Stability and Lebesgue constants in Good interpolation points RBF - PowerPoint PPT Presentation

Stability and Lebesgue constants in RBF interpolation De Marchi Motivations Stability and Lebesgue constants in Good interpolation points RBF interpolation Results Numerical examples Stability of kernel-based interpolation Stefano De Marchi 1 Results Numerical examples 1D RBF having u j 1 Dept. of Computer Science, University of Verona with compact support http://www.sci.univr.it/~demarchi Future work References G¨ ottingen, 20 September 2008

Stability and Outline Lebesgue constants in RBF interpolation De Marchi Motivations Motivations Good interpolation points Good interpolation points Results Results Numerical examples Numerical examples Stability of kernel-based interpolation Stability of kernel-based interpolation Results Numerical examples Results 1D RBF having u j Numerical examples with compact support Future work 1D RBF having u j with compact support References Future work References

Stability and Motivations Lebesgue constants in RBF interpolation De Marchi Motivations ◮ Stability is very important in numerical analysis: Good interpolation desirable in numerical computations, it depends on the points Results accuracy of algorithms [4, Higham’s book]. Numerical examples ◮ In polynomial interpolation, the stability of the process Stability of kernel-based interpolation can be measured by the so-called Lebesgue constant, i.e Results the norm of the projection operator from C ( K ) Numerical examples 1D RBF having u j (equipped with the uniform norm) to P n ( K ) (or itselfs) with compact ( K ⊂ R n ), which estimates also the interpolation error. support Future work ◮ The Lebesgue constant depends on the interpolation References points via the fundamental Lagrange or cardinal polynomials.

Stability and Our approach Lebesgue constants in RBF interpolation De Marchi Motivations Good interpolation points Results Numerical examples 1. Good interpolation points [DeM. RSMT03; DeM. Stability of Schaback Wendland AiCM05]. kernel-based interpolation 2. Cardinal functions bounds [DeM. Schaback AiCM08] . Results Numerical examples 3. Lebesgue constants estimates and growth [DeM. 1D RBF having u j with compact Schaback AiCM08; Bos DeM. EJA08 (1d)]. support Future work References

Stability and Notations Lebesgue constants in RBF ◮ X = { x 1 , ..., x N } ⊆ Ω ⊆ R d , distinct; data sites ; interpolation ◮ { f 1 , ..., f N } , data values ; De Marchi ◮ Φ : Ω × Ω → R symmetric positive definite kernel Motivations the RBF interpolant Good interpolation points Results Numerical examples Stability of kernel-based N interpolation � s f , Φ := α j Φ( · , x j ) , (1) Results Numerical examples j =1 1D RBF having u j with compact support Letting V X = span { Φ( · , x ) : x ∈ X } , s f , X can be written in Future work terms of cardinal functions, u j ∈ V X , u j ( x k ) = δ jk , i.e. References N � s f , X = f ( x j ) u j . (2) j =1

Stability and Error estimates Lebesgue constants in RBF interpolation De Marchi ◮ Take V Ω := span { Φ( · , x ) : x ∈ Ω } on which Φ is the Motivations Good interpolation reproducing kernel. clos ( V Ω ) = N Φ (Ω), the native points Hilbert space to Φ. Results Numerical examples ◮ f ∈ N Φ (Ω), using (2) and the reproducing kernel Stability of kernel-based property of Φ on V Ω , applying the Cauchy-Schwarz interpolation Results inequality, we get Numerical examples 1D RBF having u j with compact support Future work | f ( x ) − s f , X ( x ) | ≤ P Φ , X ( x ) � f � Φ (3) References P Φ , X : power function.

Stability and A power function expression Lebesgue constants in RBF interpolation De Marchi Letting det ( A Φ , X ( y 1 , ..., y N )) = det (Φ( y i , x j )) 1 ≤ i , j ≤ N , then Motivations Good interpolation points Results u k ( x ) = det Φ , X ( x 1 , . . . , x k − 1 , x , x k +1 , . . . , x N ) Numerical examples , (4) Stability of det Φ , X ( x 1 , . . . , x N ) kernel-based interpolation Results Numerical examples Letting u j ( x ) , 0 ≤ j ≤ N with u 0 ( x ) := − 1 and x 0 = x , then 1D RBF having u j with compact support Future work P 2 Φ , X ( x ) = u T A Φ , Y u , References (5) where u T = ( − 1 , u 1 ( x ) , . . . , u N ( x )) , Y = X ∪ { x } .

Stability and The problem Lebesgue constants in RBF interpolation De Marchi Motivations Good interpolation points Results Numerical examples Stability of Are there any good points for kernel-based interpolation Results approximating all functions in the native Numerical examples 1D RBF having u j space? with compact support Future work References

Stability and Our approach Lebesgue constants in RBF interpolation De Marchi Motivations Good interpolation points Results Numerical examples Stability of 1. Power function estimates. kernel-based interpolation Results 2. Geometric arguments. Numerical examples 1D RBF having u j with compact support Future work References

Stability and Literature Lebesgue constants in RBF interpolation De Marchi ◮ A. Beyer: Optimale Centerverteilung bei Interpolation Motivations mit radialen Basisfunktionen. Diplomarbeit , Universit¨ at Good interpolation points G¨ ottingen, 1994. Results Numerical examples He considered numerical aspects of the problem . Stability of kernel-based ◮ L. P. Bos and U. Maier: On the asymptotics of points interpolation Results which maximize determinants of the form Numerical examples det( g ( | x i − x j | )), in Advances in Multivariate 1D RBF having u j with compact Approximation (Berlin, 1999), support They investigated on Fekete-type points for univariate Future work RBFs, proving that if g is s.t. g ′ (0) � = 0 then points References that maximize the Vandermonde determinant are the ones asymptotically equidistributed.

Stability and Literature Lebesgue constants in RBF interpolation De Marchi ◮ A. Iske:, Optimal distribution of centers for radial basis Motivations function methods. Tech. Rep. M0004, Technische Good interpolation points Universit¨ at M¨ unchen, 2000. Results He studied admissible sets of points by varying the Numerical examples Stability of centers for stability and quality of approximation by kernel-based interpolation RBF, proving that uniformly distributed points gives Results Numerical examples better results. He also provided a bound for the 1D RBF having u j � so-called uniformity : ρ X , Ω ≤ 2( d + 1) / d , d = space with compact support dimension. Future work ◮ R. Platte and T. A. Driscoll:, Polynomials and potential References theory for GRBF interpolation , SINUM (2005),they used potential theory for finding near-optimal points for gaussians in 1d.

Stability and Main result Lebesgue constants in RBF interpolation De Marchi Idea: data set for good approximation for all f ∈ N Φ (Ω) should have regions in Ω without large holes. Motivations Assume Φ, translation invariant, integrable and its Fourier Good interpolation points transform decays at infinity with β > d / 2 Results Numerical examples Theorem Stability of kernel-based [DeM., Schaback&Wendland, AiCM 2005.] For every α > β interpolation Results there exists a constant M α > 0 with the following property: if Numerical examples ǫ > 0 and X = { x 1 , . . . , x N } ⊆ Ω are given such that 1D RBF having u j with compact support for all f ∈ W β 2 ( R d ) , � f − s f , X � L ∞ (Ω) ≤ ǫ � f � Φ , (6) Future work References then the fill distance of X satisfies 1 α − d / 2 . h X , Ω ≤ M α ǫ (7)

Stability and Remarks Lebesgue constants in RBF interpolation De Marchi Motivations 1. The interpolation error can be bounded by Good interpolation points Results � f − s f , X � L ∞ (Ω) ≤ C h β − d / 2 Numerical examples � f � W β 2 ( R d ) . (8) X , Ω Stability of kernel-based interpolation Results Numerical examples 2. M α → ∞ when α → β , so from (8) we cannot get 1D RBF having u j h β − d / 2 ≤ C ǫ but as close as possible. with compact X , Ω support 3. The proof does not work for gaussians (no compactly Future work supported functions in the native space of the References gaussians).

Stability and Lebesgue constants in RBF interpolation To remedy, we made the additional assumption that De Marchi Motivations X is already quasi-uniform ,i.e. h X , Ω ≈ q X , Ω . Good interpolation points Results ◮ As a consequence, P Φ , X ( x ) ≤ ǫ . The result follows from Numerical examples Stability of the lower bounds of P Φ , X (cf. [Schaback AiCM95] kernel-based interpolation where they are given in terms of q X ). Results Numerical examples ◮ Quasi-uniformity brings back to bounds in term of h X , Ω . 1D RBF having u j with compact support Future work Observation : optimally distributed data sites are sets that References cannot have a large region in Ω without centers, i.e. h X , Ω is sufficiently small.

Stability and On computing near-optimal points Lebesgue constants in RBF interpolation De Marchi Motivations Good interpolation points Results Numerical examples We studied two algorithms. Stability of kernel-based interpolation 1. Greedy Algorithm (GA) Results Numerical examples 2. Geometric Greedy Algorithm (GGA) 1D RBF having u j with compact support Future work References