Stability and Lebesgue constants in RBF interpolation

Transcription

1 constants in RBF Stefano 1 1 Dept. of Computer Science, University of Verona Göttingen, 20 September 2008 Good

2 Outline Good Good

3 Stability is very important in numerical analysis: desirable in numerical computations, it depends on the accuracy of algorithms [4, Higham s book]. In polynomial, the stability of the process can be measured by the so-called constant, i.e the norm of the projection operator from C(K) (equipped with the uniform norm) to P n (K) (or itselfs) (K R n ), which estimates also the error. The constant depends on the via the fundamental Lagrange or cardinal polynomials. Good

4 Our approach 1. Good [DeM. RSMT03; DeM. Schaback Wendland AiCM05]. 2. Cardinal functions bounds [DeM. Schaback AiCM08]. 3. constants estimates and growth [DeM. Schaback AiCM08; Bos DeM. EJA08 (1d)]. Good

5 Notations X = {x 1,...,x N } Ω R d, distinct; data sites; {f 1,...,f N }, data values; Φ : Ω Ω R symmetric positive definite kernel the RBF interpolant s f,φ := N α j Φ(,x j ), (1) j=1 Letting V X = span{φ(,x) : x X }, s f,x can be written in terms of cardinal functions, u j V X, u j (x k ) = δ jk, i.e. Good N s f,x = f (x j )u j. (2) j=1

6 Error estimates Take V Ω := span{φ(,x) : x Ω} on which Φ is the reproducing kernel. clos(v Ω ) = N Φ (Ω), the native Hilbert space to Φ. f N Φ (Ω), using (2) and the reproducing kernel property of Φ on V Ω, applying the Cauchy-Schwarz inequality, we get f (x) s f,x (x) P Φ,X (x) f Φ (3) Good P Φ,X : power function.

7 A power function expression Letting det (A Φ,X (y 1,...,y N )) = det (Φ(y i,x j )) 1 i,j N, then u k (x) = det Φ,X(x 1,...,x k 1,x,x k+1,...,x N ) det Φ,X (x 1,...,x N ), (4) Letting u j (x), 0 j N with u 0 (x) := 1 and x 0 = x, then Good P 2 Φ,X (x) = ut A Φ,Y u, (5) where u T = ( 1,u 1 (x),...,u N (x)), Y = X {x}.

8 The problem Are there any good for approximating all functions in the native space? Good

9 Our approach 1. Power function estimates. 2. Geometric arguments. Good

10 Literature A. Beyer: Optimale Centerverteilung bei Interpolation mit radialen Basisfunktionen. Diplomarbeit, Universität Göttingen, He considered numerical aspects of the problem. L. P. Bos and U. Maier: On the asymptotics of which maximize determinants of the form det(g( x i x j )), in Advances in Multivariate Approximation (Berlin, 1999), They investigated on Fekete-type for univariate RBFs, proving that if g is s.t. g (0) 0 then that maximize the Vandermonde determinant are the ones asymptotically equidistributed. Good

11 Literature A. Iske:, Optimal distribution of centers for radial basis function methods. Tech. Rep. M0004, Technische Universität München, He studied admissible sets of by varying the centers for stability and quality of approximation by RBF, proving that uniformly distributed gives better results. He also provided a bound for the so-called uniformity: ρ X,Ω 2(d + 1)/d, d= space dimension. R. Platte and T. A. Driscoll:, Polynomials and potential theory for GRBF, SINUM (2005),they used potential theory for finding near-optimal for gaussians in 1d. Good

12 Main result Idea: data set for good approximation for all f N Φ (Ω) should have regions in Ω without large holes. Assume Φ, translation invariant, integrable and its Fourier transform decays at infinity with β > d/2 Theorem [DeM., Schaback&Wendland, AiCM 2005.] For every α > β there exists a constant M α > 0 with the following property: if ǫ > 0 and X = {x 1,..., x N } Ω are given such that f s f,x L (Ω) ǫ f Φ, for all f W β 2 (Rd ), (6) then the fill distance of X satisfies Good 1 h X,Ω M α ǫ α d/2. (7)

13 Remarks 1. The error can be bounded by f s f,x L (Ω) C h β d/2 X,Ω f W β 2 (Rd ). (8) 2. M α when α β, so from (8) we cannot get C ǫ but as close as possible. h β d/2 X,Ω 3. The proof does not work for gaussians (no compactly ed functions in the native space of the gaussians). Good

14 To remedy, we made the additional assumption that X is already quasi-uniform,i.e. h X,Ω q X,Ω. As a consequence, P Φ,X (x) ǫ. The result follows from the lower bounds of P Φ,X (cf. [Schaback AiCM95] where they are given in terms of q X ). Quasi-uniformity brings back to bounds in term of h X,Ω. Observation: optimally distributed data sites are sets that cannot have a large region in Ω without centers, i.e. h X,Ω is sufficiently small. Good

15 On computing near-optimal We studied two algorithms. 1. Greedy Algorithm (GA) 2. Geometric Greedy Algorithm (GGA) Good

16 The Greedy Algorithm (GA) At each step we determine a point where the power function attains its maxima w.r.t. the preceding set. starting step: X 1 = {x 1 }, x 1 Ω, arbitrary. iteration step: X j = X j 1 {x j } with P Φ,Xj 1 (x j ) = P Φ,Xj 1 L (Ω). Good

17 The Geometric Greedy Algorithm (GGA) This algorithm works quite well for subset Ω of cardinality n with small h X,Ω and large q X. The are computed independently of the kernel Φ. starting step: X 0 = and define dist(x, ) := A, A > diam(ω). iteration step: given X n Ω, X n = n pick x n+1 Ω \ X n s.t. x n+1 = max x Ω\Xn dist(x,x n ). Then, form X n+1 := X n {x n+1 }. Good

18 Remarks on convergence Practical experiments show that the GA fills the currently largest hole in the data point close to the center of the hole and converges at least like P j L (Ω) C j 1/d, C > 0. Defining the separation distance for X j as q j = 1 2 min x y X j x y 2 and the fill distance as h j = max x Ω min y Xj x y 2 then, we can prove that Good h j q j 1 2 h j h j, j 2 i.e. the GGA produces quasi-uniformly distributed in the euclidean metric.

19 Connections with discrete Leja sequences Let Ω N be a discretization of a compact domain of Ω R 2 and let x 0 arbitrarily chosen in Ω. The x n = max x Ω N \{x 0,...,x n 1 } are a Leja sequence on Ω. min x x k 2 0 k n 1 Hence, the construction technique of GGA is conceptually similar to finding Leja sequences : both maximize a function of distances. The construction of the GGA is independent of the Euclidean metric. If µ is any metric on Ω, the GGA algorithm produces asymptotically equidistributed in that metric. In [Caliari,DeM.,Vianello AMC2005] the GGA was used with the Dubiner metric on the square. Good

20 How good are the point sets computed by GA and GAA? We could check these quantities: Interpolation error Uniformity: in particular, the GGA maximizes ρ X,Ω = q X h X,Ω, since it works well with subset Ω n Ω with large q X and small h X,Ω. constant Good Λ N := max λ N(x) = max x Ω x Ω N u k (x). k=1

21 : details 1. We considered a discretization of Ω = [ 1,1] 2 with random. 2. The GA run until P X,Ω η, η a chosen threshold. 3. The GGA, thanks to the connection with the Leja extremal sequences, run once and for all. We extracted 406 from random on Ω = [ 1,1] 2, 406 = dim(π 27 (R 2 )). Good

22 GA: Gaussian Gaussian with scale 1, 48, η = The error in the right hand figure is P N 2 L (Ω) which has a decay as a function of the number N of data. As determined by the regression line in the figure, the decay is like N Error Good N

23 GA: Wendland C 2 Wendland function scale 15, N = 100 to depress the power function down to The error decays like N 1.9 as determined by the regression line in the figure Error 10 1 Good N

24 GGA: Gaussian error decay when the Gaussian power function is evaluated on the data supplied by the geometric greedy method up to X 48. The final error is larger by a factor of 4, and the estimated decrease of the error is only like N Error Good N

25 GGA: Wendland The error factor is only 1.4 bigger, while the estimated decay order is Error Good N

26 Gaussian Below: 65 for the gaussian with scale 1. Left: their separation distances; Right: the (+) are the one computed with the GA with η = 2.0e 7, while the (*) the one computed with the GGA geometric greedy geometric greedy Good

27 Inverse multiquadrics Below: 90 for the IM with scale 1. Left: their separation distances; Right: the (+) are the one computed with the GA with η = 2.0e 5, while the (*) the one computed with the GGA geometric greedy geometric greedy Good

28 Wendland Below: 80 for the Wendland s RBF with scale 1. Left: their separation distances; Right: the (+) are the one computed with the GA with η = 1.0e 1, while the (*) the one computed with the GGA geometric greedy geometric greedy Good

29 constants for the near-optimal for the gaussian. Left: the growth of the data-dependent. Right: the growth of the data-independent constant for Gaussian constant for Gaussian Good

30 constants for the near-optimal for the inverse multiquadrics. Left: the growth of the data-dependent. Right: the growth of the data-independent constant for Inverse Multiquadrics constant for Inverse Multiquadrics Good

31 constants for the near-optimal for the Wendland s rbf. Left: the growth of the data-dependent. Right: the growth of the data-independent constant for Wendland constant for Wendland Good

32 A comparison of constants growth for on the square: RND (random ), EUC (data-independent ), DUB (Dubiner ) constants 1e+15 1e+10 1e+05 RND (2.3) n EUC 4.0 (2.3) n DUB 0.4 n 3 Good degree n

33 f 1 (x, y) = exp( 8 x 2 8 y 2 ) and f 2 (x, y) = x 2 + y 2 xy, on Ω = [ 1, 1]. G-G65 GGA-G65 G-W80 GGA-W80 G-IMQ90 GGA-IMQ90 f f Table: Errors in L 2 -norm for by the Gaussian. When errors are > 1.0 we put. G-G65 GGA-G65 G-W80 GGA-W80 G-IMQ90 GGA-IMQ90 f f Table: Errors in L 2 -norm for by the Wendland s function. Good G-G65 GGA-G65 G-W80 GGA-W80 G-IMQ90 GGA-IMQ90 f f Table: Errors in L 2 -norm for by the inverse multiquadrics.

34 Remarks 1. The GGA is independent on the kernel and generates asymptotically equidistributed optimal sequences. It still inferior to the GA that considers the power function. 2. The computed by the GGA is such that h Xn,Ω = max x Ω min y Xn x y 2. In [Caliari,DeM,Vianello2005], we proved that they are quasi-uniform in the Dubiner metric and connected to Leja sequences. 3. So far,we have no proof of the fact the GGA generates a sequence with h n Cn 1/d, as required by asymptotic optimality. 4. We could look for data-dependent adaptive strategies for reconstruction of functions from spans of translates of kernels using new techniques known from learning theory and algorithms, applying optimization techniques for data selection (proposed by Robert... not yet implemented!). Good

35 Initial ideas Given the recovery process f s f,x, where s f,x = N j=1 f (x j)u j for some u j : Ω R d R we look for bounds of the form s f,x L (Ω) C(X) f l (X). (9) C(X), the stability constant, can be bounded below as N C(X) u j (x) j=1 L (Ω) (10) Good i.e. by the constant Λ X := max x Ω N u j (x). j=1

36 Remarks on Polynomial Interpolation 1. Looking for upper bounds for C(X) and/or Λ X is a classical problem.in recovering by polynomials, upper bounds for the constant exist, leading to the problem of finding near-optimal. 2. For P.I., near-optimal X of cardinality N, have Λ X that, in 1D behaves like log(n) and in 2D on the square like log 2 (N).An important set of near-optimal in the square for P.I., are the Padua [Bos,Caliari,DeM,Vianello,Xu JAT06, NM07], Good

37 Padua 1 0,5 0-0, ,5 0 0, Figure: (Left) Padua for N = 13 and the generating curve. (Right) Padua for N = Good

38 Padua MP EMP PD constants 100 MP (0.7 n+1.0) 2 EMP (0.4 n+0.9) 2 PD (2/π log(n+1)+1.1) 2 Good degree n Figure: (Left): Morrow-Patterson, Extended Morrow-Patterson and Padua for N = 8. (Right) constants growth

39 Stability bounds for multivariate kernel based recovery processes are missing. How can we proceed to derive them? Good

40 Recovering by kernels Given a kernel Φ : Ω Ω R (positive definite), construct N s f,x := α j Φ(,x j ) (11) j=1 from V X := span {Φ(,x) : x X } of translates of Φ so that Good f (x k ) = s f,x (x k ), 1 k N (12) with matrix A Φ,X = (Φ(x k,x j )), 1 j,k N.

41 The matrix A Φ,X 1. Unfortunately the kernel matrix has bad condition number if the data locations come close, i.e. if q X is small. 2. Then, the coefficients of the representation (11) get very large even if the data values f (x k ) are small, and simple linear solvers will fail. Users often report that the final function s f,x of (11) behaves nicely in spite of the large coefficients, and using stable solvers (for instance Riley s algorithm) lead to useful results even in case of unreasonably large condition numbers [Fasshauer s talk] 3. The interpolant can be stably calculated (in the sense of (9)), while the coefficients in the basis supplied by the Φ(x,x j ) are unstable. Good

42 Error estimates and (in)stability 1. h X,Ω and q X are used for standard error and stability estimates for multivariate interpolants. The inequality holds in most cases. q X h X,Ω 2. If of X nearly coalesce, q X can be much smaller than h X,Ω, causing instability of the standard solution process. Point sets X are called quasi uniform with uniformity constant γ > 1, if holds the inequality Good 1 γ q X h X,Ω γq X.

43 Kernels and Fourier transforms To generate interpolants, we allow (conditionally) positive definite translation-invariant kernels Φ(x,y) = K(x y) for all x,y R d, K : R d R which are reproducing in their native Hilbert space N Φ which we assume to be norm equivalent to some Sobolev space W2 τ (Ω) with τ > d/2. The kernel will then have a Fourier transform satisfying 0 < c(1 + ω 2 2) τ ˆK(ω) C(1 + ω 2 2) τ (13) Good at infinity. This includes polyharmonic splines, thin-plate splines, the Sobolev/Matérn kernel, and Wendland s compactly ed kernels.

44 Theorem 1 Theorem The classical constant for with Φ on N = X data locations in a bounded Ω R d has a bound of the form Λ X C ( ) τ d/2 hx,ω N. (14) For quasi-uniform sets, with uniformity bounded by γ < 1, this simplifies to Λ X C N. Each single cardinal function is bounded by u j L (Ω) C q X ( hx,ω q X ) τ d/2, (15) Good which, in the quasi-uniform case, simplifies to u j L (Ω) C.

45 Corollary Corollary Interpolation on sufficiently many quasi uniformly distributed data is stable in the sense of s f,x L (Ω) C ( ) f l (X) + f l2(x) (16) and with a constant C independent of X. s f,x L2(Ω) Ch d/2 X,Ω f l 2(X) (17) In the right-hand side of (17), l2 is a properly scaled discrete version of the L 2 norm. Good Proofs have been done by resorting to classical error estimates. An alternative proof based on sampling inequality [Rieger, Wendland NM05], has been proposed in [Schaback, DeM. RR59-08,UniVR].

46 Proof sketch 1. Bound u j. Using standard error estimates ([Corol ,Wendland s book]), we get ( ) ( ) ( ) xj xj u j L (Ω) 1+ I X Ψ Ψ 1+C h τ d/2 q X q X X,Ω Ψ. L (Ω) q X N (18) Ψ C, having in the unit ball and such that Ψ(0) = 1, Ψ L (Ω) = 1 (i.e. a bump function). 2. Estimate the native space norm of Ψ( q X ) getting ( ) 2 Ψ q X Thus, the estimates easily follow. N C 1 q d τ/2 X Ψ 2 L 2. Good

47 Proof sketch Finally, for the constant, observe that p f,x (x) = ( ) N j=1 f (x x xj j)ψ q X Then I X p f,x L (Ω) p f,x L (Ω) + I X p f,x p f,x L (Ω). p f,x L (Ω) f l (X), since p f,x is a sum of functions with nonoverlapping s. I X p f,x p f,x L (Ω) Ch τ d/2 X,Ω p f,x N. Then, it remains to estimate p f,x N. For τ N, we have p f,x N Cq d 2τ X Ψ W τ 2 ( N ) 1/2 f (x j ) 2 i=1 Good Cq d 2τ X Ψ W τ 2 N f l (X).

48 Kernels 1. Matérn/Sobolev kernel (finite smoothness, definite positive) Φ(r) = (r/c) ν K ν (r/c), of order ν. K ν is the modified Bessel function of second kind. Examples were done with ν = 1.5 at scale c = 20,320. Schaback call them Sobolev splines. 2. Gauss kernel (infinite smoothness, definite positive) Good Φ(r) = e νr, ν > 0. Examples with ν = 1 at scale c = 0.1, 0.2, 0.4.

49 constants against n Interior Corner 10 2 against n Interior Corner Good Figure: constants for the Matérn/Sobolev kernel (left) and Gauss kernel (right)

50 functions Good Figure: Lagrange basis function on 225 data, Gaussian kernel with scale 0.1 (left) and scale 0.2 (right). See how scaling influences the Lagrange basis.

51 functions Good Figure: Gauss kernel with scale 0.4: function on 225 regular. The maximum of the function is attained near the corners for large scales, while the behavior in the interior is as stable as for kernels with limited smoothness.

52 functions Good Figure: Matérn/Sobolev kernel with scale 320. function on 225 scattered (left) and on 225 equidistributed (right).

53 Lagrange basis functions Data Points Figure: Matern/Sobolove kernel with scale 320: Lagrange basis (left) on 225 random (right) Good

54 Lagrange basis functions Good Figure: Lagrange basis (left) and function (right) for 168 scattered data on the circle, Gaussian kernel with scale 0.4

55 Lagrange basis functions Data Points Good Figure: Data for the previous figure

56 constants Here we collect some computed constants on a grid of centers consisting of 225 pts on [ 1, 1] 2. The constants were computed on a finer grid made of 7225 pts. Matérn and Wendland had scaled by 10, IMQ and GA scaled by 0.2. Matern W2 IMQ GA First line contains the max of functions. The second are the estimated constants, by the function computed by the formula [Wendland s book, p. 208] 1 + N (uj (x)) 2 PΦ,X 2 (x) λ min (A Φ,X {x} ), x X. i=1 Good in a neighborhood of the point that maximizes the classical constant.

57 Remarks on the finite smooth case 1. In all examples, our bounds on the constants, are confirmed. 2. In all experiments, the constants seem to be uniformly bounded. 3. The maximum of the function is attained in the interior. Good

58 Remarks on the infinite smoothness... things are moreless specular The constants do not seem to be uniformly bounded. 2. In all experiments, the function attains its maximum near the corners (for large scales). 3. The limit for large scales is called flat limit which corresponds to the Lagrange basis function for polynomial (see Larsson and Fornberg talks, [Driscoll, Fornberg 2002], [Schaback 2005],...). Good

59 A possible solution Schaback, in a recent paper with S. Müller [Müeller, Scahaback JAT08], studied a Newton s basis for overcoming the ill-conditioning of linear systems in RBF. The basis is orthogonal in the native space in which the kernel is reproducing and more stable. Good

60 The case φ(x) = x This is based on the work [Bos,DeM. EJA2008]. Sites x 1 < x 2 < < x n belong to some interval [a, b] Interpolation problem (correct): find coefficients a j R n a j x x j = y j, (19) j=1 for function values y j in two ways 1. solve the linear system with Vandermonde matrix; 2. give formulas for the associated cardinal functions u j. Good

61 Formulas for the cardinal functions The cardinal functions are the classical hat functions given as follows. When x j is an interior point 0 if x x j 1 x x j 1 if x x u j (x) = j x j 1 j 1 < x x j x j+1 x, 2 j n 1, (20) if x x j+1 x j < x x j+1 j 0 if x > x j+1 while for the boundary x 1 and x n u 1 (x) = { x2 x x 2 x 1 if x 1 x x 2 0 if x > x 2 ; 0 if x x n 1 u n(x) = x x n 1 xn x n 1 if x n 1 < x x n. (21) (22) Good

62 Formulas for the cardinal functions These hat functions has another interesting property: u j is a combination of just 3 translates, x x j 1, x x j and x x j+1. This also holds for u 1 and u n identifying x 0 = x n and x n+1 = x 1. For instance, for x j an interior point u j (x) = 1 2(x j x j 1 ) x x x j+1 x j 1 j 1 2(x j+1 x j )(x j x j 1 ) x x j + 1 Good 2(x j+1 x j ) x x j+1 2 j n 1, (23) Remark: (23) is defined for all x R, but is identically zero outside [x j 1,x j+1 ]. The boundary x 1 and x n are again slightly different.

63 The case φ (x) = λ 2 φ(x) Assume λ C. We proved 1. u j still a combination of 3 consecutive translates of φ( x ) and [x j 1,x j+1 ]. 2. uniqueness of this class of functions. λ = 0 is essentially φ(x) = x, then assume λ 0. Hence, for some a,b C. φ(x) = ae λx + be λx (24) Good

64 Formulas for the cardinal functions Observe that the problem for functions of the form n s(x) = a j φ( x x j ) (25) j=1 is correct provided b a and ae λxn ±be λx 1. Theorem For φ(x) of the form (24) we have Good det ([φ( x i x j )] 1 i,j n ) = (b a) n 2 e 2λ n 1 n j=1 x j (e 2λx j+1 e 2λx j ) ( ) b 2 e 2λx 1 a 2 e 2λxn. j=1

65 Formulas for the cardinal functions Proposition For φ(x) of the form (24) with a,b so that the problem is correct, we have for 2 j n 1, u j (x) = A 1 φ( x x j 1 ) + A 2 φ( x x j ) + A 3 φ( x x j+1 ) where e λxj 1 e λx j A 1 = (e 2λx j e 2λx, j 1 )(b a) (e 2λxj+1 e 2λxj 1 )e 2λx j A 2 = (e 2λxj+1 e 2λxj )(e 2λx j e 2λx, j 1 )(b a) e λxj e λx j+1 A 3 = (e 2λxj+1 e 2λxj. )(b a) Good

66 Formulas for the cardinal functions These u j are identically zero outside the interval [x j 1,x j ]. Similar formulas hold for u 1 and u n. Good

67 Cardinal functions nodes u 1 u 2 u 3 u 4 u nodes u 1 u 2 u 3 u 4 Good u Figure: Cardinal functions for the nodes [1, 2, 3.5, 6, 7.5], a = 2, b = 3, and λ = 1 (left), λ = i (right)

68 Uniqueness of the class Theorem Suppose that φ : R + R is analytic. Suppose further that for any x 1 < x 2 < < x n, the cardinal functions for of the form (25) can be given as a linear combination of three consecutive translates, i.e., there exist constants α j, β j and γ j such that u j (x) = α j φ( x x j 1 ) + β j φ( x x j ) + γ j φ( x x j+1 ), 2 j n 1. Suppose further that u j has in the interval [x j 1,x j+1 ]. Then there exists a λ C such that Good φ (x) = λ 2 φ(x).

69 Formulas for the cardinal functions Theorem Suppose that x 1 < x 2 < x n and that φ(x) = ae λx + be λx is such that the problem is correct. Then, independently of the values of a and b, e 2λx e 2λx j 1 if x [x u j (x) = e λ(x e 2λx j e 2λx j 1 j 1,x j ] j x) e 2λx e 2λx j+1 if x [x e 2λxj e 2λx j+1 j,x j+1 ] 2 j n 1, 0 otherwise and similarly for u 1, u n. Good

70 The constant Since u j are positive functions Proposition Suppose that x 1 < x 2 < x n and that φ(x) = ae λx +be λx for λ R, is such that the problem is correct. Then, independently of the values of a and b, n j=1 In particular, u j (x) = eλx + e λ(x j+x j+1 x) e λx j + e λx, x [x j+1 j,x j+1 ]. max x 1 x x n n u j (x) = 1. j=1 Good

71 The Case of λ Complex Consider λ = i with a = i/2 and b = i/2 so that g(x) = sin(x). If we make the restriction that x n x 1 < π, one can prove that problem is correct.it follows u j (x) 0 on [x 1,x n ] with x n x 1 < π. n u j (x) = cos(x x j+x j+1 cos( x j+1 x j 2 ) j=1 2 ), x [x j,x j+1 ]. The maximum is clearly attained at the midpoint x = (x j + x j+1 )/2 at which Good n u j (x) = j=1 1 cos( x j+1 x j 2 ).

72 The Case of λ Complex Hence Λ n := max x 1 x x n n u j (x) j=1 1 = max 1 j n 1 cos( x j+1 x j 2 ) 1 = x cos(max j+1 x j 1 j n 1 2 ). (26) Good

73 The Case of λ Complex Theorem Suppose that φ(x) = sin(x). Then, among all distributions of a = x 1 < x 2 < < x n = b in the interval [a,b] with b a < π, the one for which Λ n is uniquely minimized is the equally spaced one, i.e, for x j = a + (j 1)(b a), 1 j n. (n 1) Good

74 The Case of λ Complex Good Figure: functions for λ = i and equally spaced (Left) and non-equally spaced [ ] (Right)

75 Work to do 1. In 1d, we studied the case φ(x) = x 3 and discovered, for nearly equidistributed point set, a behavior similar to that of periodic cubic splines(?) [F. Schurer, Indag. Math.30 (1968)] giving Λ n < 1 4 ( ). More investigations are then necessary! 2. Study better the behavior of the cardinal functions u j : why do they concentrate around x j and decay at infinity? 3. Efficient computations (for overtaking ill-conditioning and instability) using Nick s Trefethen definition 10 digits, 5 sec. and 1 page!). Good

76 Important references Bos L. and S., Univariate Radial Basis Functions with Compact Support Cardinal Functions, East J. Approx., Vol. 14(1) 2008, De Boor, C. and Ron A., The least solution for the polynomial problem, Math. Z., Vol , Driscoll T. and Fornberg B, Interpolation in the limit of increasingly flat radial basis functions, Computers Math. Appl , Nicholas J. Higham, Accuracy and Numerical Algorithms, SIAM, Philadelphia, 1996 S. Jokar and B. Meheri, function for multivariate by RBF, Appl. Math. Comp. 187(1) 2007, S. Müller and R. Schaback, A Newton basis for Kernel Spaces, To appear on J. Approx. Theory 2008, available at R. Schaback s home page. H. Wendland, Scattered Data Approximation, Cambridge Monographs on Applied and Computational Mathematics, Vol. 17, Good H. Wendland, C. Rieger, Approximate with applications to selecting smoothing parameters,, Num. Math ,

77 DWCAA09 First announcement 2nd Dolomites Workshop on Constructive Approximation and Applications Alba di Canazei, 3-9 Sept Keynote speakers (confirmed so far!): Carl de Boor, Robert Schaback, Nick Trefethen, Holger Wendland, Yuan Xu Sessions on: Polynomial and rational approximation (Org.: J. Carnicer, A. Cuyt), Approximation by radial bases (Org.: A. Iske, J. Levesley), Quadrature and cubature (Org. B. Bojanov, E. Venturino, Approximation in linear algebra (Org. C. Brezinski, M. Eiermann). Good

78 Thank you for your attention! Good