Multivariate Splines for

Data Fitting and Approximation

Ming-Jun LaiDepartment of Mathematics

the University of GeorgiaAthens, GA 30602

May 8, 2009

The following methods for fitting a given set of data are available in theliterature.

• Minimal Energy Method and its Extensions;

• Continuous and Discrete Least Squares Method;

• Penalized Least Squares Spline Method;

• L1 Spline Method and its variants;

• Least Absolute Deviation Method;

• L1 Smoothing Spline Method;

• Possible Research Projects.

We shall give a review of these methods. We need to explain severalfundamental questions concerning each method: if a method has a solutionor not (i.e., the existence and uniqueness), how to compute that solution(i.e., numerical algorithms), whether the solution surface resembles the givendata (i.e., approximation properties), and what to do when the amount ofdata is very large.

1

1 Minimal Energy Method and Its Exten-

sions

Let E(f) be the thin-plate energy functional

E(f) =

∫

Ω

(

(

∂2

∂x2f

)2

+ 2

(

∂2

∂x∂yf

)2

+

(

∂2

∂y2f

)2)

dxdy.

Let Λ(f) = s ∈ Srd(∆), s(xi, yi) = fi, i = 1, · · · , N. Find Sf ∈ Λ(f)

such thatE(Sf ) = minE(s), s ∈ Λ(f).

The following result was proved in [von Golitschek, Lai, and Schumaker’02]and in [Awanou, Lai, Wenston’06] by different methods.

Theorem 1.1 If Λ(f) is not empty, there exists a unique interpolatory splinein Sr

d(∆).

Once we have an interpolatory surface, we would like to know how thesurface resembles the given data. Let W 2

∞(Ω) be the Sobolev space of allfunctions whose second derivatives are essentially bounded over Ω. |f |2,∞,Ω

is the maximal norm of all second order derivatives of f over Ω. The followingresults can be found in [von Golitschek, Lai, and Schumaker’02].

Theorem 1.2 Suppose zi = f(xi, yi), i = 1, · · · , N , for f ∈ W 2∞(Ω). Let

d ≥ 3r + 2, and let ∆ be a triangulation of the data sites (xi, yi), i =1, · · · , N. Then

‖sf − f‖L∞(Ω) ≤ C|∆|2|f |2,∞,Ω.

Our next concern is how to compute interpolatory minimal energy splinesusing a spline space of arbitrary degree d and arbitrary smoothness r withd ≥ 3r + 2. The following computational scheme was described in [Awanou,Lai and Wenston’06].

(1) Express each s ∈ S−1d (∆) in B-form (cf. [de Boor’87]), i.e.,

s(x, y)|t =∑

i+j+k=d

ctijkB

d,tijk(x, y),

where Bd,tijk are Bernstein-Bezier basis functions defined only on t. Let

c = (ctijk, i + j + k = d, t ∈ ∆) be a coefficient vector for s.

2

(2) When s ∈ Srd(∆), there are smoothness conditions over interior edges

of ∆ (cf. [Farin’86]). The smoothness conditions are linear. Put allsmoothness conditions together to write

Hc = 0,

for a matrix H, i.e., s ∈ Srd(∆) iff Hc = 0.

(3) Compute the energy functional E(s) = cTEc for an energy matrix Ewhich is a diagonally block matrix.

(4) The interpolatory conditions can be written Ic = f for a matrix I anda vector f containing all data values zi.

(5) The minimal energy method for interpolatory splines is equivalent tofinding c such that

mincTEc, subject to Hc = 0, Ic = f.

(6) By the Lagrange multiplier method, we solve

E HT IT

H 0 0I 0 0

c

αβ

=

00f

.

(7) To solve this system, we use the following iterative method introducedin [Awanou, Lai, Wenston’06]:

(

E +1

ǫ

[

HT IT]

[

HI

])

c(1) =1

ǫIT f ,

(

E +1

ǫ

[

HT IT]

[

HI

])

c(k+1) = Ec(k) +1

ǫIT f ,

for k = 1, 2, · · · and ǫ > 0, e.g., ǫ = 10−6.

We need to show that the iterative method above is convergent. To thisend, recall that a matrix A is positive definite with respect to B if cT Ac ≥ 0and if Ac = 0 and Bc = 0 for some c, then c = 0. In [Awanou and Lai’05],we proved the following (cf. [Awanou, Lai, and Wenston’06] for a similarresult).

3

Theorem 1.3 Suppose that E is positive definite with respect to [H, I]T .Then the above iteration converges, and

‖c(k+1) − c‖ ≤ Cǫk, ∀k ≥ 1.

When the number of data sites is large, e.g., N > 1000, a computer maynot be powerful enough to solve the linear system. A domain decompositiontechnique for computing an approximation of the minimal energy spline in-terpolation was proposed in [Lai and Schumaker’09]. The ideas of domaindecomposition for scattered data fitting can be explained as follows.

Let D1(t) be the union of all triangles in ∆ which share a vertex or edgewith t, and Dk+1(t) the union of all triangles sharing a vertex or edge withtriangles in Dk(t). For k ≥ 1, we compute a minimal energy interpolatoryspline Sf,t,k ∈ Λ(f) such that

EDk(t)(Sf,t,k) = minEDk(t)(s), s ∈ Λ(f |t),

EDk(t)(s) =

∫

Dk(t)

(

(

∂2

∂x2f

)2

+ 2

(

∂2

∂x∂yf

)2

+

(

∂2

∂y2f

)2)

.

The following result was established in [Lai and Schumaker’03].

Theorem 1.4 Suppose that f ∈ C2(Ω). For d ≥ 3r+2, there is a 0 < ρ < 1such that

‖Sf − Sf,t,k‖L∞(t) ≤ Cρk|f |2,∞,Ω

for k ≥ 1, where C is a constant dependent on d, β.

This result shows that a (global) minimal energy spline interpolation Sf

can be approximated by local minimal energy spline interpolations Sf,t,k forall t ∈ . That is, for each triangle t, one can use a local minimal energyspline interpolation Sf,t,k to replace the global one Sf |t within some tolerance.In the following we give a numerical example.

Example 1.1 We are given a set of data shaped like a cone in Fig. 1. Thereare about 900 points in 3D Euclidean space. A Delaunay triangulation of thegiven data locations is shown in Fig. 2. A piecewise linear interpolation isgiven in Fig. 3. We use C1 quintic spline functions and find the minimalenergy interpolatory spline surface as shown in Fig. 4. It is clear that thesurface is smooth although there are a few bumpy spots which indicate imper-fect data values.

4

−10−5

05 −15

−10−5

010

15

20

25

30

35

Figure 1: A set of scattered data (courtesy Tom Grandine).

Figure 2: A triangulation of the given data locations.

5

Figure 3: A piecewise linear interpolation.

Figure 4: A C1 quintic spline interpolation.

6

−600−400

−2000

200400

600

400450

500550

600650

700750

400

450

500

550

600

650

700

Figure 5: A set of data points (courtesy Gerald Farin).

2 Extensions of Minimal Energy Method

We now outline some extensions to incomplete data interpolation, Hermitedata interpolation, hole filling, and spherical scattered data interpolation.

Example 2.1 When a given data set is incomplete, i.e., values at some gridlocations are not given as shown in Fig. 5, we can still use the minimal en-ergy method with the assumption that the spline coefficients at those verticeswhich have no given data values are free. The computation is exactly asabove. Indeed, the interpolation conditions Ic = f have fewer entries thanthe standard one.

We use C1 quintic spline to find an interpolatory surface using the mini-mal energy method. It is clear from Fig. 6 that the surface is smooth.

Example 2.2 When a given data set contains Hermite data values

(xi, yi, Dαf(xi, yi), |α| ≤ r, i = 1, · · · , N,

we can use the minimal energy method to find a spline function Hf in Srd(∆)

to interpolate all the given data values including derivatives, i.e..

DαHf(xi, yi) = Dαf(xi, yi), |α| ≤ r, i = 1, · · · , N.

The existence, uniqueness, and approximation properties of Hf have beendiscussed in [Zhou, Han and Lai’07].

7

Figure 6: A C1 quintic spline surface with the given data locations.

Example 2.3 When the given data values as well as normal derivative val-ues are all on the boundary of a surface hole, we can use the minimal energymethod to find a C1 spline surface patch to mend the hole (cf. [Chui andLai’00]). See the example in Fig. 7.

Example 2.4 When the given data values are over the spherical domain,we can use the spherical splines [Alfeld, Neamtu, Schumaker’96] and theminimal energy method to find a Cr interpolatory spline surface. The frame-work of minimal energy interpolatory splines in the bivariate setting has beengeneralized to the spherical setting (cf. [Baramidze, Lai and Shum’06]). Thecomputational algorithm is similar to the one for bivariate polynomial splines.In Fig. 8, we present a set of normalized scattered data values over the sur-face of the earth. They are simulated measurements from a German satelliteCHAMP launched on 2000. We use C1 quintic spherical splines to find aninterpolant. More detail will be given in Dr. Baramidze’s lecture.

8

Figure 7: Hole filling using C1 quintic splines.

9

Figure 8: Normalized simulated geopotential measurements (top) and C1

quintic spherical spline interpolation (bottom)

10

3 Continuous and Discrete Least Squares Fit-

ting

The discrete least squares method is one of the classical methods for datafitting. Instead of polynomial fitting, we use multivariate splines. Let ℓ(f) =∑N

i=1 |f(xi, yi)|2. We look for Sf ∈ Srd(∆) such that

ℓ(Sf − f) = minℓ(s − f), s ∈ Srd(∆).

Sf is called the discrete least squares fit of the given data (xi, yi, fi), i =1, · · · , N with fi = f(xi, yi).

To show the existence and uniqueness of the solution Sf , we need toassume

A1‖s‖L∞(T ) ≤√

∑

(xi,yi)∈T

|s(xi, yi)|2

for all s ∈ Srd(∆) and all triangle T ∈ ∆ (cf. [von Golitschek and Schu-

maker’02a]).

Theorem 3.1 Suppose that the above constant A1 is strictly positive. Thenthere exists a unique spline fit Sf ∈ Sr

d(∆).

Let√

∑

(xi,yi)∈T

|s(xi, yi)|2 ≤ A2‖s‖L∞(T )

for all T ∈ ∆ and s ∈ Srd(∆). It is easy to see that A2 must be less than

or equal to the maximal number of points per triangle. The following resultwas established in [von Golitschek and Schumaker’02a].

Theorem 3.2 Assume that f ∈ W m+1∞ (Ω). Then

‖Sf − f‖L∞(Ω) ≤ CA2

A1|∆|m+1|f |m+1,∞,Ω

for a constant C dependent on β, d.

Furthermore, we can show the following

11

Corollary of Theorem 6. Under the same assumptions above, for |α| ≤m + 1,

‖Dα(Sf − f)‖L∞(Ω) ≤ CA2

A1

|∆|m+1−|α||f |m+1,∞,Ω

for a constant C dependent only on β and d.

This can be proved by using a polynomial approximation property andMarkov’s inequality. Details are omitted here.

Our next question is how to compute discrete least squares fits. Recallthat we write each s ∈ S−1

d (∆) in the B-form

s(x, y)|t =∑

i+j+k=d

ctijkB

d,tijk(x, y)

with coefficient vector c = (ctijk, i + j + k = d, t ∈ ∆).

We put all smoothness conditions of Srd(∆) together as

Hc = 0.

Let L be an observation matrix. It is easy to see

ℓ(s − f) = cTLLTc − 2cTLf + fT f .

The discrete least squares spline is the solution of

mincTLLTc − 2cTLf , subject to Hc = 0.

By the Lagrange multipliers method, we solve[

LLT HT

H 0

] [

c

α

]

=

[

Lf

0

]

.

The ALW iteration introduced in the previous subsection can be appliedto solve the above linear system. As before the iterative solutions convergethe exact solution.

When the number of data sites is large, especially when the number oftriangles is large, a computer may not be powerful enough to solve the asso-ciated linear system. We again propose a domain decomposition techniquefor computing an approximation of the discrete least squares spline (cf. [Laiand Schumaker’03]). That is, for k ≥ 1, we compute Sf,t,k such that

ℓDk(t)(Sf,t,k − f) = minℓDk(t)(s − f), s ∈ Srd(∆),

12

ℓDk(t)(s − f) =∑

(xi,yi)∈Dk(t)

|s(xi, yi) − f(xi, yi))2.

We have the following (cf. [Lai and Schumaker’09])

Theorem 3.3 Suppose that Srd(∆) with d ≥ 3r + 2 over a β quasi-uniform

triangulation ∆. Suppose that data values are obtained from a continuouslydifferentiable function f ∈ Cm+1(Ω). Suppose that A1 > 0 and A2 < ∞are constants such that A2/A1 is independent of ∆. Then there is a positiveρ < 1 such that

‖sf − Sf,k‖L∞(t) ≤ Cρk(k + 2)|∆|m+1|f |m+1,∞,Ω

for k ≥ 1, where C is a constant dependent only on d, β and A2/A1.

Continuous least squares fitting is to compute the least squares approx-imation of any given function. Let f ∈ C(Ω) be a given function. We lookfor spline function Sf ∈ Sr

d(∆) such that∫

Ω

|f(x, y)− Sf(x, y)|2dxdy = mins∈Sr

d(∆)

∫

Ω

|f(x, y) − s(x, y)|2dxdy.

The existence and uniqueness of continuous least squares spline Sf are wellknown. We now explain how to compute Sf .

We write each s ∈ S−1d (∆) in the B-form

s(x, y)|t =∑

i+j+k=d

ctijkB

d,tijk(x, y)

with coefficient vector c = (ctijk, i + j + k = d, t ∈ ∆).

We put all smoothness conditions of Srd(∆) together as

Hc = 0.

Let M be the mass matrix, i.e., M = [mij ]1≤i,j≤n with

mij =

∫

Ω

φiφjdxdy

where φ1, · · · , φn = Bd,tijk, i + j + k = d, t ∈ ∆. The continuous least

squares spline is the solution of

mincT Mc − 2cTMf + fT Mf , subject to Hc = 0,

13

where we approximate f by sumni=1fiφi and denote f = (f1, · · · , fn)

T .By the Lagrange multipliers method, we solve

[

M HT

H 0

] [

c

α

]

=

[

Mf

0

]

.

Thus, this computation can be done easily.

14

3.1 Penalized Least Squares Spline Method

Recall that E(f) denotes a thin-plate energy functional of f and ℓ(s) =∑N

i=1(s(xi, yi) − fi)2 as before. Fix λ > 0. Define P (s) = ℓ(s) + λE(s). The

PLS spline is the minimization solution Sf,λ ∈ Srd(∆) such that

P (Sf,λ) = minP (s), s ∈ Srd(∆).

We refer to [Awanou, Lai, and Wenston’06] for a proof of the following.

Theorem 3.4 Suppose that N ≥ 3, and there exist three data sites, say(xi, yi), i = 1, 2, 3, which are not colinear. Then there exists a unique Sf,λ inSr

d(∆) solving the above minimization problem.

We certainly want to know if the penalized least squares fitting surfaceresembles the given data or not. Since f − Sf,λ = f − Sf,0 + Sf,0 − Sf,λ,we need to estimate Sf,0 − Sf,λ. To do so, we introduce the following twoquantities: (cf. [von Golitschek and Schumaker’02b])

K1 = supE(s)1/2

ℓ(s)1/2, s ∈ Sr

d(∆), s 6= 0

and

K2 = sup‖s‖L∞(Ω)

ℓ(s)1/2, s ∈ Sr

d(∆), s 6= 0.

Then in [von Golitschek and Schumaker’02b], von Golitschek and Schumakerproved the following

Theorem 3.5 Let Sf,λ be the Penalized Least Squares spline in Srd(∆) with

d ≥ 3r + 2. Assume that K1 and K2 are finite. Then

‖Sf,λ − Sf,0‖L∞(Ω) ≤ K2

√λE(Sf,0) min1, K1

√λ.

We now work on estimating K1 and K2. It is easy to get

E(s) ≤∑

T∈∆

AT‖s‖22,∞,T ≤

∑

T∈∆

AT

ρ4T

‖s‖2L∞(T ) ≤

β2

(ρ∆)2

ℓ(s)

A21

.

It follows that K1 ≤ βA1ρ∆

.

15

Since ‖s‖L∞(Ω) = ‖s‖L∞(T ) for a triangle T ,

‖s‖L∞(Ω) ≤1

A1

√

∑

(xi,yi)∈T

|s(xi, yi)|2 ≤1

A1ℓ(s)1/2.

It follows that

K2 ≤1

A1.

Theorem 3.6 Let Sf,λ be the PLS spline in Srd(∆) with d ≥ 3r+2. Suppose

that f ∈ W m+1∞ (Ω) with 1 ≤ m ≤ d. Then

‖Sf,λ − f‖L∞(Ω) ≤ C1|∆|m+1|f |m+1,∞,Ω + λC|f |2,∞,Ω

A21(ρ∆)2

,

where C1 > 0, C2 > 0 are constants dependent on A2/A1, β and d.

To see that the convergence is linear in λ, we present some numericalexperiments: For λi = 1/210+i, the maximum errors of Sf,λi

to f are

λ2 λ3 λ4 λ5

S15(∆) 5.466e − 4 2.800e − 4 1.421e − 4 7.819e − 5

S16(∆) 5.451e − 4 2.762e − 4 1.408e − 4 7.318e − 5

As we see the condition for the existence of penalized least squares splinefits is much weaker than that for the existence of the discrete least squaresspline fits. However, the approximation result on penalized least squaresspline fits is dependent on a very strong condition on the data sites, i.e.,A1 > 0. It is interesting to see if one can remove this condition while provingthat the penalized least squares fits resemble the shape of the data.

Recall c is the coefficient vector of a spline s ∈ S−1d (∆), H is the smooth-

ness matrix such that Hc = 0 if and only if s ∈ Srd(∆), E is the energymatrix, and L is the observation matrix. Then the PLS spline is the mini-mization solution

mincTLLTc − 2cTLf + λcTEc, subject to Hc = 0.

By the Lagrange multipliers method, we solve[

LLT + λE HT

H 0

] [

c

α

]

=

[

Lf

0

]

.

16

We apply the ALW iteration introduced before.When the number of triangles is large, a computer may not be powerful

enough to find the PLS splines. We use a domain decomposition technique forcomputing an approximation of the PLS spline (cf. [Lai and Schumaker’03]).For k ≥ 1, we compute a PLS spline Sf,t,k such that

PDk(t)(Sf,t,k) = minPDk(t)(s), s ∈ Srd(∆),

where

PDk(t)(s) =∑

(xi,yi)∈Dk(t)

|s(xi, yi) − f(xi, yi)|2 + λE(s|Dk(t)).

Here Dk(t) = stark(t) for each triangle t ∈ . We have the following result(cf. [Lai and Schumaker’03]).

Theorem 3.7 Suppose that Srd(∆) with d ≥ 3r + 2 over a β quasi-uniform

triangulation ∆. Suppose that data values are obtained from a continuouslydifferentiable function f ∈ Cm+1(Ω). Suppose that A1 > 0 and A2 < ∞are constants such that A2/A1 is independent of ∆. Then there is a positiveρ < 1 such that

‖sf − Sf,k‖L∞(t) ≤ Cρk((k + 2)3/2|∆|m+1|f |m+1,∞,Ω + λ|f |2,∞,Ω)

for k ≥ 1, where C is a constant dependent only on d, β and A2/A1.

17

3.2 L1 Spline Methods

L1 spline methods for data fitting were proposed in [Lavery’2000]. He usedC1 cubic spline curves and bivariate C1 cubic Sibson’s elements for scattereddata in 1D and grid data in 2D, respectively. Lai and Wenston in 2004generalized the study to the scattered data in the bivariate setting. Recallthat

Λ(f) = s ∈ Srd(∆), s(xi, yi) = f(xi, yi), i = 1, · · · , N.

Let E1(s) be the L1 energy functional, i.e.,

E1(f) =

∫

Ω

(∣

∣

∣

∣

∂2

∂x2f

∣

∣

∣

∣

+ 2

∣

∣

∣

∣

∂2

∂x∂yf

∣

∣

∣

∣

+

∣

∣

∣

∣

∂2

∂y2f

∣

∣

∣

∣

)

dxdy.

Find Sf ∈ Λ(f) such that

E1(Sf ) = minE1(s), s ∈ Λ(f).

Sf is called the L1 interpolatory spline of the given data (xi, yi, f(xi, yi)),i = 1, · · · , N. A proof of the following theorem can be found in [Lai andWenston’04]. This can be seen from the fact that the minimization functionalis convex. However, the functional is not strictly convex and hence, thesolution may not be unique.

Theorem 3.8 Suppose that Λ(f) is not empty. Then there exists at leastone Sf solving the above minimization problem.

The interpolatory surfaces which minimize the L1 energy functional areindeed different from the usual L2 minimal energy splines. Figures 9 and10 show their differences. (These figures are borrowed from [Lai and Wen-ston’04].)

It is necessary to show that L1 interpolatory splines resembles the shapeof the given data. Lai in [Lai’07] proved the following

Theorem 3.9 Suppose that f ∈ C2(Ω). Let Sf be the L1 interpolatory splineof the data (xi, yi, f(xi, yi)), i = 1, · · · , N . Then

‖Sf − f‖L1(Ω) ≤ C|∆|2|f |2,∞,Ω,

for a constant C dependent only on β and d.

18

−2

−1

0

1

2

−2

−1

0

1

2−0.5

0

0.5

1

1.5

L1−spline interpolation

Figure 9: L1 interpolatory spline (the top row) and minimal energy interpo-latory spline (the bottom row)

19

Figure 10: L1 interpolatory spline (the top row) and minimal energy inter-polatory spline (the bottom row)

20

3.3 Least Absolute Deviation

For a given data set (xi, yi, f(xi, yi)), i = 1, · · · , N, let

ℓ1(s) =N∑

i=1

|s(xi, yi)|.

We find Sf ∈ Srd(∆) such that

ℓ1(Sf − f) = minℓ1(s − f), s ∈ Srd(∆).

Sf is the least absolute deviation(LAD) from the given data (cf. [Bloomfieldand Steiger’83]).

Since the minimization functional is convex, there always exist a mini-mizer Sf (cf. [Lai and Wenston’04]). Next we would like to know how wellthe LAD surface resembles the given data. Let F1 and F2 be positive numberssuch that

F1‖s‖L∞(T ) ≤∑

(xi,yi)∈T

|s(xi, yi)| ≤ F2‖s‖L∞(T )

for all s ∈ Srd(∆) and for all T ∈ ∆. We have the following (cf. [Lai’07]).

Theorem 3.10 Suppose that two constants F1 > 0 and F2 < ∞ such thatF2/F1 independent of ∆. Suppose that f ∈ W m+1

∞ (Ω) for 0 ≤ m ≤ d. Then

‖Sf − f‖L1(Ω) ≤ C|∆|m+1|f |m+1,∞,Ω

for a positive constant C dependent on F2/F1, β and d.

3.4 L1 Smoothing Splines

L1 smoothing splines are Sf ∈ Srd(∆) which minimizes

ℓ1(Sf − f) + λE1(Sf ) = minℓ1(s − f) + λE1(s), s ∈ Srd(∆).

Since the minimization functional is convex, there exists at least one Sf

solving the above minimization problem. We next need to show that Sf

approximates f as the size of the triangulations goes to zero (cf. [Lai’07]).

21

Theorem 3.11 Under the same assumptions as Theorem 3.10,

‖Sf − f‖L1(Ω) ≤ C|∆|m+1|f |m+1,∞,Ω + λCf

F1|∆|2

for a positive constant C dependent on F2/F1, β and d.

Algorithms computing these three L1 spline methods were discussed in[Lai and Wenston’04]. The main ideas are

1) use discontinuous piecewise polynomial functions and set the smoothnessconditions as side constraints;

2) convert L1 norm minimization to a linear programming problem;

3) use Karmarkar’s algorithm to solve the linear programming problem.

4 Possible Research Projects

Let me give a list of possible projects:

• 1. Instead of minimal energy method for scattered data interpolation,we can solve the following unconstrained minimization problem:

mins∈Sr

d(∆)

E(s) +1

2λ

N∑

j=1

|f(xi, yi) − s(xi, yi)|2,

where λ > 0 is a small parameter and E(f) is the thin-plate energyfunctional. If λ is very small, the solution Sf satisfies the interpola-tion conditions approximately. We should study its existence, unique-ness, characteristic conditions, stability, extremal value, computationalmethod, ...

• 2. Instead of E(f), we may use E1(f) in the previous problem. TheL1 norm of the second order derivatives of s. That is, we study

mins∈Sr

d(∆)

E1(s) +1

2λ

N∑

j=1

|f(xi, yi) − s(xi, yi)|2.

22

• 3. Again we replace E1(f) by the L1 norm of the first order derivativesof spline functions s

mins∈Sr

d(∆)

‖∇s‖1 +1

2λ

N∑

j=1

|f(xi, yi) − s(xi, yi)|2.

Then we study its properties of existence, uniqueness, resemblance,characterization, stability, extremal value, and computational method,and etc..

• 4. Generalize all the scattered data fitting/interpolation to 4D dataor data values over 3D points using trivariate splines. I will explainmore and show you my matlab programs for 4D data interpolation andfitting on next Monday morning.

• 5. Use 2D and 3D splines for statistical applications. See Bree Et-tinger’s lecture in the afternoon.

References

[1] P. Alfeld, M. Neamtu and L. L. Schumaker(1996), Bernstein-Bezier poly-nomials on spheres and sphere-like surfaces, Computer Aided GeometricDesign, 13, 333–349.

[2] Alfeld, P., M. Neamtu and L. L. Schumaker(1996), Fitting scattered dataon sphere-like surfaces using spherical splines, J. Comp. Appl. Math.,73, 5–43.

[3] Awanou, G. and M. J. Lai(2005), On convergence rate of the augmentedLagrangian algorithms for non symmetric saddle point problems, AppliedNum. Math., 54, 122–134.

[4] Awanou, G., M. J. Lai, and P. Wenston, The multivariate spline methodfor numerical solution of partial differential equations and scattered datainterpolation, in Wavelets and Splines: Athens 2005, edited by G. Chenand M. J. Lai, Nashboro Press, Nashville, TN, 2006, 24–74.

[5] Baramidze, V., M. J. Lai, and C. K. Shum (2006), Spherical Splines forData Interpolation and Fitting, SIAM J. Scientific Computing 28 (2006),241–259

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[6] Bloomfield, P. and W. L. Steiger, Least Absolute Deviation: Theory,Applications, and Algorithms, Birkhauser, Boston, 1983.

[7] de Boor, C., B-form basis, in Geometric Modeling, edited by G. Farin,SIAM Publication, Philadelphia, 1987, 131-148.

[8] Ettinger, B., Guillas, S. and M. J. Lai, Bivariate Splines for FunctionalRegression Modelswith Application to Ozone Concentration Forecasting, 2007.

[9] Farin, G., Triangular Bernstein-Bezier patches, Comput. Aided Geom.Design, 3 (1986), 83-127.

[10] Chui, C. K. and Ming-Jun Lai, Filling polygonal holes using C1 cubictriangular spline patches, Computer Aided Geometric Design 17(2000),297-307.

[11] Fasshauer, G. and L. L. Schumaker, Scattered data fitting on the sphere,in Mathematical Methods for Curves and Surfaces II, M. Daehlen, T.Lyche, L. Schumaker, Vanderbilt University Press, 1998, 117–166.

[12] von Golitschek, M., M. J. Lai, L. L. Schumaker, Error bounds for min-imal energy bivariate polynomial splines, Numer. Math. 93(2002), 315–331.

[13] von Golitschek, M. and L. L. Schumaker, Bounds on projections ontobivariate polynomial spline spaces with stable local bases, Const. Approx.18 (2002), 241–254.

[14] von Golitschek, M. and L. L. Schumaker, Penalized least squares fitting,Serdica 18 (2002), 1001–1020.

[15] Guillas, S. and M. J. Lai, Approximation of functional regression modelswith bivariate splines, submitted, 2007.

[16] Lai, M. J., Convergence of three L1 spline methods for scattered datainterpolation and fitting, Journal of Approximation Theory, 145(2007),196–211.

[17] Lai, M. J., Bivariate spline approximation of kernel functions,manuscript, 2007.

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[18] Lai, M. J. and L. L. Schumaker, Domain decomposition technique forscattered data interpolation and fitting, unpublished manuscript, 2003.

[19] Lai, M. J. and L. L. Schumaker, Spline Functions on Triangulations,Cambridge Univ. Press, Cambridge, U.K. 2007.

[20] Lai, M. J. and P. Wenston, L1 Spline Methods for Scattered Data Inter-polation and Approximation, Advances in Computational Mathematics21 (2004), 293–315.

[21] Lavery, L., Shape-preserving, multiscale fitting of univariate data by cu-bic L1 smoothing splines, Comp. Aided Geom. Design, 17 (2000), 715–727.

[22] Zhou, T., D. Han, M. J. Lai, Energy Minimization Method for ScatteredData Hermite Interpolation, to appear in Applied Num. Math, 2007.

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