In this note we state some problems on approximation by univariate splines with free knots, bivariate segment approximation and tensor product splines with variable knot lines. There is a vast literature on approximation and interpolation by univariate splines with fixed knots (see e.g. the books of de Boor [1], Braess [2], DeVore & Lorentz [4], Powell [20], Schumaker [21], Nürnberger [13] and the book of Chui [3] on multivariate splines). On the other hand, numerical examples show that in general, the error is much smaller if variable knots are used for the approximation of functions instead of fixed knots. This is true for univariate splines as well as for bivariate splines. But approximation by splines with free knots leads to rather difficult nonlinear problems.[...

Nürnberger, Günther

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Bivariate Segment Approximation

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Bivariate Segment
Approximation
Günther Nürnberger
Reihe Mathematik Nr. 208/1996
•
Bivariate Segment Approximation
G. Nürnberger
Fakultät für Mathematik und Informatik
Universität Mannheim, D-68131Mannheim, Germany
In this note we state some problems on approximation by univariate splines with free
knots, bivariate segment approximation and tensor product splines with variable knot lines.
There is a vast literature on approximation and interpolation by univariate splines with
fixed knots (see e.g. the books of de Boor [1]' Braess [2]' DeVore & Lorentz [4]' Powell [20]'
Schumaker [21]' Nürnberger [13] and the book of Chui [3] on multivariate splines). On the
other hand, numerical examples show that in general, the error is much smaller if variable
knots are used for the approximation of functions instead of fixed knots. This is true for
univariate splines as weIl as for bivariate splines. But approximation by splines with free
knots leads to rather difficult nonlinear problems.
We first consider best approximation by univariate splines with free knots. We denote
by Sm,k the (non-convex) set of splines of degree m with k free knots. Given a function
1 E G[a, b], we call SI E Sm,k a best approximation of 1 if 111 - sll = min 111- sll,
SESm,k
where Ilhll = max Ih(t)1 denotes the maximum norm of hE G[a, b]. This problem was first
tE[a,b]
investigated by Johnson [5] in 1960. Since then, many results were proved on this topic by
using advanced methods (see e.g. the books [2]' [4], [13]' [21] and the recent results in [6]'
[10]' [11]' [12]' [14]' [15]' [16]' [17]' [22]). Nevertheless, there is no complete solution of the
following problem.
Problem 1. Give characterizations of best approximations and unique best approximations
from Sm,k'
It is weIl known that, in contrast to the fixed knot case, best approximations from Sm,k
cannot be characterized by alternation properties of the error function alone. But at present,
it seems not to be known what the additional conditions could be which completely charac-
terize best approximations.
The deeper reason that nonlinear problems of this type are unsolved is that there is no
complete theory of nonlinear optimization and no algorithm for computing global minima of
arbitrary nonlinear optimization problems.
On the other hand, an algorithm was developed for computing good spline approximations
with free knots (Nürnberger, Sommer & Strauß [19]' Meinardus, Nürnberger, Sommer &
Strauß [8]). The spline approximations are computed in two steps.
Given a function 1E G[a, b], first a segment approximation problem is solved. We denote
by PPm,k the (non-convex) set of piecewise polynomials of degree m with k free knots. In
the first step, a best approximation of 1 from PPm,k is computed such that the errors on
all knot-intervals are the same. Therefore, the corresponding knots, denoted by Xl, ... , Xk
(which in general, are nonuniform), reflect the critical parts ofthe function I. The algorithm
converges by starting with an arbitrary set of knots.
Then in the second step, by applying aRemez type algorithm (Nürnberger & Som-
mer [18]), a best approximation of 1 from the space Sm(XI,"', Xk) of splines of degree m
with k fixed knots is computed. Numerical examples show that in this way a good approxi-
1
r------------------------------ --, .
mation from Sm,k is obtained (see [8]' [13]). The resulting error can be compared with the
error for PPm,k which is a lower bound.
In the following, we consider similar problems in the bivariate case. First, we formulate
a general spline approximation problem for variable knot lines.
Let a rectangle T = [a, b] x [c, d] be subdivided by knot lines x = Xi and y = Yi, i = 1, ... , k,
into (k+ 1)2 subrectangles. Such a partition is called 0/ type I. Moreover, let an approximation
or interpolation method be given which yields for each function / E C(T) an approximation
A(f) from the tensor product spline space Sm (Xl, ... , Xk) ~ Sm (Yl, ... , Yk) (see e.g. [1]' [13]).
We consider the case when the knot lines are variable.
Problem 2. Determine a partition of type I for which the error 11/ - A(f)IIT is relatively
smalI.
As in the univariate case, it is natural to get a good partition as in Problem 2 by solving
segment approximation problems for lln ~ lln (where lln denotes the space of univariate
polynomials of degree n) of the following type.
We subdivide the rectangle T by k horizontal lines into k + 1 strips. Then we subdivide
each strip by k verticalline segments into k + 1 subrectangles. The partitions of each strip
are different, in general. Such a partition of T is called 0/ type II. If we subdivide T by k
vertical lines into k + 1 strips, and subdivide each strip by k horizontal line segments into
k + 1 subrectangles, then the partition is called 0/ type III.
We associate to each subrectangle Tj of a partition of type I, II or III areal number d(Tj)
which may be the error 11/ - B(fHIT; of some approximation or interpolation method, where
B(f) is from the tensor product space lln ~ lln' The real number mayaIso be a lower or
upper bound of this error. We consider the case when the line segments of the partition are
variable.
Problem 3. Describe and determine the partitions of type I, II or III for which max d(Tj)
J
is minimal.
There may be relations between optimal partitions of type I, II and III.
Problem 4. Do the horizontal lines of an optimal partition of type II combined with the
vertical lines of an optimal partition of type III yield an optimal partition of type I (in the
sense of Problem 3)?
In arecent paper, Meinardus, Nürnberger & Walz [9] developed an algorithm for com-
puting optimal partitions of type II and III for the best approximation error and related
functionals (Problem 3). They also showed that for some of these cases, Problem 4 has a
positive answer. Their numerical examples show that tensor product polynomial interpola-
tion at uniform points on the subrectangles is a suitable method for segment approximation
(although this method does not quite fit into the general setting of the convergence results).
An optimal partition of type I in the sense of Problem 3 yields a good partition for tensor
product spline interpolation in the sense of Problem 2.
Although, first results on the above problems are known, several problems for various
approximation methods with different function classes are unsolved at present. Efficient
methods for solving these problems would be important for applications in practice.
References
[1] C. DE BOOR (1978): A Practical Guide to Splines. New York: Springer.
[2] D. BRAESS (1986): Nonlinear Approximation Theory. Berlin: Springer.
[3] C. K. CHUI (1988): Multivariate Splines. Philadelphia: CBMS-SIAM.
2
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•
•
[4] R. A. DEVORE, G. G. LORENTZ (1993): Constructive Approximation. New York:
Springer.
[5] R. S. JOHNSON (1960): On monosplines of least deviation. Trans. Amer. Math. Soc.,
96: 458-477.
[6] H. KAWASAKI (1994): A seeond-order properly of spline funetions with one free knot. J.
Approx. Theory, 78: 293-297.
[7] G. MEINARDUS (1967): Approximation of F\mctions: Theory and Numerical Methods.
Berlin: Springer.
[8] G. MEINARDUS, G. NÜRNBERGER, M. SOMMER, H. STRAUSS (1989): Algorithms for
pieeewise polynomials and splines with free knots. Math. Comp., 53: 235-247.
[9] G. MEINARDUS, G. NÜRNBERGER, G. WALZ (preprint): Bivariate segment approxi-
mation and splines.
[10] B. MULANSKY (1992): Chebyshev approximation by spline functions with free knots.
IMA J. Numer. Anal., 12: 95-105.
[11] B. MULANSKY (1992): Neeessary eonditions for loeal best Chebyshev approximations by
splines with free knots. In: Numerical Methods of Approximation Theory (D. Braess, L.
L. Schumaker, eds.). ISNM, 105. Basel: Birkäuser. pp. 195-206.
[12] G. NÜRNBERGER (1987): Strongly unique spline approximation with free knots. Constr.
Approx., 3: 31-42.
[13] G. NÜRNBERGER (1989): Approximation by Spline Functions. Berlin: Springer.
[14] G. NÜRNBERGER (1992): The metrie projeetion for free knot splines. J. Approx. Theory,
71: 145-153.
[15] G. NÜRNBERGER (1994): Strong unieity in nonlinear approximation and free knot
splines. Constr. Approx., 10: 285-299.
[16] G. NÜRNBERGER (1994): Approximation by univariate and bivariate splines. In: Sec-
ond International Colloquium on Numerical Analysis (D. Bainov, V. Covachev, eds.).
Utrecht: VSP. pp. 143-153.
[17] G. NÜRNBERGER, L. L. SCHUMAKER, M. SOMMER, H. STRAUSS (1989): Uniform
approximation by generalized splines with free knots. J. Approx. Theory, 59: 150-169.
[18] G. NÜRNBERGER, M. SOMMER (1983): ARemez type algorithm for spline funetions.
Numer. Math., 41: 117-146.
[19] G. NÜRNBERGER, M. SOMMER, H. STRAUSS (1986): An algorithm for segment approx-
imation. Numer. Math., 48: 463-477.
[20] M. J. D. POWELL (1981): Approximation Theory and Methods. Cambridge University
Press.
[21] L. L. SCHUMAKER (1981): Spline Functions: Basic Theory. New York: Wiley-
Interscience.
[22] M. SOMMER (1992): Segment approximation using linear funetionals. In: Numerical
Methods of Approximation Theory (D. Braess, L. L. Schumaker, eds.). ISNM, 105.
Basel: Birkäuser. pp. 331-346 .
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