ON THE METHOD OF ADDING-REMOVING KNOTS FOR SOLVING THE SMOOTHING - - PDF document

ON THE METHOD OF ADDING-REMOVING KNOTS FOR SOLVING THE SMOOTHING PROBLEM WITH OBSTACLES

Evely Leetma, Peeter Oja

Let xi, i = 0, . . . , N, be given points on the interval [a, b] and fi, i = 0, . . . , N, measured values at knots xi. In particular case we have N = 22, xi = 1 + i

2

and fi presented on the figure

A cubic spline interpolant f, satisfying condi- tions f(xi) = fi, i = 0, . . . , 22, f′′(1) = f′′(12) = 0

For given weights p0 > 0, . . . , pN > 0, p > 0 we consider a minimization problem min

f∈H2(a,b)

 p b

a

• f′′(x)
• 2 dx +

N

• i=0

pi (f(xi) − fi)2

  ,

where H2(a, b) is the Sobolev space. The solution in the case of equal weights p0 = . . . = p22 = p = 1

The solution in the case of non-equal weights p10 = p12 = p14 = 10

The solution in the case of non-equal weights p10 = p12 = p14 = 100

Let us consider a smoothing problem with obs- tacles as follows min

f ∈ H2(a, b) |f(xi) − fi| ≤ εi, i = 0, . . . , N

b

a

• f′′(x)
• 2 dx.

For given natural numbers r and n, 2r > n ≥ 1, we denote L(r)

2 (Rn) = {f : Rn → R | Dαf ∈ L2(Rn), |α| = r}

as the Beppo-Levi space, where α = (α1, . . . , αn), αi ≥ 0 and |α| = α1+. . .+αn. We define an operator T : L(r)

2 (Rn) → L2(Rn) × . . . × L2(Rn),

Tf =

  

• r!

α!Dαf

• |α| = r

   ,

where α! = α1! · . . . · αn!. We also need the semi-inner product f, gL(r)

2 (Rn) = Tf, Tg =

• |α|=r
• Rn

r! α! Dαf Dαg dX and the corresponding seminorm Tf =

• Tf, Tf.

Let I, I0 ⊂ I and I1 = I \ I0 be finite sets of

• indexes. For given data Xi ∈ Rn, i ∈ I, zi ∈ R,

i ∈ I0, and αi, βi ∈ R, αi < βi, i ∈ I1, we consi- der a smoothing problem with obstacles as the problem min

f∈Ωαβ

• |α|=r
• Rn

r! α! (Dαf)2 dX, (1) where Ωαβ =

• f ∈ L(r)

2 (Rn) | f(Xi) = zi, i ∈ I0,

αi ≤ f(Xi) ≤ βi, i ∈ I1

• .

The solution of this problem is a natural spline.

Let Pr−1 be the space of polynomials of order ≤ r − 1 on Rn. A function S(X) = Q0(X) +

• i∈I

diG(X − Xi), X ∈ Rn, with Xi ∈ Rn, Q0 ∈ Pr−1 and

• i∈I

diQ(Xi) = 0 ∀Q ∈ Pr−1, is called the natural spline. Function G is the fundamental solution of the

• perator ∆r, where ∆ is the n-dimensional

Laplace operator. For n odd G(X) = crnX2r−n and for n even G(X) = crnX2r−n ln X where crn > are some constants and X = (x2

1 + . . . + x2 n)1/2.

It is known that any natural spline belongs to L(r)

2 (Rn).

A natural spline S ∈ Ωαβ is the solution of the problem (1) if and only if in the knots of I1 the spline S satisfies the conditions di = 0, if αi < S(Xi) < βi, (−1)rdi ≥ 0, if S(Xi) = αi, (−1)rdi ≤ 0, if S(Xi) = βi. (2) According to (2) the solution S ∈ Ωαβ is in form S(X) = Q0(X) +

• i∈I0∪Mα∪Mβ

diG(X − Xi), satisfying conditions S(Xi) = zi, i ∈ I0, S(Xi) = αi, i ∈ Mα ⊂ I1, S(Xi) = βi, i ∈ Mβ ⊂ I1, with (−1)rdi > 0 for i ∈ Mα and (−1)rdi < 0 for i ∈ Mβ.

The method of adding-removing knots consists

• f three steps.

At first we construct natural spline satisfying conditions S0(Xi) = zi, i ∈ I0. If I0 = ∅, we take S0 ∈ Pr−1 arbitrarily. At second step we add all knots in which the spline S0 does not satisfy the conditions of obs-

• tacles. If (2) is satisfied we have the solution
• f initial problem. In opposite case we go to

the step 3. At third step we repeatedly remove knots not satisfying conditions (2) until we get spline S1 satisfying (2) and hence optimal on his own set of knots. We continue the second step with S1 if the conditions of obstacles are not satisfied. As result, we get splines S0, S1, S2, . . ., optimal on their own sets of knots.

Examples Let us consider one-dimensional case (n = 1) and r = 1, then corresponding natural splines are linear splines S(x) = c +

N

• i=0

di(x − xi)+, with knots x0, . . . , xN where x+ =

x, if x ≥ 0,

0, if x < 0. There is an easy way to determine the signs of coefficients di, i = 1, . . . , N.

d0 > 0 d1 < 0 d2 < 0 d3 > 0

b b b b

c ¯ c x0 x1 x2 x3 x4 . . . xN S(x)

Let I0 = {1, 2} and I1 = {3, 4, 5, 6}. The con- ditions of interpolation and obstacles are given in figure

b b b b b b b b b b

x1 x3 x4 x5 x6 x2

S0 S0

1

S1 S2

Ignatov, M. I. and Pevnyj, A. B. Natural Splines of Several Variables. Nauka, Leningrad, 1991 (in Russian): The method is finite, i.e. ∃k such that Sk is the solution of initial problem. Proof is based on a false lemma. In the case of cubic splines (n = 1, r = 2) we have an example of cycling.