Integrated multiquadric radial basis function approximation methods

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E L S E V I E R

An International Journal

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computers &

.c,=.c~ C@o,.=cT.

mathematics

with applications Computers and Mathematics with Applications 51 (2006) 1283-1296

www.elsevier.com/locate/camwa

Integrated Multiquadric

Radial Basis

Function A p p r o x i m a t i o n M e t h o d s

S. A. SARRA

D e p a r t m e n t of M a t h e m a t i c s , M a r s h a l l U n i v e r s i t y O n e J o h n M a r s h a l l Drive Huntington, WV 25755, U.S.A.

A b s t r a c t - - P r o m i s i n g numerical results using once and twice integrated radial basis functions have been recently presented. In this work we investigate the integrated radial basis function (IRBF) concept in greater detail, connect to the existing RBF theory, and make conjectures about the properties of IRBF approximation methods. The IRBF methods are used to solve PDEs. @ 2006 Elsevier Ltd. All rights reserved.

K e y w o r d s - - R a d i a l basis function, Interpolation, Numerical partial differential equations.

1 . I N T R O D U C T I O N

Over the last several decades r a d i a l basis functions ( R B F s ) have b e e n found to be widely successful for the i n t e r p o l a t i o n of s c a t t e r e d d a t a . More recently, the R B F m e t h o d s have emerged as an i m p o r t a n t t y p e of m e t h o d for the n u m e r i c a l solution of p a r t i a l differential e q u a t i o n s ( P D E s ) [1,2]. R B F m e t h o d s can be as a c c u r a t e as s p e c t r a l m e t h o d s w i t h o u t being t i e d to a s t r u c t u r e d com- p u t a t i o n a l grid. This leads to ease of a p p l i c a t i o n in c o m p l e x g e o m e t r i e s in any n u m b e r of space dimensions.

A radial basis function O(r)

is a continuous u n i v a r i a t e function t h a t has been radialized b y

c o m p o s i t i o n w i t h the E u c l i d e a n n o r m on IK d. R B F s m a y or m a y not c o n t a i n a free p a r a m e t e r called the

shape parameter

which we d e n o t e by c. Well-known R B F s w i t h o u t a s h a p e p a r a m e t e r are t h e

polyharrnonic splines

r k, k ~t 2N,

~ ( r ) = r 2 k l o g r , k C N . (1)

T h e p o l y h a r m o n i c splines are a l t e r n a t i v e l y known as surface splines or in the case of ~b(r) = r 2k log r, t h i n - p l a t e splines. This class of R B F t y p i c a l l y a p p r o x i m a t e s w i t h a l g e b r a i c convergence rates. In this work we are i n t e r e s t e d in R B F s t h a t c o n t a i n a s h a p e p a r a m e t e r , b u t we will make some connections between t h e two types. Some c o m m o n l y used R B F s containing a shape p a r a m e t e r are listed in Table 1. This class can e x h i b i t s p e c t r a l r a t e s of convergence. T h e previously m e n t i o n e d R B F s have global s u p p o r t . However, this is not a necessity as c o m p a c t l y

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1284 S. A. SARRA

Table 1. Global, infinitely s m o o t h RBFs.

Name of RBF Definition

Multiquadric (MQ) Inverse Quadratic (IQ) Gaussian (GA)

¢(~, 6 =

1 ¢ ( r , c ) -- (1 +c2r 2) ¢(~, c) = e - ~ ' ~

supported RBFs, such as the Wendland functions [3], are also well known. T h e interested reader is referred to the recent book by B u h m a n n [4] for more basic details about RBFs.

R B F methods for P D E s are based on a scattered d a t a interpolation problem. Let ~ be a finite distinct set of points in II{ a, which are traditionally called center's in the language of RBFs. The idea is to use linear combinations of translates of one function ¢(r) of one real variable which is centered at { e F_ to approximate a function f as

N

s(x) = ~ ~ ¢ (llx - ~ 119, ~ • R ~. (2)

i=0

The most attractive feature of the R B F methods is t h a t the location of the centers can be chosen arbitrarily in the domain of interest. The interpolation problem is to find expansion coefficients, hi, so t h a t

s]= = fl= (3)

for given d a t a

flz-

T h a t is, they are obtained by solving the linear system

H A = f, (4)

where the elements of the interpolation matrix H are hi,j = ¢([[(i - ( j 112) for i , j = O, 1 , . . . , N . For the R B F s in Table 1 the interpolation matrix H can be shown to be invertible [5]. However, the interpolation matrix is often very ilLcondidoned and it m a y be difficult to deal with the system numerically. T h e conditioning of H is measured by the condition number defined as

~ ( H ) = [[Sl[ [ [ H - I [ [ - ~m~x, (5)

O'mi n

where a are the singular values of H. T h e reason t h a t H is in practice usually ill-conditioned is due to what Schaback [6] has assigned the name of the uncertainty principle. T h e uncertainty principle states t h a t with R B F approximation methods we cannot simultaneously have good conditioning and good accuracy.

In general, small shape parameters produce the most accurate results, but are also associated with a poorly conditioned interpolation matrix. For fixed values of the shape parameter, the conditioning of the interpolation matrix deteriorates as the separation distance between centers, defined as

1

qs = ~ min~¢j II~ - %112, (6)

decreases. Obviously, by increasing the number of centers used, we decrease q~. M a n y researchers [7-9] have searched for an algorithm to specify an "optimal" shape parameter, i.e., a value t h a t produces maximal accuracy while maintaining numerical stability. However, deter- mining the optimal shape parameter is still an open question and it is often chosen by brute force in applications. The shape parameter is modified until the resulting interpolation matrix has a condition number below a threshold. Using 32-bit double precision floating point arithmetic, standard linear algebra packages typically began to fail when ~ ( H ) = O(lOe17), which we refer to as the floating point limit.

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I n t e g r a t e d M u l t i q u a d r i c R a d i a l B a s i s F u n c t i o n i 2 8 5

Recently, some promising numerical results [10-12] using once and twice integrated radial basis functions have been presented. In these works, the authors argued heuristically t h a t the smoothing process of integration led to more accurate m e t h o d s t h a n those using the standard nonintegrated RBFs. Since the convergence rates of the MQ R B F s drop with increasing order of differentiation [13,14], integration of the basis functions counters this trend. Subsequently, other researchers have used the idea. In [15] twice integrated MQ R B F s were used to solve one- dimensional b o u n d a r y value problems. In [16] a volumetric integral R B F m e t h o d is formulated for time-dependent PDEs. Unfortunately, in [15] the methods were only discussed for a fixed constant value of the shape parameter. We are interested in the properties of the I R B F methods over the entire range of the shape parameter. In [10-12] a larger range of the shape parameter was considered but the limiting cases c ~ 0 and c ~ ec were not discussed. In this work we take a closer look at integrated R B F s and examine their properties over the full range of the shape parameter including the limiting cases. Additionally, we discuss the invertibility of the interpolation matrix for the integrated RBFs.

The focus of this work is on the integrated family of R B F s based on the MQ R B F which we refer to as the parent of the integrated family. The MQ was chosen because it is very popular in applications and m a n y theoretical results exist for the MQ [8]. We note t h a t integrated R B F families based on other parent R B F s such as the IQ and G A have m a n y similar properties as the MQ family, but in several aspects there are marked differences.

2. I N T E G R A T E D R B F S

Let the notation Cn(r, c) represent a global infinitely differentiable R B F containing a shape parameter c t h a t has been integrated (n > 0) or differentiated (n < 0) n times with respect to r. We refer to these R B F s as integrated R B F s (IRBFs), as it is the case n > 0 t h a t we are interested in for use in approximations. I R B F s are easily found using a c o m p u t e r algebra system. For reference, we list the first four members of the integrated R B F family based on the MQ R B F

¢1 = crvZf + c2r2 + s i n h - l ( c r ) 2C ¢2 = (--2 + c2r 2) v/1 + c2r 2 + 3or sinh-l(cT) 6c 2 ¢3 ~- C T ~ (--13 + 2c2r ') + 3 (--1 + 4c2r 2) s i n h - l ( c r ) 48c 8 ¢4 = ~ (16 -- 83c2r 2 + 6c4r 4) + 15cr (--3 + 4c2r 2) sinh l(cr) 720c 4

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(8) (9) (10)

For small c, members of the MQ I R B F family have the series expansion oo

Cn(r) = E amr'~+2mc2m' (11)

where M = - I n / 2 ] . expansion

and ¢5 has

For example, ¢4 of the I M Q (integrated multiquadric) family has the 1 r 2 r 4 r 6 c 2 ¢ 4 _ + + + . . . (12) 45c 4 6c 2 ~-~ r F 3 7 ,5 c 2 r 7

¢~-

- - + + + . . . (13) 45c 4 18c 2 1 ~ 5 - ~ "

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1286 S . A . S A R R A 10 0 - 2 - 4 - 6

<i,4(r)

c = 0 . 2 5 c=0.5=. 0.5 1 1.5 r

(a) Graph of evenly integrated IMQ ¢4(r) for various values of the shape parameter.

,S(r)

5000

I

000

3000 2000 1000

o /

0.5 1 1.5 2 r

(b) Graph of oddly integrated IMQ ¢5(r) for various values of the shape parameter.

Figure 1.

As c approaches zero the even (n even) IMQs approach a large c o n s t a n t a n d are infinite for c = 0. T h e odd IMQs approach a large multiple of 4@) = r. For large c the odd IMQs are a p p r o x i m a t e l y

4~(r) ~ cr ~+1 +

lrn-~

logr, n odd, (14)

C a n d the even IMQs are approximately

4 n ( r ) ~ c r n + l , n even. (15)

T h a t is, for large c, the odd IMQs are a p p r o x i m a t e l y a p o l y h a r m o n i c spline p e r t u r b e d by a multiple of a n even power of 7- a n d the even IMQs are a large multiple of a p o l y h a r m o n i c spline. Figure 1 illustrates the shape of 4 4 a n d 4 5 for various values of t h e shape parameter.

Derivatives of mth order with respect to xk, k = 1 , . . . , d, of i n t e r p o l a n t (2) m a y be calculated in a straightforward m a n n e r at centers ~j C ~ as

N

rf~ Tr~

o--~

d~,)=

V'A. m"

:.,

i=0

E l e m e n t a r y calculus allows us to find derivatives of a n y order, a n d in a n y n u m b e r of space dimensions, in t e r m s of the other m e m b e r s of the I R B F family. For example, the first derivative with respect to xk m a y be calculated as

C94 n

__

4(~-1) (r, c)

°qr

Oxk

cgxk'

where 7- =

Ilxl12.

Likewise, the second derivative with respect to xk is

02:

02T

( aT ~

-- 4(n-1)(T,C)~-X2 -t- 4(n--2)(r,c)

\-~-~Xk )

"

Om~

Next we discuss t h e stability a n d t h e a p p r o x i m a t i o n properties of t h e I R B F m e t h o d s based on the MQ RBF. Most of the properties of the I R B F m e t h o d s d e p e n d on w h e t h e r the basis functions have been i n t e g r a t e d an even or an odd n u m b e r of times. I n the n u m e r i c a l examples which follow, we measure the error over a range of shape p a r a m e t e r s in a d o m a i n fl in the m a x i m u m n o r m as

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Integrated Multiquadric Radial Basis Function 3 . B E H A V I O R A S A F U N C T I O N

O F T H E S H A P E P A R A M E T E R

1287

3.1. A p p r o x i m a t i o n

T h e limiting case as c ~ 0 has previously been e x a m i n e d in [17] for R B F s c o n t a i n i n g a shape p a r a m e t e r t h a t has a series expansion of form (11) with n = 0. A m o n g m a n y other RBFs, the analysis of [17] includes the MQ, IQ, a n d GA. In the limit c ~ 0 the R B F s become a constant, a s i t u a t i o n t h a t is referred to as the fiat limit. In [17] it was shown t h a t for a distinct set of centers in one dimension, the R B F i n t e r p o l a n t exists as c --~ 0 a n d is equal to the Lagrange i n t e r p o l a t i n g p o l y n o m i a l on t h a t set of nodes given the invertibility of two associated matrices. Although n o t proven, it was conjectured t h a t the associated matrices are n o n s i n g u l a r for all s t a n d a r d choices of ¢(r), including the IQ, MQ, a n d GA. I n two dimensions, it was found t h a t the limiting i n t e r p o l a n t m a y n o t exist, especially on t e n s o r - p r o d u c t grids. However, when the limit does exist, it depends on ¢ ( r ) a n d is a m u l t i v a r i a t e finite-order polynomial. T h e analysis of the two-dimensional case was continued in [18] a n d extended by Schaback in [19]. T h e limiting case as c -~ oo has not received much a t t e n t i o n in the l i t e r a t u r e except for t h e acknowledgment t h a t R B F s with "large" shape parameters produce poor a p p r o x i m a t i o n s which are comparable to low-order local methods.

To n u m e r i c a l l y explore tile I R B F m e t h o d s with shape p a r a m e t e r s for which the i n t e r p o l a t i o n m a t r i x is too poorly conditioned to use s t a n d a r d methods, we use the C o n t o u r - P a d ~ (CP) algo- r i t h m [20]. T h e CP algorithm s t a b l y calculates the R B F a p p r o x i m a t i o n , for a small n u m b e r of centers, for all values of c including at the limiting value c = 0. Details of the algorithm m a y be found in [20].

We now consider the quality of approximations produced b y the I R B F for b o t h small a n d large c. T h e cases of n even a n d n odd are different a n d are discussed separately.

In one dimension, the result of T h e o r e m 3.1 in [17] holds if we replace the hypothesis t h a t the basis functions have the small c expansion (11) with n = 0 w i t h the same hypothesis with n _> 0 a n d n is even. T h u s if some auxiliary conditions are satisfied as specified in [17], the evenly integrated IMQ a p p r o x i m a n t s are equivalent to the Lagrange a p p r o x i m a n t in the limit c --4 0.

LLI 10 0 10 -s 10 -7 10 -8

~2

0 0.2 0.4 0.6 0.8 1 C

(a) n even with the direct method (dotted) and Contour-Pad6 algorithm (solid).

10 -1

¢7

¢=r

. . .

10 -2 10-3 II~

¢

.,'"'"

10-41

. . . : ' , • . • . ' . ' . ' . ' . ' .

"

" " : : " .

¢4

lO-St • ..".":. .. .. . . . ' :

."

1 0 - 3 i , , , , 0 0.2 0.4 0.6 0.8 1 C

(b) n odd compared with linear splines ¢(r) = r. Figure 2. Small shape parameter in ld. Max error vs. shape parameter from interpolating f(x) = cos(~rx) with N = 10 uniformly spaced centers to x0 = 0.25 using IMQs Cn Note the different scales.

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1288 S.A. SARRA 1 0 - 2 10 -= 10 -4 10 ~ 10 -a 1 0 - l o 10 -12 ¢4 . . . ' " " '

(a) Cn for n even. solid. 10 ~" 10 "~ ¢3 ¢7 10 ~ 10 -lq 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 1000 2000 3000 c o

Direct dotted, Contour-Pad~ (b) ¢'~ for n odd.

4000

Figure 3. Scattered 2d derivative approximation. Errors at the point (0, 0.2) (shaded point in Figure 4) from calculating ~ for f ( x , y) = e x p ( x / 2 + y / 4 ) on the unit circle with 54 centers distributed as in Figure 4.

0 . 8 0 0.6 0 0.4 0.2 o • -0.2 -0.4 -0.6 -0.8 -1 -1 0 0 0 o w -0.5 O o o o o j ) o o o / 0 0.5 1 X

Figure 4. 54 scattered nodes one the unit circle.

Thus, for even I M Q i n t e r p o l a t i o n m e t h o d s which include the p a r e n t R B F , we get something familiar as c approaches zero. As c --* 0 the condition n u m b e r of the i n t e r p o l a t i o n m a t r i x for the even IMQs becomes u n b o u n d e d which prevents the evaluation of the i n t e r p o l a n t by s t a n d a r d methods. For small N the C o n t o u r - P a d 5 algorithm is applicable a n d we use it in such an example with results shown in Figure 2a to illustrate the equivalence of all the even I M Q i n t e r p o l a n t s as c -~ 0. For d > 1, we no longer have equivalence with the Lagrange a p p r o x i m a n t as c ~ 0. This is illustrated in Figure 3 with the aide of the C P algorithm for the a p p r o x i m a t i o n of a partial derivative at scattered centers on the u n i t circle. W h e n the a p p r o x i m a n t exists, it depends on the p a r t i c u l a r even I M Q family m e m b e r t h a t is used.

For large c, the even IMQ a p p r o x i m a n t s behave as expccted as their large c approximations, the p o l y h a r m o n i c splines (15). This large c behavior is i l l u s t r a t e d by an example in Figure 5a.

As c ~ 0 the odd members of the IMQ family approach a m u l t i p l e of the linear p o l y h a r m o n i c spline ¢(r) = 7". For small c > 0 the odd IMQ m e t h o d s a p p r o x i m a t e with the accuracy of the linear splines. A n example i l l u s t r a t i n g the sm~ll shape accuracy of the odd IMQs is in Figure 2b.

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Integrated Multiquadric Radial Basis Function 1289 1 0 -1 10 -2 "G" N" 10-a 10 -4 ~/r #02# 3 ~4/r5 ~8/r9 ~6/r7 10 -1 10 -2 10 -3 10 ~ ,(r) = r ~(r) = r s log(r) N O o ( r ) = c 2 r 10 + r 8 log(r) 50 100 150 2 0 0 2 5 0 C

(a) r~ even (solid) compared with polyharmonic splines r ~+1 (dotted).

0 50 100 150 2 0 0 250

C

(b) IMQ ¢9 compared with the accuracy of its large c approximation (17) which reduces to a thin plate spline for c = 0 and for small c compared with the accuracy of ¢ ( r ) = r.

Figure 5. Interpolation m a x i m u m error vs. shape p a r a m e t e r for "large" s h a p e parameters with IMQs ¢'~. The function f ( x ) = exp(x 3) -I- cos(2x) at N -- 20 equally

spaced interpolation points on [ - 1 , 1] was interpolated to a finer evenly spaced grid with N = 100.

For large c, the o d d I M Q m e t h o d s a p p r o x i m a t e as if t h e basis functions are as in e q u a t i o n (14). If e q u a t i o n (14) is m u l t i p l i e d b y c we get

O n ,-~ C2r n + l q- r n - 1 log(r), n odd, (17)

which reduces to a p o l y h a r m o n i c spline for c = 0. B y p e r t u r b i n g t h e p o l y h a r m o n i c spline by a multiple of an even power of r, we find t h a t the result is often c o n s i d e r a b l y m o r e accurate, at the expense of p o o r e r conditioning, t h a n if t h e p o l y h a r m o n i c spline alone were employed. A n e x a m p l e of the large c accuracy of the odd I M Q s as well as t h e a c c u r a c y of the p e r t u r b e d

ILl 101 10 ° 10 -1 10 -2 10 -3 10 ~ 0 • ¢ 8 \ \ \ \ \ 10 20 30 40 50 C

Figure 6. ld derivative approximation of f ( x ) = exp(x 3) + cos(2x) with N = 30

equally spaced collocation points on [ - 1 , 1]. First derivative m a x i m u m error vs. shape p a r a m e t e r with 6 o (solid), ¢8 (dotted), ¢9 (dashed).

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10 -1 10

-2I

0 -3 ~-1

10

'~ 0.5

¢

,2

,4

1290 S . A . S A R R A 1 0 -1 10 -2 10 -3 10 4 ¢ - - r

¢5

1 1.5 2 2.5 3 0 2 0 4 0 6 0 c c

¢9

F i g u r e 7. I n t e r p o l a t i o n m a x i m u m e r r o r vs. s h a p e p a r a m e t e r w i t h C n n e v e n (left) a n d o d d ( r i g h t ) . T h e f u n c t i o n f ( x ) ~ e x p ( x 3) -t- c o s ( 2 x ) a t N = 20 e q u a l l y s p a c e d interpolation points on [-1, 1] was interpolated to a finer evenly spaced grid with N : 100.

polyharmonic spline (17) is illustrated for interpolation in Figure 5b. The large c accuracy of the odd IMQs is further illustrated for derivative approximation in Figure 6 and for scattered partial derivative approximation in Figure 3b.

For interpolating or derivative approximation of many functions, the IMQ approximations produce higher quality approximations over a wider range of shape parameters when compared to ¢% The parent RBFs, ¢0, typically produce their smallest errors over a very small range of shape parameters which borders on a region of numerical instability where the condition number of the interpolation matrix exceeds the floating point limit. Since a theory which predicts the optimal shape parameter in advance does not exist, in practice a shape parameter is used which produces an interpolation matrix with condition numbers safely away from the region of instability and which does not produce the "optimal" results that are obtainable in floating point arithmetic. Often less tweaking of the shape parameter in required to achieve adequate results when using tile IMQs than the MQ. Illustrating the greater accuracy of the IMQ when compared to the MQ over a wide range of the shape parameter is the example in Figure 6. The figure displays the maximum error as a function of the shape parameter from differentiating a smooth function in ld with ¢0, Cs, and ¢9 from the IMQ family. Results from interpolating a smooth function with a wide range of shape parameters are given in Figure 7.

However, at the expense of the potentially greater accuracy of the IMQs, we should expect poorer conditioning as is predicted by the uncertainty principle.

3.2. Conditioning

For a fixed value of tile shape parameter, both even and odd IMQs have interpolation matrices with condition numbers that are an increasing function of the nmnber of centers (decreasing separation distance (6)). A comparison of condition numbers with fixed shape and variable N is made among the evenly integrated MQs in Figure 8b and among oddly integrated

IMQs

in Figure 9b.

Even IMQs are associated with interpolation matrices that have condition numbers that are decreasing functions of the shape parameter for a fixed set of centers. The condition numbers become unbounded as c -~ 0. For n even, numerical evidence indicates that the condition number of the interpolation matrix for all c is bounded below by the condition number of the interpolation

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Integrated Multiquadric Radial Basis Function 1291 i i t 10 is 101° 105 ~8/r9 • . ~. ;,.. . . ~~ "~" L2 . . . O)6jr 7 . . . , " - 2 . . . *'it . . . . \ . . . \ ~

~2/r3

¢°/r 5 10 15 20 25 30 C

i015

101° 10 s | /

so loo 1so 200 2so zoo 3so 400

N

(a) ~b ~, n even, *v(H) vs. shape (dashed), polyharmonic splines r ~+1 (dotted) for 100 uniformly placed centers on the square domain [-1, 1] 2.

(b) ~(H) vs. N for e = 2.5. Dashed line is 10e17 at which standard linear algebra packages begin to fail. Figure 8. 10 is ~5 .~" 101o ,3 10 s i i i 10 T M 101° 105 ,3 f A j 10 ° 102 104 50 100 150 200 250 300 350 400 c N

(a) ~ , n odd, ~(H) vs. shape for 100 uniformly placed centers on the square domain [-1, 1] 2. Dot- ted line is tile condition number of linear polyharmonic splines qS(r).

Figure 9.

(b) ~(H) vs. N for c = 1000. Dashed line is 10e17.

m a t r i x of t h e p o l y h a r m o n i c spline, qS(r) = 7- '~+t, t h a t is a s s o c i a t e d w i t h t h e l a r g e c a p p r o x i m a t i o n of t h e e v e n I M Q s . T h u s , if t h e c o n d i t i o n n u m b e r of t h e a s s o c i a t e d p o l y h a r m o n i e spline e x c e e d s t h e f l o a t i n g p o i n t l i m i t for a p a r t i c u l a r set of centers, t h e I R B F will n o t b e a p p l i c a b l e w i t h a n y v a l u e of c. B o u n d s for i n t e r p o l a t i o n m a t r i x c o n d i t i o n n u m b e r s of t h e p o l y h a r m o n i c splines as well as o t h e r R B F s are d i s c u s s e d in [4,21-25]. T h e i n t e r p o l a t i o n m a t r i x c o n d i t i o n n u m b e r s v e r s u s t h e s h a p e p a r a m e t e r for several e v e n l y i n t e g r a t e d I M Q s are c o m p a r e d F i g u r e 8a. F o r reference, t h e c o n d i t i o n n u m b e r s of t h e i n t e r p o l a t i o n m a t r i c e s of t h e a s s o c i a t e d p o l y h a r m o n i e splines are also s h o w n .

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1292 S . A . SARRA

O d d IMQs are associated with interpolation matrices t h a t have condition numbers t h a t are increasing functions of the shape parameter for a fixed number of centers. The condition numbers become u n b o u n d e d as c --* oc. For c sufficiently greater t h a n zero, numerical evidence indicates t h a t the condition numbers of the odd IMQs are bounded below by the condition n u m b e r of the linear polyharmonic splines, ~(r) = r. T h e condition number for odd I M Q interpolation matrices is compared with t h a t of linear polyharmonic splines in Figure 9a.

4 . S O L V A B I L I T Y

T h e theoretical results on the solvability of system (4) are based on the concept of a condition- alIy positive definite function. A function ¢(r) is conditionally positive definite of order rn in R d for d > 1 if and only if

( - 1 ) e d ~ e 6 ( v / ~ ) > _ 0 , f o r t > 0 and g > _ m .

If in addition, ~ ¢ ( v / F ) ¢; constant, then ¢(r) is conditionally strictly positive definite of order m and system (4) is uniquely solvable.

The result with m = 0, which establishes the solvability of system (4) for the IQ and the GA, was shown by Schoenberg in [26]. Micchelli [5] extended this result to establish the invertibility of H for R B F s t h a t are conditionally strictly positive definite of order one, such as the MQ. This is easily verified for the M Q since ~b(x/7 ) = ~/1 + e2r and

d e F(g -- 1/2)e 2e

( - 1 ) ~ 7 ¢ ( 4 7 )

= 2 4 ~ ( 1 + d ~ ) ~ _ 1 / 2 > 0 ,

f o r ~ > 0

and e>_l.

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Micchelli further extended the theory in [5] to include R B F s t h a t are conditionally strictly positive definite of order m _> 2. This was accomplished by placing some mild restrictions on the center locations and by adding polynomial terms to the basic R B F interpolation problem. W i t h the addition of polynomial terms we have the augmented R B F interpolant

N M

s(x) - ~

~,¢ (ll~ - xirr) + ~ ~jvj(x),

x e ~d.

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i=O j = l

The additional constraints are N

~ ~ipj(~j) = 0,

j = 1,2,... ,M,

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~=0

w h e r e { p j g x VtM k }Jj=l is a basis for the M-dimensional space, l i d , of algebraic d-variate polynomials t h a t are of degree less t h a n or equal to m. Examples of strictly positive definite R B F s of order m are the polyharmonic splines ¢(r) = ( - 1 ) ' ~ r k where 2rn - 2 < k < 2rn and

¢(~) = ( - 1 ) " > 2"~-2 log ~.

We can verify t h a t the IMQs are not conditionally strictly positive definite of order m for any m < [ ( n + 2 ) / 2 ] . It is difficult to show rigorously t h a t the I M Q s are conditionally strictly positive definite of order m > l ( n + 2 ) / 2 j as we are unable to obtain a simple expression for the derivatives of en (,/7) as is possible for the MQ in equation (18) due to the more complicated structure of the IRBFs. Graphically, it can be verified t h a t the IMQs are conditionally strictly positive definite of order m = [(n + 2)/2j. This is the same order m as the associated polyharmonic spline of the even I M Q s and the polyharmonic spline portion of the large e approximation (17) of the odd IMQs.

In our numerical experiments, a singular H was never encountered when the I M Q m e t h o d s were not augmented with polynomials. W h e n polynonfial terms were added, t h e y did not significantly

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Integrated Multiquadric Radial Basis Function 1293 improve, a n d in m a n y cases lessened, t h e a c c u r a c y of t h e a p p r o x i m a n t s u n l e s s t h e f u n c t i o n b e i n g a p p r o x i m a t e d was a p o l y n o m i a l .

5. N U M E R I C A L R E S U L T S

As a numerical example, we take test problem 1 from [27], the two-dimensional Poisson problem

~(~)

:

g(~),

o n 0 a , (21) A u ( x ) = f ( x ) , i n f/, o n t h e u n i t circle ft w i t h t h e e x a c t s o l u t i o n 65 7~1 = (65 -~- (X 1 - - 0.2) 2 -~- (X 2 -1- 0.1)2) .

K a n s a ' s asymmetric collocation m e t h o d [1] on N -- 54 centers distributed as in Figure 4 is used to solve the problem. The linear system which arises when using the asymmetric m e t h o d to solve problem (21) is of the form

A~ = F, w h e r e

The set of collocation points E is split into a set Z of interior points, and B of b o u n d a r y points. The two blocks of the matrix A are generated as

¢ ~ j = ¢ ( l l x ~ - x j l t 2 ) , x ~ e u , x j e ~ , A ¢ ~ j = A ¢ (llx~ - x j l l 2 ) , x~ c z, xj c z.

After the coefficients A are found, the approximate solution is c o m p u t e d via (2). The nonsin- gularity results for the interpolation matrices do not carry over to the asymmetric collocation method. T h e addition of polynomial terms no longer guarantees the invertibility of the coefficient matrix. For certain configurations, it was demonstrated in [28] t h a t the collocation matrix m a y

u" 1 0 -2 10 ~ 10 4 10 -s 10 -1° 10 -12 , 2 10 -z 10 ~ 10" 1 0 - e 10 -1 ~3 ~7 10 -1 0.2 0.4 0.6 0.8 1 1.2 1.4 10 4 c c

(a) n odd. (b) n even.

Figure 10. 2d Poisson problem on scattered centers, M a x errors vs. shape parameter from solving A u = f with center distribution pictured in Figure 8 on the unit circle using C n from the [ M Q family.

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1294 S . A . SARRA

be singular. Despite the fact t h a t Kansa's m e t h o d cannot be shown to be well posed, it rarely produces singular matrices in practice and has been widely successful in applications [29].

The m a x i m u m errors resulting from the asynnnetric collocation m e t h o d applied to test problem 1 with solution us using I M Q s over a wide range of shape parameters are shown in Figure 10. T h e results using ¢~, n even, in the left image shows t h a t the I M Q s produce a smaller error over a wide range of the shape parameter t h a n the s t a n d a r d R B F produces. As c gets small and the region of ill-conditioning is approached all the even members of the family approach approximately the same m i n i m u m error. The IMQs realize their smallest error for larger c t h a n the MQ. In this particular example, the oddly integrated results shown in the loglog plot in Figure 10b are particularly impressive. I M Q Cr produces the smallest error of 6 . 5 e - l l with e = 3.5e+5 which is several decimal places more accurate t h a n the best evenly integrated result using ¢4 with c = 0.32 which produces an error of 7.7e-9. Ill-conditioning prevented obtaining better results with ¢6 or Cs t h a n we obtained with ¢4.

In [27], five additional functions were used as exact solutions to (21) which required very different optimal values of the shape parameter due to the varying smoothness of the solutions. T h e I R B F results for test problems two to six were not quite as impressive, as the I M Q minimum errors were not substantially better t h a n the standard MQ results as in problem 1. However, tile IMQs typically achieved a smaller error over a larger range of the shape parameter t h a n did the MQ. This is perhaps the major advantage of the I R B F s as R B F methods are typically not employed in applications using the optimal shape parameters, but using some value of the parameter safely away from the region of ill-conditioning. Obviously, finding the optimal shape parameter in applications by brute force, using standard algorithms or the Contour-Pad~ algorithm, is impossible as an exact solution is seldom known.

6. C O N C L U S I O N S

Radial basis function approximation methods t h a t use integrated families of R B F s have been described. T h e MQ R B F was used as the parent of the family, b u t other R B F s such as the IQ or GA could be used as well. Members of the I R B F family are found by integrating the parent R B F n times with respect to r. The properties of the resulting m e t h o d s depend on whether n is even or odd. We did not consider the case n > 9, but we do not discount the utility of doing so. For n even, the methods behave as those based on the parent RBF. T h a t is, t h e y are generally most accurate and most poorly conditioned for small values of the shape p a r a m e t e r c. For n odd, the methods are most accurate and most poorly conditioned for large c. A s u m m a r y of properties is listed in Table 2.

Nmnericai results indicate t h a t when compared to the MQ method, the I M Q approximation methods m a y produce significantly more accurate results over a wide range of shape parameters. T h e I M Q methods are also more poorly conditioned t h a n the MQ method. While it is largely uncertain which choice of R B F will produce the most accurate result for a particular

T a b l e 2. S u m m a r y o f I R B F p r o p e r t i e s . n E v e n n O d d ~c(H) a s c --~ O, N fixed ~ ( H ) a s c --* e c , N fixed l d i n t e r p o l a n t as c --* 0 I n t e r p o l a n t a s c --* 0, d > 1 L a r g e c i n t e r p o l a n t o o ~(H) for ¢(r) - rn+l ¢ 0 = L a g r a n g e i n t e r p o l a n t D e p e n d s o n Cn ¢(~) = <~+1 ~ ( H ) f o r ¢ ( r ) = r o ~ ¢(~) = ,. ¢(~) = ¢ ( T ) -~ e 2 r nq-1 -[- 7 " n - 1 l o g r

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Integrated Multiquadric Radial Basis Function 1295 a p p r o x i m a t i o n p r o b l e m , R B F s t h a t h a v e b e e n i n t e g r a t e d s e v e r a l t i m e s a p p e a r t o b e s u p e r i o r t o s t a n d a r d n o n i n t e g r a t e d R B F s w h e n t h e f u n c t i o n b e i n g a p p r o x i m a t e d is s u f f i c i e n t l y s m o o t h . T h e I R B F s m a y b e m o s t v a l u a b l e for a p p r o x i m a t i n g s m o o t h f u n c t i o n s i n h i g h e r d i m e n s i o n s w h e r e t h e I R B F s c o u l d p o s s i b l y u s e s i g n i f i c a n t l y f e w e r c e n t e r s t h a n t h e i r p a r e n t s t o a c h i e v e t h e s a m e a p p r o x i m a t i o n a c c u r a c y .

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