Computing bivariate splines in scattered data fitting and the finite ...

Numer Algor (2008) 48:237–260DOI 10.1007/s11075-008-9175-xORIGINAL PAPERComputing bivariate splines in scattered data fittingand the finite-element methodLarry L. SchumakerReceived: 8 October 2007 / Accepted: 25 January 2008 /Published online: 21 May 2008© Springer Science + Business Media, LLC 2008Abstract A number of useful bivariate spline methods are global in nature,i.e., all of the coefficients of an approximating spline must be computed at onetime. Typically this involves solving a system of linear equations. Examplesinclude several well-known methods for fitting scattered data, such as theminimal energy, least-squares, and penalized least-squares methods. Finiteelementmethods for solving boundary-value problems are also of this type.It is shown here that these types of globally-defined splines can be efficientlycomputed, provided we work with spline spaces with stable local minimaldetermining sets.Keywords Bivariate splines · Data fitting · Finite-element method ·Scattered data1 IntroductionBivariate splines defined over triangulations are important tools in severalapplication areas including scattered data fitting and the numerical solutionof boundary-value problems by the finite element method. Methods for computingspline approximations fall into two classes:1. Local methods, where the coefficients of the spline are computed one at atime or in small groups,2. Global methods, where all of the coefficients of the spline have to becomputed simultaneously, usually as the solution of a single linear systemof equations.L. L. Schumaker (B)Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USAe-mail: larry.schumaker@vanderbilt.edu.

238 Numer Algor (2008) 48:237–260In this paper we focus on global methods, and in particular those that arisefrom minimizing a quadratic form, possibly with some constraints. The purposeof this paper is to show how such minimization problems can be efficientlysolved for spline spaces that possess stable local minimal determining sets(MDS) (see Section 2). In particular, we show how our approach appliesto three commonly used scattered data fitting methods: the minimal energymethod, the discrete least-squares (DLSQ) method, and the penalized leastsquares(PLSQ) method. In addition, we discuss how it works for solvingboundary-value problems involving partial differential equations.The standard approach to solving global variational problems involvingpiecewise polynomials on triangulations is to use Lagrange multipliers toenforce interpolation and smoothness conditions, see Remark 1. This resultsin a linear system of equations for the coefficients of the spline. This approachcan be used even when the dimension of the approximating spline space is notknown. In this case the linear system may be singular, and we can only find anapproximate solution. This leads to spline fits which only satisfy the desiredsmoothness conditions approximately. In addition, the Lagrange multiplierapproach generally leads to very large systems of equations, which is its maindisadvantage. In comparison, the method proposed here• involves much smaller systems of equations,• is generally much faster,• produces spline fits lying in the prescribed spline spaces.Our method can be used with spline spaces of any degree and smoothness,as long as they have stable local MDS. There are many examples of such spacesin [14]. We discuss the method for bivariate splines, but the approach appliesto spherical splines and trivariate splines as well, see Remarks 2 and 3.The paper is organized as follows. In Section 2 we recall some of thebasic theory of bivariate splines, including the concepts of MDS and stablelocal bases. We also recall how to compute with bivariate splines using theBernstein–Bézier representation. In Section 3 we discuss the computationof splines solving general quadratic minimization problems. Three explicitexamples of such problems are described in Sections 4–6, namely, minimalenergy, DLSQ, and PLSQ spline fitting. The Galerkin method for solvingelliptic boundary-value problems is treated in Section 7. Section8 is devotedto numerical experiments to illustrate the performance of our method. InSection 9 we describe a useful algorithm for computing all of the Bernsteinbasis polynomials (or their directional derivatives) at a given point in anefficient manner. Finally, we collect several remarks in Section 10.2 PreliminariesIn working with bivariate splines, we follow the notation used in the book [14].For convenience, we review some key concepts here. Given d > r ≥ 0, anda

Numer Algor (2008) 48:237–260 239triangulation △ of a domain ⊂ IR 2 , the associated space of bivariate splinesof smoothness r and degree d is defined to beS r d (△) := {s ∈ Cr () : s| T ∈ P d , all T ∈△}.Here P d is the ( )d+22 -dimensional space of bivariate polynomials of degree atmost d. Given r ≤ ρ ≤ d, we also work with the superspline spaceS r,ρd (△) := { s ∈ Sd r (△) : s ∈ Cρ (v), all v ∈ V } ,where V is the set of vertices of △,andwheres ∈ C ρ (v) means that all polynomialpieces of s on triangles sharing the vertex v have common derivatives upto order ρ at v.2.1 Bernstein–Bézier methodsWe make use of the Bernstein–Bézier representation of splines. Given d and△, letD d,△ := ∪ T∈△ D d,T be the corresponding set of domain points, where foreach T := 〈v 1 ,v 2 ,v 3 〉,{D d,T := ξijk T := iv }1 + jv 2 + kv 3.di+ j+k=dThen every spline s ∈ Sd 0 (△) is uniquely determined by its set of coefficients{c ξ } ξ∈Dd,△ ,ands| T := ∑c ξ Bξ T ,ξ∈D d,Twhere { }BξT are the Bernstein basis polynomials associated with the triangle T.Suppose now that S(△) is a subspace of Sd 0(△). ThenasetM ⊆ D d,△of domain points is called a MDS for S(△) provided it is the smallest setof such points such that the corresponding coefficients {c ξ } ξ∈M can be setindependently, and all other coefficients of s can be consistently determinedfrom smoothness conditions, i.e., in such a way that all smoothness conditionsare satisfied, see p. 136 of [14]. The dimension of S(△) is then equal tothe cardinality of M. Clearly, M = D d,△ is a MDS for Sd 0 (△), and thus thedimension of Sd 0(△) is n V + (d − 1)n E + ( )d−1 nT ,wheren2V , n E , n T are thenumber of vertices, edges, and triangles of △.For each η ∈ D d,△ \ M, letƔ η be the smallest subset of M such that c η canbe computed from the coefficients {c ξ } ξ∈Ɣη by smoothness conditions. Then Mis called local provided there exists an integer l not depending on △ such thatƔ η ⊆ star l (T η ), all η ∈ D d,△ \ M, (1)where T η is a triangle containing η. Recall that given a set U ⊂ , then star(U)is the set of triangles in △ intersecting U, while star l (U) := star(star l−1 (U)).

240 Numer Algor (2008) 48:237–260M is said to be stable provided there exists a constant K depending only on land the smallest angle in the triangulation △ such that2.2 Stable local bases|c η |≤K maxξ∈Ɣ η|c ξ |, all η ∈ D d,△ \ M. (2)Suppose M is a stable local MDS for S(△). For each ξ ∈ M,letψ ξ be the splinein S(△) such that c ξ = 1 while c η = 0 for all other η ∈ M. Thenthesplines{ψ ξ } ξ∈M are clearly linearly independent and form a basis for S(△).Thisbasisis called the M-basis for S(△), see Sect. 5.8 of [14]. It is stable and local in thesense that for all ξ ∈ M,1) ‖ψ ξ ‖ ≤ K,2) supp ψ ξ ⊆ star l (T ξ ),whereT ξ is a triangle containing ξ,where l is the integer constant in (1), and the constant K depends only on land the smallest angle in △. Here and in the sequel, for any set U ⊂ IR 2 , ‖·‖ Udenotes the ∞-norm over points in U. There are many spaces with stable localbases. For example, the spaces Sd 0 (△) have stable local bases with l = 1. Thesame is true for the superspline spaces S r,2r4r+1(△) for all r ≥ 1. Therearealsoseveral families of macro-element spaces defined for all r ≥ 1 with the sameproperty, see [14].2.3 Computational methodsWe now list several useful techniques for working with splines numerically.1. Using the well-known smoothness conditions for two polynomial patchesto join together smoothly across an edge of a triangulation (see Theorem2.28 of [14]), for each η ∈ D d,△ \ M, we can find real numbers { a η }ξ ξ∈Ɣ ηsuch that for every spline s ∈ S(△) with coefficients {c β } β∈M ,c η = ∑ ξ∈Ɣ ηa η ξ c ξ . (3)Note that for spaces with locally supported bases, the number of nonzeroa η ξ in (3) will generally be quite small. If we intend to use the spline spaceS(△) for several fitting problems, these weights can be precomputed andstored.2. To store a particular spline s in the space S(△), we simply store itscoefficient vector c := (c ξ ) ξ∈M . Recall that the length of this vector is justthe dimension of S(△).3. To evaluate s at a point (x, y) in a triangle T, weusetheformulae(3) tocompute the B-coefficients of s corresponding to all domain points in T.Then we can use the well-known de Casteljau algorithm to find s(x, y),seep. 26 of [14].

Numer Algor (2008) 48:237–260 2414. If we need to evaluate s at many points, for example to display the surfacecorresponding to s, it may be most efficient to compute and store all of theB-coefficients {c ξ } ξ∈Dd,△ of s using (3). The total number of B-coefficientsis equal to the dimension of Sd 0(△), whichisnco := n V + (d − 1)n E +)nT ,wheren V , n E , n T are the numbers of vertices, edges, and triangles( d−12of △.3 Computing splines solving quadratic minimization problemsSuppose S(△) ⊆ Sd 0 (△) is a space of splines with a stable local MDS M. Let{ψ ξ } ξ∈M be the associated M-basis. In this paper we focus on variational splineproblems where the coefficients of the desired spline are the solution of asystem of equations of the form∑〈ψ ξ ,ψ η 〉c ξ = r η , η ∈ M, (4)ξ∈Mwhere 〈·, ·〉 is some appropriate inner-product. We suppose the inner-productis such that〈ψ, φ〉 = ∑ 〈ψ| T ,φ| T 〉, all ψ, φ ∈ S(△). (5)T∈△Note that if S(△) has a local basis, then only a small number of terms in (5)will be nonzero.To find a coefficient vector c := {c ξ } ξ∈M satisfying (4), we have to computethe matrixM := [〈ψ ξ ,ψ η 〉] ξ,η∈Mand the vector r = (r η ) η∈M . This matrix will be sparse whenever S(△) has alocal basis. To describe an efficient algorithm for finding M, we make thefollowing observation.Lemma 1 For every ξ ∈ M and every triangle T ∈△,ψ ξ | T = ∑a η ξ BT η , (6)η∈D d,Twhere a η ξ are the weights in formula (3).Proof The spline ψ ξ is the spline with c ξ = 1, andc β = 0 for all other β ∈ M.Fix a domain point η in T. Thenby(3), the coefficient c η of ψ ξ is given byc η = a η ξ .⊓⊔Algorithm 1 For all T ∈△,• Compute the matrix [〈 Bα T, 〉]BT β α,β∈D d,T.• For all ξ,η ∈ M,〈ψ ξ ,ψ η 〉←〈ψ ξ ,ψ η 〉+ ∑ 〈 〉α,β∈D d,Ta α ξ aβ η BTα , BβT .

242 Numer Algor (2008) 48:237–260This algorithm works by looping through the triangles of △. The entries inM are obtained by an accumulation process. If the value r η on the right-handside of (4) involves an inner-product of a given function with ψ η ,itcanbecomputed by adding an additional accumulation step to the algorithm. We discussthis algorithm in more detail for the particular variational spline methodstreated below.4 Minimal energy interpolating splinesIn this section we discuss the minimal energy method for computing aninterpolating spline. Suppose we are given values { f i } n di=1associated with a setof n d ≥ 3 abscissae A := {(x i , y i )} n di=1in the plane. The problem is to constructa smooth function s that interpolates this data in the sense thats(x i , y i ) = f i , i = 1,...,n d . (7)To solve this problem, suppose △ is a triangulation with vertices at the pointsof A, and suppose S(△) is a spline space defined over △ with dimension n ≥ n d .Then the set of all splines in S(△) that interpolate the data is given by( f ) ={s ∈ S(△) : s(x i , y i ) = f i , i = 1, ··· , n d }.Given a spline s ∈ ( f ), we measure its energy using the well-known thinplateenergy functional∫[E(s) = (Dxx s) 2 + 2(D xy s) 2 + (D yy s) 2] dxdy. (8)Definition 1 The minimal energy (ME) interpolating spline is the spline s e in such that E(s e ) = min{E(s), s ∈ ( f )}.It follows from standard Hilbert space approximation results that if ( f ) isnot empty, then there exists a unique ME-spline which is characterized by theproperty〈s e ,ψ〉 E = 0, all ψ ∈ (0), (9)where∫〈φ,ψ〉 E = [D xx φ D xx ψ + 2D xy φ D xy ψ + D yy φ D yy ψ]dxdy.To compute the minimal energy spline, we now suppose that there exists aMDS M for S(△) such that the corresponding M-basis {ψ i }i=1 n is stable andlocal. For all standard spline spaces, we may choose M to include the set A,

Numer Algor (2008) 48:237–260 243and in fact, can assume that the first n d points in M are (x i , y i ), i = 1,...,n d .Now suppose s is written in the formn∑s = c i ψ i . (10)i=1Let c := (c 1 ,...,c nd ) T and c := (c nd +1,...,c n ) T .Theorem 1 The spline s is the ME interpolating spline if and only if c =( f 1 ,..., f nd ) T ,andMc = r, (11)whereM ij := 〈ψ nd +i,ψ nd + j〉 E , i, j = 1,...,n − n d ,r j :=∑n di=1f i 〈ψ i ,ψ nd + j〉 E , j = 1,...,n − n d .Proof By our assumptions on the M-basis, s belongs to ( f ) if and only if c =( f 1 ,..., f nd ) T .Sinceψ nd +1,...,ψ n span (0), it follows that (9) is equivalentto〈∑ nd〉c i ψ i ,ψ j = 0, j = n d + 1,...,n, (12)i=1which can immediately be rewritten as (11).EThe uniqueness of the ME spline insures that the matrix M in (11) isnonsingular. Thus, to compute the coefficient vector of the ME spline, wesimply need to compute M and r, andthensolve(11) forc. We now discussthe assembly of M and r, which we base on the algorithm of Section 3. First,we observe that in this case the inner-product is an integral, and thus clearlysatisfies (5). To carry out Algorithm 1, all we need to do is compute theinner-products 〈 Bα T, 〉BT β E . Derivatives of the BT α can be expressed in termsof Bernstein basis polynomials of lower degree, see Lemma 2.11 of [14]. Butinner-products of Bernstein basis polynomials of any degree can be computedexplicitly, see Theorem 2.34 in the book [14]. For tables of the inner-productsfor degrees d = 1, 2, 3, see pages 46–47 of the book.⊓⊔5 Discrete least squares splinesIn this section we discuss a well-known method for fitting bivariate scattereddata in the case when the number n d of data locations is very large. We assumethat the data are the measurements { f i := f (x i , y i )} n di=1of an unknown functionf defined on . To create a spline fit, we work with a spline space S(△) definedon a triangulation △ with a set V of vertices which are not necessarily at the

244 Numer Algor (2008) 48:237–260points of A. Typically we choose the number of vertices to be (much) smallerthan n d .Definition 2 The DLSQ spline fit of f is the spline s l ∈ S(△) that minimizesn d‖s − f ‖ 2 A := ∑[s(x j , y j ) − f j ] 2 .It is well known that if S(△) satisfies the propertyj=1s(x i , y i ) = 0, i = 1,...,n d , implies s ≡ 0, (13)then there is a unique DLSQ spline s l fitting the data. This requires that thenumber n d of data points be at least equal to the dimension n of the splinespace. In practice n d will typically be much larger than n. However, in generalthis is not sufficient – we need the data points to be reasonably well distributedamong the triangles of △ in order to make (13) hold.It follows from standard Hilbert space approximation results that thereexists a unique DLSQ-spline s l which is characterized by the propertywhere〈s l − f,ψ〉 A = 0, all ψ ∈ S(△), (14)∑n d〈φ,ψ〉 A := φ(x i , y i )ψ(x i , y i ).i=1To compute the least-squares spline s l , we now write it in the form (10),where {ψ i } n i=1 isastablelocalM-basis for S(△). Letc := (c 1,...,c n ) be thevector of coefficients.Theorem 2 Thesplines l is the DLSQ spline fit of f if and only ifMc = r, (15)whereM ij := 〈ψ i ,ψ j 〉 A ,r j := 〈 f,ψ j 〉 A,i, j = 1,...,n,j = 1,...,n.Proof Since ψ 1 ,...,ψ n span S(△), it follows that (14) is equivalent to〈 n∑〉c i ψ i − f,ψ j = 0, j = 1,...,n, (16)i=1which can immediately be rewritten as (15).AThe assumption (13) insures that the matrix M in (15) is nonsingular. Thus,to compute the coefficient vector of the least-squares spline, we simply needto compute M and r, andthensolve(15) forc. Note that here the matrix⊓⊔

Numer Algor (2008) 48:237–260 245M is of size n × n in contrast to the minimal-energy case where it is of size(n − n d ) × (n − n d ). For spline spaces S(△) with stable local MDS M, thecomputation of M can be done efficiently using Algorithm 1. To apply it, weneed to compute the inner-products 〈 Bα T, 〉BT β . For this, we simply need theAvalues of the Bernstein basis polynomials at all of the points of A that lie inT. These values can also be used to compute the vector r in (15). We give anefficient algorithm in Section 9 for computing the values of all Bernstein-Basispolynomials at a fixed point.6 PLSQ splinesIn this section we discuss a method for fitting bivariate scattered data which ispreferable to least-squares when the data { f i } n di=1are noisy. It is a generalizationof the least-squares fitting method in Section 5. Suppose A := {x i , y i } n di=1 ,andS(△) are as in the previous section. For each s ∈ S(△),letE(s) be its associatedthin-plate energy defined in (8).Definition 3 Given λ ≥ 0,thePLSQ spline fit of f is the spline s λ in S(△) thatminimizesE λ (s) := ‖s − f ‖ A + λE(s).It is well known that if the spline space S(△) satisfies (13), then there existsa unique PLSQ-spline s λ . Moreover, s λ is characterized by〈s λ − f, s〉 A + λ〈s λ , s〉 E = 0, all s ∈ S(△).To compute the PLSQ spline s λ , we write it in the form (10), where {ψ i } n i=1 is astable local M-basis for S(△).Letc := (c 1 ,...,c n ) be the vector of coefficients.We immediately have the following result.Theorem 3 Thesplines λ is the PLSQ fit of f if and only ifwhereMc = r, (17)M ij := 〈ψ i ,ψ j 〉 A + λ(ψ i ,ψ j 〉 E ,i, j = 1,...,n,and r is the vector in (15).The assumption (13) insures that the matrix M in (17) is nonsingular. We canefficiently compute both M and r by the methods of the previous two sections.Note that when λ = 0, we get the least-squares spline fit.

246 Numer Algor (2008) 48:237–2607 Solution of boundary-value problemsIn this section we describe how Algorithm 1 can be used to compute Galerkinapproximations to solutions of elliptic boundary-value problems.7.1 Second order problems with homogeneous boundary conditionsWe first describe the method for the simple model problem−∇·(κ∇u) = f, on ,u = 0, on ∂, (18)where f and κ are given functions on . Suppose that △ is a triangulation of,andletS(△) be a bivariate spline space defined on △ with a stable MDS M.LetU 0 := {s ∈ S(△) : s(x, y) = 0, all (x, y) ∈ ∂}.We look for an approximation s g of u in U 0 . Suppose ψ 1 ,...,ψ n is a stable localbasis for U 0 , and suppose s g is written in the form (10). Then by the Galerkinmethod, see [8], the coefficients of s g should be chosen to be the solution of thesystem of equationswhere∫M ij =〈ψ i ,ψ j 〉 G:=for i, j = 1,...,n, and∫r i =〈f,ψ i 〉 2 :=Mc = r, (19)κ(x, y)∇ψ i (x, y) ·∇ψ j (x, y)dxdy,f (x, y)ψ i (x, y)dxdy,i = 1,...,n.Since ∇ψ i ·∇ψ j = D x ψ i D x ψ j + D y ψ i D y ψ j , the computation of M involvescomputing inner-products of first derivatives of the Bernstein basis polynomials.If κ ≡ 1, these inner-products can be computed explicitly. For general κand for the computation of the vector r, we will need to use a quadrature rule.We use Gaussian quadrature, see Remark 7.We emphasize that for this problem, to use Algorithm 1 we need a stablelocal minimal determining set for the subspace of splines U(0). Inparticular,all spline coefficients associated with domain points on the boundary of mustbe set to 0. Thus, for example, if the approximating spline space is S 1,25(△),thenfor each boundary vertex v whose exterior angle is less than π, only one of thedomain points in the disk D 2 (v) of radius 2 around v can be included in M.Recall, that for the full space, we chose six points in each D 2 (v), see Fig. 1.When the exterior angle at v is equal to π, we must choose three points inD 2 (v) to include in M.

Numer Algor (2008) 48:237–260 247.................Fig. 1 MDS for S 1,25 (△) and S 2,49 (△)7.2 Second order problems with inhomogeneous boundary conditionsOnly minor modifications are required to solve the inhomogeneous boundaryvalueproblem−∇·(κ∇u) = f, on ,u = g, on ∂, (20)where f and κ are given function on , andg is a given function on ∂.Asbefore, we choose a spline space S(△) defined on a triangulation △ of with astable local M-basis {ψ i }i=1 n . As is well known, see e.g. [8], in this case we lookfor an approximation to u of the form s = s b + s h ,wheres b is a spline suchthat s b ≈ g on ∂,ands h is the Galerkin approximation of the solution of theboundary-value problem with homogeneous boundary conditions and righthandside f − s b .Writings h in the form (10), we can compute its coefficientsfrom the equationswhere M is as in (19) andMc = ˜r,˜r j =〈f,ψ j 〉 2−〈s b ,ψ j 〉 G,j = 1,...,n.It is straightforward to construct s b . We first choose its Bernstein–Béziercoefficients associated with domain points on the boundary edges of △ so thats b does a good job of approximating g on ∂. For example, we may interpolateg at an appropriate number of points on each edge of △. Thenwesetasmany remaining coefficients to zero as possible while observing all smoothnessconditions.7.3 Fourth order problemsAs an example of a fourth order problem, consider the biharmonic equation△ 2 u = f, on , (21)

248 Numer Algor (2008) 48:237–260subject to the boundary conditionsu = g, on ∂,∂u= h,∂non ∂, (22)where ∂n stands for the normal derivative to the boundary. As before, weconstruct an approximation s g := s b + s h to u by first constructing a splines b that approximately satisfies the boundary conditions. We then computes h as the Galerkin spline for the corresponding problem with homogeneousboundary conditions. Starting with a spline space S(△), we define the subspaceU 0 := {s ∈ S(△) : s satisfies the boundary conditions (22) withg ≡ h ≡ 0.}Assuming ψ 1 ,...,ψ n is a stable local basis for U 0 , and writing s h in the form(10), the Galerkin method provides the following linear system of equationsfor the coefficients of s h :where∫M ij :=Mc = r,△ψ i △ψ j dxdy,i, j = 1,...,n,and∫r j =〈f,ψ j 〉 2− △s g △ψ j dxdy, j = 1,...,n.Note that although we are allowed to use C 0 splines in the Galerkin methodfor solving second order boundary-value problems, for fourth order problemsthe well-known conformality conditions require that we use a space of splinesthat is C 1 at least.8 ExamplesIn this section we give several numerical examples. Given a triangulation △,we write V and E for the sets of vertices and edges of △. We also write n V , n E ,and n T for the numbers of vertices, edges, and triangles of △. Weworkwiththe following spline spaces:1) Sd 0(△). The dimension of this space is n V + (d − 1)n E + ( )d−1 nT .Astable2local MDS is given by the full set D d,△ of domain points. This spaceapproximates smooth functions up to order O(|△| d+1 ), see Remark 5.2) S 1,25(△). This space has dimension 6n V + n E , and as shown in Theorem 6.1of [14], a stable local MDS M is given by the set of domain pointsM := ⋃ M v ∪ ⋃ M e ,v∈Ve∈E

Numer Algor (2008) 48:237–260 249where for each vertex v of △, M v is the set of six domain points inD 2 (v) ∪ T v for some triangle attached to v. For each edge e of △, M econsists of the domain point ξ T e122 for some triangle T e containing the edgee and with first vertex opposite e. Figure 1 (left) shows the points in Mfor a typical triangulation, where points in the sets M v are marked withblack dots, while those in the sets M e are marked with triangles. SinceS 1,25(△) has a stable local basis, it approximates smooth functions up toorder O(|△| 6 ), see Remark 5.3) S 2,49(△). This space has dimension 15n V + 3n E + n T , and as shown inTheorem 7.1 of [14], a stable local MDS M is given by the set of domainpointsM := ⋃ v∈VM v∪ ⋃ e∈EM e ∪ ⋃ T∈△M T ,where for each vertex v, M v is the set of fifteen domain points in D 4 (v) ∪T v for some triangle attached to v. Foreachedgee of △, M e consists ofthe three domain points ξ T e144 ,ξT e243 ,ξT e234 for some triangle T e containing theedge e and with first vertex opposite e. For each triangle T, M T consists ofthe domain point ξ333 T . Figure 1 (right) shows M for a typical triangulation.Here squares are used to mark points in the set M T .SinceS 2,49(△) has astable local basis, it approximates smooth functions up to order O(|△| 10 ),see Remark 5.8.1 Minimal energy fitExample 1 We construct a C 1 surface based on measurements of the height ofa nose cone at 803 points in the domain shown in Fig. 2. The figure shows theDelaunay triangulation △ nose associated with the data points.Fig. 2 The minimal energy fit of Example 1

250 Numer Algor (2008) 48:237–260Discussion: For this application we choose the minimal energy interpolatingspline in the space S 1,25(△ nose ). This triangulation has n E = 2357 edges andn T = 1555 triangles. Thus, the dimension of the space is 7175, and finding theminimal energy fit requires solving a system of 6372 equations. The associatedmatrix is sparse with only 243,402 of the 40,602,384 entries being nonzero. Thecomputation took 34 seconds on a desktop, but see Remark 8. The resultingspline fit is shown in Fig. 2 (right).To further illustrate the behavior of minimal energy spline fits, we give asecond example involving a known function where we can compute errors. Weuse the well-known Franke functionF(x, y) = 0.75 exp(−0.25(9x − 2) 2 − 0.25(9y − 2) 2 )+ 0.75 exp(−(9x + 1) 2 /49 − (9y + 1) 2 /10)+ 0.5exp(−0.25(9x − 7) 2 − 0.25(9y − 3) 2 )− 0.2exp(−(9x − 4) 2 − (9y − 7) 2 ) (23)defined on the unit square. The surface corresponding to F is shown in Fig. 3.Example 2 We construct minimal energy spline fits of F from the spacesS 1,25(△ k ) based on type-I triangulations △ k with k = 9, 25, 81, and 289 vertices.Discussion: Type-I triangulations are obtained by first forming a rectangulargrid, and then drawing in the northeast diagonals. We show the results of ourcomputations in the table in Fig. 4, where the columns labelled nt and nsys givethe number of triangles in △ k and the size of the linear system being solved,respectively. The columns labelled e ∞ and e 2 give the maximum error and theRMS errors measured over a grid of 640,000 points. The last column containsthe computational times in seconds. To illustrate the quality of the fits, wegive contour plots for each of the four cases. Clearly, the fit on △ 9 is not verygood, and completely misses the peaks and valley in the function. On the otherhand, the contour plot for △ 289 is almost identical to the contour plot of fitself. In comparing Figs. 3 and 4, note that the 3D view of F in Fig. 3 is frombehind the surface so that the peaks do not obscure the valley. The results hereare very comparable to those obtained for the space of C 2 quintic splines onFig. 3 The Franke function

Numer Algor (2008) 48:237–260 251Fig. 4 Minimal energy fits of F from S 1,25 (△ k ), see Example 2Powell-Sabin splits, see Table 3 of [11], which were computed by the Lagrangemultiplier method of [4].8.2 Least-squares fittingIn this section we give examples of least-squares fitting with splines. Let F bethe Franke function (23) defined on the unit square.

252 Numer Algor (2008) 48:237–260Fig. 5 Least-squares fits of F from S 1,25 (△ k ) and S 2,49 (△ k ), see Example 3Example 3 We construct least-squares fits of F using the spaces S 1,25(△ n )and S 2,49(△ k ) defined on type-I triangulations based on measurements onrectangular grids of nd := 289, 1089 and 4225 points.Discussion: The table in Fig. 5 shows the results of our computations, wherethe columns labelled e ∞ and e 2 give the maximum error and RMS errors on agrid of 640,000 points. The column labelled nsys gives the size of the linearsystem being solved. The last column contains the computational times inseconds. We show contour plots of the fits based on 289 data points, wherethe plot on the left is the fit from S 1,25(△ 9 ) while the one on the right is the fitfrom S 2,49(△ 9 ).

Numer Algor (2008) 48:237–260 253Fig. 6 PLSQ fits of F with λ = .01,.005,.001, 08.3 Penalized least-squaresLet F be the Franke function (23) defined on the unit square. We approximateF based on measurements on a grid of 1089 points. However, here we addrandom errors ε i to the measurements f i , where the ε i are uniformly distributedin [−.1,.1]. This is a rather significant amount of noise since the values of F liein the interval [−.2, 1.1].Example 4 We compute the PLSQ spline fits s λ ∈ S 1,25(△ 25 ) for λ=.01,.005,.001 and 0.Discussion: The maximum errors for these choices of λ were .127, .111, .08,and .10. The corresponding surfaces are shown in Fig. 6. Clearly, the value ofλ makes a big difference. If it is too small, the fit follows the noise, and is notvery smooth. If it is too large, some of the shape is lost.

254 Numer Algor (2008) 48:237–260Fig. 7 Spline approximations of the BVP of Example 58.4 Solution of boundary-value problemsExample 5 Compute a solution to the boundary-value problem (20)withf (x, y) := 4cos(x 2 + y 2 ) − 4(x 2 + y 2 ) sin(x 2 + y 2 )+ 10 cos(25(x 2 + y 2 )) − 250 sin(25(x 2 + y 2 )),g(x, y) := sin(x 2 + y 2 ) + .1sin(25(x 2 + y 2 )),and κ ≡ 1 on the unit square .Discussion:In this case, the solution of the boundary-value problem (20)isu(x, y) := sin(x 2 + y 2 ) + .1sin(25(x 2 + y 2 )).

Numer Algor (2008) 48:237–260 255Table 1 Spline approximationsof the BVP ofExample 6k nt dim e ∞ e 2 timeS 1,25 9 8 18 3.4(−5) 1.1(−5) .0125 32 106 3.7(−8) 8.9(−9) .0781 128 498 6.1(−10) 1.3(−10) .46We computed spline approximations of u by the Galerkin method using splinespaces defined on type-I triangulations △ k with k = 81, 289, 1089 and 4225vertices. The results are shown in the tables in Fig. 7, wherek and nt are thenumber of vertices and triangles, and dim is the dimension of the associatedspace. We give results not only for S 1,25(△ k ), but also for S 0 1 (△ k) for comparisonpurposes. As above, e ∞ and e 2 are the maximum and RMS errors on a grid of25,600 points, and the last column shows the computational time in seconds.The figure on the left shows the traditional C 0 linear fit with k = 1089, while theone on the right shows the fit with S 1,25(△ k ) with k = 289. In addition to havingmuch smaller errors, it is much smoother. The results here can be comparedwith those in Table 13 of [4] which were computed with C 1 quintic splines, butwith f and g multiplied by 10.Finally, we give an example involving a fourth-order differential equation.Example 6 Compute a solution to the fourth-order boundary-value problem(21–22)withf (x, y) := 4exp(x + y),g(x, y) = h(x, y) := exp(x + y),on the unit square .Discussion: The solution of this boundary-value problem is u(x, y) :=exp(x + y). We computed spline approximations of u by the Galerkin methodusing the space S 1,25(△ k ) defined on type-I triangulations with n = 9, 25, and 81vertices. The errors are shown in Table 1 along with the computational times.For a comparison with an approximation computed using S5 1(△ 25) using theLagrange multiplier method, see Example 26 in [4].9 Computing Bernstein basis polynomials and their derivativesIn this section we describe algorithms for efficiently evaluating Bernstein basispolynomials of degree d and their derivatives. Fix a triangle T := 〈v 1 ,v 2 ,v 3 〉with vertices v i = (x i , y i ) for i = 1, 2, 3. The associated Bernstein basis polynomialsare defined byBijk d d !(x, y) :=i ! j ! k ! b i 1 b j 2 b k 3 , i + j + k = d, (24)

256 Numer Algor (2008) 48:237–260Fig. 8 One step ofAlgorithm 2where b 1 , b 2 , b 3 are the barycentric coordinates of (x, y) relative to the triangleT. Specifically,∣ ∣b 1 (x, y) := 11 1 1 ∣∣∣∣∣ 1 1 1 ∣∣∣∣∣ detxx 2 x 3 , det :=x 1 x 2 x 3 , (25)∣ yy 2 y 3∣ y 1 y 2 y 3with similar definitions for b 2 and b 3 . This shows that b 1 , b 2 , b 3 are linearfunctions of x and y.The following algorithm simultaneously computes the values of all of theBernstein basis polynomials {B d ijk } i+ j+k=d at a fixed point (x, y). Suppose thebarycentric coordinates of (x, y) relative to T are b 1 , b 2 , b 3 .Algorithm 2 Computation of all Bernstein basis polynomials at a point (x, y).• Set B(0, 0) = 1• For k = 1 to dFor i = k to 0For j = i to 0B(i, j ) = b 1 B(i, j ) + b 2 B(i − 1, j ) + b 3 B(i − 1, j − 1)A simple inductive proof shows that the following lemma holds.Lemma 2 Algorithm 2 produces the values⎛B d ⎞d,0,0(x, y)⎜B d d−1,1,0⎝(x, y) Bd d−1,0,1(x, y)⎟··· ···⎠(x, y) Bd 0,1,d−1 (x, y) ··· Bd 0,0,d (x, y).B d 0,d,0We illustrate one step of this algorithm (k = 2) in Fig. 8. It shows how thesix Bernstein basis polynomials of degree 2 are computed from the three ofdegree 1. The idea is to use the inverted triangle with b 1 , b 2 , b 3 at its verticesas a mask which is applied at six different positions to the triangular array of1st degree Bernstein basis polynomials on the left to get the triangular array ofBernstein basis polynomials of degree 2 on the right. Applying the mask againwould lead to an array containing the 10 values of the cubic Bernstein basispolynomials, etc.

Numer Algor (2008) 48:237–260 257For the applications in Section 7, we also need an algorithm for evaluatingderivatives of the Bernstein basis polynomials at a given point (x, y). We nowshow that Algorithm 2 can be easily adapted to produce the values of arbitrarydirectional derivatives of the Bernstein basis polynomials. Recall (see e.g. [14])that if u = u 1 − u 0 is a vector in IR 2 , then its directional coordinates (a 1 , a 2 , a 3 )relative to T are just the differences of the barycentric coordinates of u 1 andu 0 relative to T. This implies a 1 + a 2 + a 3 = 0. The following lemma showshow Algorithm 2 can be used to compute the values of the set of functions{D m u Bd ijk (x, y)} i+ j+k=d at a fixed point (x, y) whose barycentric coordinatesrelative to T are b 1 , b 2 , b 3 .Lemma 3 Fix 0 ≤ m ≤ d. Suppose that in carrying out Algorithm 2, in the firstm times through the k-loop we use a 1 , a 2 , a 3 in place of b 1 , b 2 , b 3 .Thenthealgorithm produces (d−m) ! times the matrixd !⎛D m ⎞u Bd d,0,0(x, y)⎜D m u⎝Bd d−1,1,0 (x, y) Dm u Bd d−1,0,1(x, y)⎟··· ···⎠(x, y) Dm u Bd 0,1,d−1 (x, y) ··· Dm u Bd 0,0,d (x, y).D m u Bd 0,d,0Proof It is well known that[]D u Bijk d = d a 1 B d−1i−1, j,k + a 2 B d−1i, j−1,k + a 3 B d−1i, j,k−1,see e.g. Lemma 2.11 of [14], The result then follows from a minor variant ofthe inductive proof of Lemma 2.⊓⊔This result can be generalized to compute arbitrary mixed directional derivatives.In particular, suppose we are given m directional vectors u 1 ,...,u mwhose direction coordinates relative to T are a (ν) := (a (ν)1 , a(ν) 2 , a(ν) 3 ),forν =1,...,m. Then we can compute (d − m) ! /d ! times the matrix of values[D u1 ...D um Bijk d (x, y)] by running Algorithm 2, but using a(ν) in place of(b 1 , b 2 , b 3 ) in the ν-th pass through the k-loop for ν = 1,...,m.Since we are interested in computing the derivatives in the directions ofthe Cartesian axes, we now give formulae for the direction coordinates of thespecial directional derivatives D x and D y .Lemma 4 Suppose T := 〈v 1 ,v 2 ,v 3 〉 where v i := (x i , y i ) for i = 1, 2, 3. Then thedirection coordinates of D x area x 1 = (y 2 − y 3 )/det, a x 2 = (y 3 − y 1 )/det, a x 3 = (y 1 − y 2 )/det, (26)while those of D y area y 1 = (x 3 − x 2 )/det, a y 2 = (x 1 − x 3 )/det, a y 3 = (x 2 − x 1 )/det, (27)where det is the determinant in (25).

258 Numer Algor (2008) 48:237–260These quantities are just the derivatives of the barycentric coordinatefunctions b 1 , b 2 , b 3 with respect to x and y.10 RemarksRemark 1 My first work on programs to compute minimal energy splines wasdone with Ewald Quak in the mid 1980’s during his stay at Texas A&M.Our programs used C 1 cubic and quartic splines, although even to this daythe dimension of S3 1 (△) has not been determined for arbitrary triangulations△. The Bernstein–Bézier representation was used, and smoothness conditionswere simply incorporated with Lagrange multipliers. The codes were not madepublic, but the details of how to compute energies were published in [15].This approach was later used in [7] in connection with the construction ofminimal energy surfaces with C 1 cubic parametric splines, where in somecases certain C 2 smoothness conditions were also incorporated via Lagrangemultipliers. The Lagrange multiplier approach was further developed in [4],where the Bernstein–Bézier representation is based on the space of piecewisepolynomials rather than on Sd 0 (△) as in our earlier papers. A useful iterativemethod for solving the resulting systems of equations was also developed in[4], see also the survey [12].Remark 2 There is an extensive theory of splines defined on spherical triangulationswhich is remarkably similar to the theory of bivariate splines,see [1–3] and Chapters 13–14 of [14]. Such splines are piecewise sphericalharmonics. The techniques described here also work for solving variationalproblems involving spherical spline spaces with stable local MDS. A numberof such spaces are described in [14]. For some computational experiments withminimal energy spherical splines based on Lagrange multiplier methods, see[2] and[5].Remark 3 There has been considerable recent work on trivariate splinesdefined on tetrahedral partitions, see Chapters 15–18 of [14]. The methodsdescribed here can also be carried over to solve variational problems associatedwith trivariate splines with stable local MDS. A number of such spacesare described in [14].Remark 4 The global methods described here are not the only way to fitscattered data using splines. There are a host of local methods that work withvarious macro-element spaces. For numerical experiments with some of thesemethods in the spherical case, see [2].Remark 5 It is well known, see [13] or Chap. 10 of [14], that if a spline spaceS(△) has a stable local basis, then it approximates sufficiently smooth functionsto order O(|△| d+1 ),where|△| is the mesh size of △, i.e., the diameter ofthe largest triangle in △. However, the minimal energy interpolation methoddoes not attain this optimal approximation order for d ≥ 2 since it only

Numer Algor (2008) 48:237–260 259reproduces linear functions. Indeed, the results of [11] show that the orderof approximation using minimal energy splines is only O(|△| 2 ). This can beseen in the table in Fig. 4, which is based on a nested sequence of type-Itriangulations whose mesh sizes decrease by a factor of .5 at each step. Forerror bounds for DLSQ and PLSQ splines, see [9] and[10], respectively.Remark 6 Our method can also be applied to compute minimal energy fitsusing the spaces Sd 0 (△). However, it doesn’t really make sense to minimizeenergy for splines in C 0 spline spaces, since minimizing energy leads to splineswhich are as close to piecewise linear as possible. Even though Sd 0 (△) is a muchlarger space than S 1,25(△), the minimal energy fit from Sd 0 (△) is much worse thatthe one from S 1,25(△).Remark 7 A Gaussian quadrature rule for approximating integrals of functionsover a triangle T is defined by a set of positive numbers {w k , r k , s k , t k } m k=1where r k + s k + t k = 1 for all k. Then for any function g defined on T, itsintegral is approximated by∫m∑g(x, y)dxdy ≈ w k g ( xk T , ) yT k , (28)Twherexk T = r kx 1 + s k x 2 + t k x 3 , yk T = r ky 1 + s k y 2 + t k y 3 .For each m, thereisad m such that the corresponding Gaussian quadraturerule integrates all polynomials up to degree d m exactly. Formulae for variousvalues of m can be found in the literature. In our experiments we use rulesfrom [6] withm = 25 and m = 79 which integrate bivariate polynomials up todegree 10 and 20 exactly.Remark 8 The numerical experiments presented here were performed on aMacintosh G5 computer using Fortran. The programs were not optimizedfor performance, and the times reported are only meant to give a generalimpression of the speed of computation and to provide a basis for comparingdifferent methods with various parameters.Remark 9 The programs used for the experiments presented here work withspline spaces up to smoothness C 2 . However, it is straightforward to writesimilar programs for spline spaces with higher smoothness. For example, wecould work with any of the macro-element spaces discussed in [14] whicharedefined for arbitrary smoothness r.Remark 10 While it may appear that the practical use of global spline fittingmethods is limited by the need to solve linear systems of equations whichcan become very large, in Lai and Schumaker (submitted for publication) werecently described a method for decomposing large variational spline problemsinto smaller more manageable pieces.k=1

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