General Methods for Approximation and Interpolation
3.2 Piecewise polynomials
3.2 Piecewise polynomials
Spline, i.e. piecewise polynomial, methods usually require a triangulation of the set in order to define the space from which we approximate, unless the data sites are in very special positions, e.g. gridded or otherwise highly regularly distributed. The reason for this is that it has to be decided where the pieces of the piecewise polynomials lie and where they are joined together. Moreover, it then has to be decided with what smoothness they are joined together at common vertices, edges etc. and how that is done. This is not at all trivial in more than one dimension and it is highly relevant in connection with the dimension of the space. Since triangulations or similar structures (such as quadrangulations) can be very difficult to provide in more than two dimensions, we concentrate now on two-dimensional problems – this in fact is one of the severest disadvantages of piecewise polynomial techniques and a good reason for using radial basis functions (in three or more dimensions) where no triangulations are required. Moreover, the quality of the spline approximation depends severely on the triangulation itself, long and thin triangles, for instance, often being responsible for the deterioration of the accuracy of approximation.
Letwith elementsξ be the given data sites in R2. We describe the Delaunay
triangulation which is a particular technique for triangulation, and give a stan- dard example. We define the triangulation by finding first the so-called Voronoi tessellation which is in some sense a dual representation. Let, for ζ from ,
T ζ = { x ∈ R2 | x −ζ = min x −ξ ,ξ ∈ }. These T ζ are two-dimensional
tiles surrounding the data sites. They form a Voronoi diagram and there are points where three of those tiles meet. These are the vertices of the tessellation. (In degenerate cases there could be points where more than three tiles meet.) Let t ζ be any vertex of the tessellation; in order to keep the description simple,
we assume that degeneracy does not take place. Let Dζ be the set of those three
ξ such that t ζ − ξ is least. Then the set of triangles defined by the Dζ is our triangulation, it is the aforementioned dual to the Voronoi diagram. This algorithm is a reliable method for triangulation with a well-developed theory, at least in two dimensions (cf., e.g., Braess, 1997, Brenner and Scott, 1994). In higher dimensions there can be problems with such triangulations, for instance it may be difficult to re-establish prescribed boundary faces when triangula- tions are updated for new sets of data, which is important for solving PDEs numerically with finite element methods.
Now we need to define interpolation by piecewise polynomials on such a triangulation. It is elementary how to do this with piecewise linears. However, often higher order piecewise polynomials and/or higher order smoothness of the interpolants are required, in particular if there is further processing of the
42 3. General methods
approximants in applications to PDE solving etc. needed. For instance, piece- wise quadratics can be defined by interpolating at all vertices of the triangulation plus the midpoints of the edges, which gives the required six items of informa- tion per triangle. Six are needed because quadratics in two dimensions have six degrees of freedom. This provides an interpolant which is still only continuous. In order to get continuous differentiability, say, we may estimate the gradient of the proposed interpolant at the vertices, too. This can be done by taking suitable differences of the data, for example. In order to have sufficient freedom within each of the triangles, they have to be further subdivided. The subdivision into subtriangles requires additional, interior C 1 conditions.
Powell and Sabin (1977) divide the triangles into six subtriangles in such a way that the approximant has continuous first derivatives. To allow for this, the subdivision must be such that, if we extend any internal boundary from the common internal vertex to an edge, then the extension is an internal boundary of the adjacent element. Concretely, one takes the midpoint inside the big triangle to be the intersection of the normals at the midpoints of the edges. By this construction and by the internalC 1 requirement we get nine degrees of freedom
for interpolating function values and gradients at the vertices, as required. Continuity of the first derivatives across internal edges of the triangulation is easy to show due to the interpolation conditions and linearity of the gradient. Another case is the C 1-Clough–Tocher interpolant (Ciarlet, 1978). It is a
particularly easy case where each triangle of the triangulation is divided into three smaller ones by joining the vertices of the big triangle to the centroid. If we wish to interpolate by these triangles over a given (or computed) triangulation, we require function and gradient values at each of the vertices of the big triangle plus the normal derivatives across its edges (this is a standard but not a necessary condition; any directional derivative not parallel to the edges will do). Therefore we get 12 data for each of the big triangles inside the triangulation, each of which is subdivided into three small triangles. On each of the small triangles, there is a cubic polynomial defined which provides 10 degrees of freedom each. The remaining degrees of freedom are taken up by the interior smoothness conditions inside the triangle.
In those cases where the pointsform a square or rectangular grid, be it finite or infinite, triangulations such as the above are not needed. In that event, tensor- product splines can be used or, more generally, the so-called box-splines that are comprehensively described in the book by de Boor, H¨ollig and Riemenschneider (1993). Tensor-product splines are the easiest multivariate splines, but here we start by introducing the more general notion of box-splines and then we will simplify again to tensor-product splines as particular examples. Box-splines are piecewise polynomial, compactly supported functions defined by so-called
3.2 Piecewise polynomials 43
direction sets X ⊂ Zn and the Fourier transform of the box-spline B,
(3.1) Bˆ(t ) =
x ∈ X sin 12 x · t 1 2 x · t , t ∈ Rn .We recall the definition of the Fourier transform from the previous chapter. They can also be defined directly in the real domain without Fourier transforms, e.g. recursively, but for our short introduction here, the above is sufficient and indeed quite handy. In fact, many of the properties of box-splines are derived from their Fourier transform which has the above very simple form. Degree of the polynomial pieces, smoothness, polynomial recovery and linear indepen- dence are among the important properties of box-splines that can be identified from (3.1).
The direction sets X are fundamental to box-splines; they are responsible via the Fourier transform for not only degree and smoothness of the piecewise poly- nomial B but also its approximation properties and its support X [0,1]|| ⊂ Rn.
By the latter expression we mean all elements of the ||-dimensional unit cube, to which X seen as a matrix (and as a linear operator) is applied. Usually, X
consists of multiple entriesof vectors with components from{0, ±1}, but that is not a condition on X . Due to the possibly repeated entries, they are sometimes called multisets. The only condition is that always span X = Rn. If in two
dimensions, say, the vectors (10), (01), (11) are used, one speaks of a three- directional box-spline, if (−11) is added, a four-directional one, and any number of these vectors may be used. These two examples are the Courant finite element, and the Zwart–Powell element, respectively. If X contains just the two unit vectors in two dimensions, we get the characteristic function of the unit square.
In the simplest special case, X consists only of a collection of standard unit vectors of Rn, where it is here particularly important that multiple entries in the
set X are allowed. If that is so, B is a product B( y) = B1( y1)· B2( y2). . . Bn( yn)
of univariate B-splines, wherey = ( y1, y2, . . . , yn)T and the degreesi−1ofthe B-splines are defined through the multiplicityi of the corresponding unit vector
in X . When X has more complicated entries, other choices of box-splines B
occur, i.e. not tensor-products, but they are still piecewise polynomials of which we have seen two examples in the paragraph above. In order to determine the accuracy that can be obtained from approximations by B and its translates along the grid (or the h-scaled grid) it is important to find out which polynomials lie in the span of those translates. This again depends on certain properties of X , as does the linear independence of the translates of the box-spline. The latter is relevant if we want to interpolate with linear combinations of the translates of the box-spline.
44 3. General methods
Linear independence, for instance, is guaranteed if X is ‘unimodular’, i.e. the determinants of each collection of n vectors from X are either 0 or ±1 (Dahmen and Micchelli, 1983a), which is important, not only for interpolation from the space but also if we wish to create multiresolution analyses as defined in Chapter 9.
Out of the many results which are central to the theory and applications of box-splines we choose one that identifies the polynomials in the linear span of the box-splines. It is especially important to the approximational power of box-splines. Another one, which we do not prove here, is the fact that the multiinteger translates of a box-spline such as the above form a partition of unity.
Theorem 3.3. Let S be the linear span of the box-spline B defined by the
direction set X ⊂ Zn , span X = Rn. Let Pn be the space of all n-variate
polynomials. Then Pn ∩ S =
{ Z ⊂ X |span( X \ Z )=Rn} ker
z∈ Z D z,where D z , z ∈ Rn , denotes in this theorem directional derivative in the direction
of z.
For the proof and further discussion of this result, see de Boor, H¨ollig and Riemenschneider (1993). A corollary whose simple proof we present is
Theorem 3.4. Let Pk
n be all polynomials in Pn of total degree at most k, let
d := max{r |span X \ Z = Rn , ∀ Z ⊂ X with | Z | = r }. Then Pk
n ⊂ S ⇐⇒
k ≤ d.
Proof: ‘⇐=’: Let Z be a subset of X . Since
z∈ Z D z reduces the degree ofany polynomial by | Z | or less (| Z | being attained) and since by the definition of d
min
{ Z ⊂ X |span ( X \ Z )=Rn} | Z | = d + 1,
it follows that Pd
n ⊂ ker
D z, as required, for such Z .There is, again for one such Z ,| Z | = d +1 attained, whence
z∈ Z D z p = 0 for some p ∈ Pd +1n . This proves the other implication.
We remark that the numberd used in Theorem 3.4 is also related to the smooth- ness of the box-spline. The box-spline with direction set X and the quantity
d as defined above is d − 1 times continuously differentiable and its partial derivatives of the next order are bounded if possibly discontinuous.