Geometric greedy and greedy points for RBF interpolation Stefano De Marchi Department of Computer Science, University of Verona (Italy) CMMSE09, Gijon July 2, 2009 Stefano De Marchi, Department of Computer Science, University of Verona (Italy) Geometric greedy and greedy points for RBF interpolation 1/48
Motivations Stability is very important in numerical analysis: desirable in numerical computations; it depends on the accuracy of algorithms [3, Higham’s book]. In polynomial interpolation, the stability of the process can be measured by the so-called Lebesgue 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 d ), which also estimates the interpolation error. The Lebesgue constant depends on the interpolation points (via the fundamental Lagrange polynomials). Are these ideas applicable also in the context of RBF interpolation ? Stefano De Marchi, Department of Computer Science, University of Verona (Italy) Geometric greedy and greedy points for RBF interpolation 3/48
Outline Motivations Good interpolation points for RBF Results Greedy and geometric greedy algorithms Numerical examples Asymptotic points distribution Lebesgue Constants estimates References DeM: RSMT03; DeM, Schaback, Wendland: AiCM05; DeM, Schaback: AiCM09; DeM: CMMSE09 Stefano De Marchi, Department of Computer Science, University of Verona (Italy) Geometric greedy and greedy points for RBF interpolation 4/48
Notations X = { x 1 , ..., x N } ⊆ Ω ⊆ R d , distinct; data sites ; { f 1 , ..., f N } , data values ; Φ : Ω × Ω → R symmetric (strictly) positive definite kernel the RBF interpolant N � s f , X := 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. N � s f , X = f ( x j ) u j . (2) j =1 Stefano De Marchi, Department of Computer Science, University of Verona (Italy) Geometric greedy and greedy points for RBF interpolation 6/48
Error estimates Take V Ω = span { Φ( · , x ) : x ∈ Ω } on which Φ is the reproducing kernel: V Ω := N Φ (Ω), the native Hilbert space associated 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 the generic pointwise error estimate (cf. e.g., Fasshauer’s book, p. 117-118): | f ( x ) − s f , X ( x ) | ≤ P Φ , X ( x ) � f � N Φ (Ω) (3) P Φ , X : power function . Stefano De Marchi, Department of Computer Science, University of Verona (Italy) Geometric greedy and greedy points for RBF interpolation 7/48
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 ) , (4) det Φ , X ( x 1 , . . . , x N ) Letting u j ( x ) , 0 ≤ j ≤ N with u 0 ( x ) := − 1 and x 0 = x , then � u T ( x ) A Φ , Y u ( x ) , P Φ , X ( x ) = (5) where u T ( x ) = ( − 1 , u 1 ( x ) , . . . , u N ( x )) , Y = X ∪ { x } . Stefano De Marchi, Department of Computer Science, University of Verona (Italy) Geometric greedy and greedy points for RBF interpolation 8/48
The problem Are there any good points for approximating all functions in the native space? Stefano De Marchi, Department of Computer Science, University of Verona (Italy) Geometric greedy and greedy points for RBF interpolation 9/48
Our approach 1. Power function estimates. 2. Geometric arguments. Stefano De Marchi, Department of Computer Science, University of Verona (Italy) Geometric greedy and greedy points for RBF interpolation 10/48
Overview of the existing literature A. Beyer: Optimale Centerverteilung bei Interpolation mit radialen Basisfunktionen. Diplomarbeit , Universit¨ at G¨ ottingen, 1994. He considered numerical aspects of the problem . L. P. Bos and U. Maier: On the asymptotics of points which maximize determinants of the form det( g ( | x i − x j | )), in Advances in Multivariate Approximation (Berlin, 1999), They investigated on Fekete-type points for univariate RBFs, proving that if g is s.t. g ′ (0) � = 0 then points that maximize the Vandermonde determinant are the ones asymptotically equidistributed. Stefano De Marchi, Department of Computer Science, University of Verona (Italy) Geometric greedy and greedy points for RBF interpolation 11/48
Literature A. Iske: , Optimal distribution of centers for radial basis function methods. Tech. Rep. M0004, Technische Universit¨ at M¨ unchen, 2000. He studied admissible sets of points by varying the centers for stability and quality of approximation by RBF, proving that uniformly distributed points 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 interpolation , SINUM (2005), they used potential theory for finding near-optimal points for gaussians in 1d. Stefano De Marchi, Department of Computer Science, University of Verona (Italy) Geometric greedy and greedy points for RBF interpolation 12/48
Main result Idea: data sets, 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, AiCM05]. 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 for all f ∈ W β 2 ( R d ) , � f − s f , X � L ∞ (Ω) ≤ ǫ � f � Φ , (6) then the fill distance of X satisfies 1 α − d / 2 . h X , Ω ≤ M α ǫ (7) Stefano De Marchi, Department of Computer Science, University of Verona (Italy) Geometric greedy and greedy points for RBF interpolation 13/48
Remarks 1. The interpolation error can be bounded in terms of the fill-distance (cf. e.g., Fasshauer’s book, p. 121): � f − s f , X � L ∞ (Ω) ≤ C h β − d / 2 � f � W β 2 ( R d ) . (8) X , Ω provided h X , Ω ≤ h 0 , for some h 0 2. M α → ∞ when α → β , so from (8) we cannot get h β − d / 2 ≤ C ǫ but as close as possible. X , Ω 3. The proof does not work for gaussians (no compactly supported functions in the native space of the gaussians). Stefano De Marchi, Department of Computer Science, University of Verona (Italy) Geometric greedy and greedy points for RBF interpolation 14/48
To remedy, we made the additional assumption that X is already quasi-uniform, i.e. h X , Ω ≈ q X . 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. Stefano De Marchi, Department of Computer Science, University of Verona (Italy) Geometric greedy and greedy points for RBF interpolation 15/48
On computing near-optimal points We studied two algorithms. 1. Greedy Algorithm (GA) 2. Geometric Greedy Algorithm (GGA) Stefano De Marchi, Department of Computer Science, University of Verona (Italy) Geometric greedy and greedy points for RBF interpolation 17/48
The Greedy Algorithm (GA) At each step we determine a point where the power function attains its maxima w.r.t. the preceding set. That is, starting step: X 1 = { x 1 } , x 1 ∈ Ω , arbitrary . iteration step: X j = X j − 1 ∪ { x j } with P Φ , X j − 1 ( x j ) = � P Φ , X j − 1 � L ∞ (Ω) . Remark: practically we maximize over some very large discrete set X ⊂ Ω instead of Ω. Stefano De Marchi, Department of Computer Science, University of Verona (Italy) Geometric greedy and greedy points for RBF interpolation 18/48
The Geometric Greedy Algorithm (GGA) The points 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 \ X n dist( x , X n ). Then, form X n +1 := X n ∪ { x n +1 } . Notice: this algorithm works quite well for subset Ω of cardinality n with small h X , Ω and large q X . Stefano De Marchi, Department of Computer Science, University of Verona (Italy) Geometric greedy and greedy points for RBF interpolation 19/48
On the convergence of GA e GGA Experiments showed that the GA fills the currently largest hole in the data set 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 := q X j = 1 2 min x � = y ∈ X j � x − y � 2 and the fill distance as h j := h X j , Ω = max x ∈ Ω min y ∈ X j � x − y � 2 then, we proved that h j ≥ q j ≥ 1 2 h j − 1 ≥ 1 ∀ j ≥ 2 2 h j , i.e. the GGA produces point sets quasi-uniformly distributed in the euclidean metric. Stefano De Marchi, Department of Computer Science, University of Verona (Italy) Geometric greedy and greedy points for RBF interpolation 20/48
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