Spline Interpolation under Two‐Sided Restrictions on the Derivatives

Abstract

I n den letzten Johren ist die I d e r p o l d o n von Dotenmerigen verstarkt unlerBeachtung con Nebenbedingungcn betrcichtet worden. Hierzu gehort z. B. die konvexe Interpolation, welche durch die Nichtnega.ticitut der x e i t e n Ableitung charakterisiert ist. I n der vorliegenden Arbeit wird nun die Interpolation unter zweiseitigen Einschrankungen f i i r die Ableitungen untersucht. Auf diese Weise kann z. B. erfaJ0t werden, da$ die Interpolierende auf grwi,wn Teilintervalltn konrex und auj anderen konkav sein SOU. Zur Interpolation werden sowohl quadratische als auch kubische Splines ccvwpndet. I n Vernllgemeinerung friiherer Ergebnisse werden vor allem hinreichende und notwendige Existenzaussagen hergeleitet wwie Moglichkeiten zur effektiven numerischen Ermittlung van Spline-Interpolierenden aufgezeigt, deren Kriirnmung untcr den jeliigen Nebenbedingungen minimal ist. I n the present pa,per the interpolation of data sets is considered under two-sided constraints on th,e dtriiwtiws. l'hus, e.g., the interpolants m a y be convex on some pieces of the domain and concave on others. For solving these interpolation problenzv quadratic and cubic splines a.re used. I n extending earlier results neces.snry and sufficient existcncc conditims are derived as well as numerical procedures for determining interpolants with minimal curvature. B DaHHOn pa6OTe HCCJIefiyCTCH XiAaIIa HIITepnOJlH~HH MII0)fEeCTBa RalIHhlX C IlaJIMYkIeM UBYCTOPOHHHX orpaHmeHHi3 OTHocmemHo npOU360fiHblX. Cmna BXOAHT, Hanptmep , TOT cnyriaii, KorAa mrrepnonnpyio-WaH @YHKUHR fimfieTcH BbinyKnoZi Ha OAHHX, a BorHyToii Ha ApyrHx ysacmax OCH. 3 a x a~a peuaeTcH c miTepnompymWero cnnaiiaa c MmmManbriolYl KPHBH~HOB. IIOMOWblO KBanpaTHYElblX II E E Y~H Y~C K H X CnJlai8HOB. PaCIuIIpHH 6once palIHHe Pe3YJILTaTb1, BbIBOnRTCH HeO6XOAHMble 1% AOCTaTOWlbIe YCnOBHH CYIlieCTBOBaHHR, a TaHXe 113JlaraeTCH YlICJIeHHbIfi MeTOH HaXOWHeHHH 2.1. I n the extended convex interpolwtion with quadratic splines an interpolant s is to be constructed satisfying ZAMM * Z. angew. Math. Mech. 60 (1989) 10 ___-.__.. __. . . ~ ~~ ~ ~__II__-354 i.e.-1 he + (ai -6i-1) t 5 ---(tim i -] ) 2 ~i -] + ( E ( -eipl) t for 0 5 t 5 1 , i = l(1) n .These linear inequalities are fulfilled if and only if they are valid for t = 0 and for t = 1, hiSi-l (= ztmi-l 5 h + ,~i -~ , hi& (=ti -mi-l (=

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

“…Hence the steepest descent method can be used at an early stage to provide a sufficiently good starting point for the Newton method, and the Newton method will stop in a few more steps, according to Theorem 3.3. This numerical scheme is exactly what suggested in a series of papers by Schmidt and his collaborators [2,23,24,4,25,26], and their numerical observation is consistent with the convergence theory proved in this paper.…”

“…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.…”

Regularity and well-posedness of a dual program for convex best C 1-spline interpolation

“…Advantages of the dual approach include that the constrained problem (2) is solved by the unconstrained convex problem (7) and that the classical Newton matrix (Hessian of L) is tridiagonal and positive semidefinite, but at the cost of that L is only once continuously differentiable (this implies that the Hessian of L may not exist for some points). Despite this, numerical experiments [1,18,19,21] show that the Newton method terminates in a small number of steps (averaging 3-5 steps) if a sufficiently good starting point is used. In [16], by making use of the piecewise linearity property of F , we explained that the lack of twice differentiability of L does not present any difficulty in analyzing the convergence of Newton's method.…”

Powell-Sabin-Splines bei der restringierten Interpolation unregelmäßig verteilter Daten

“…In the direct approach, the interpolant is chosen from a suitable spline class depending on a finite number of parameters, e.g., polynomial splines [8], [29], [30], rational splines [24], [28], or exponential splines [34]. The nonnegativity is assured by establishing an inequality system for the free parameters, e.g., the derivatives at the knots.…”