The finite termination is even observed when starting from rough points. It is also pointed out that the Newton method alone sometimes failed to find a solution [23,26]. However, it has been unknown when the Newton method has the finite termination property or when it fails.…”

“…The problem of convex smoothing is to determine a smooth function s such that s(x i ) is an approximation to y i (instead of interpolating y i ) and s is convex. Convex smoothing is particularly useful when the feasible set (6) is empty, i.e., convex C 1 interpolation (4) does not exist, see [23,26]. See also [3] for general comments for smoothing.…”