Variably scaled kernels M. Bozzini jointed with L. Lenarduzzi, M. - PowerPoint PPT Presentation

Variably scaled kernels Variably scaled kernels M. Bozzini jointed with L. Lenarduzzi, M. Rossini, R. Schaback Maia 2013, Erice, September 25-30, 2013

Variably scaled kernels Native Spaces Background See e.g [Buhmann 2003], [Fasshauer 2007], [Wendland 2005] Native Spaces Interpolation Scale parameter Let H be an Hilbert space and Ω ⊂ R d . Variably scaled kernels Introduction ◮ The function Definition K : Ω × Ω → R Numerical Examples Improving the stability Reproduction quality is called reproducing kernel for H if K ( x, · ) ∈ H ∀ x ∈ Ω , and f ( x ) = ( f, K ( x, · )) H for all x ∈ Ω , f ∈ H. ◮ The kernel is positive definite, if for all choices of sets of knots { x 1 , . . . , x N ∈ Ω } , the kernel matrices with elements K ( x i , x j ) , 1 ≤ i, j ≤ N are positive definite.

Variably scaled kernels Background Native Spaces Interpolation ◮ For K positive definite, we define the native space Scale parameter Variably scaled kernels Introduction H ( K, Ω) = span { K ( x, · ) , x ∈ Ω } . Definition Numerical Examples Improving the stability ◮ If the kernel is radial , i.e. of the form Reproduction quality K ( x, y ) = φ ( � x − y � 2 ) for a scalar function φ : [0 , ∞ ) → R , the function φ is called a radial basis function .

Variably scaled kernels Interpolation Given a set Background X = { x 1 , . . . , x N } ⊂ Ω , Native Spaces Interpolation and the associated values Scale parameter Variably scaled kernels f = [ f ( x 1 ) , . . . , f ( x n )] T , Introduction Definition the goal is to find a continuous function Numerical Examples Improving the stability P f : R d → R Reproduction quality such that P f ( x i ) = f ( x i ) , i = 1 , . . . , N. In the RBF literature N � P f ( x ) = a i φ ( � x − x i � ) , i =1 where the coefficients are the solution of the linear system a = A − 1 f, and A ij = φ ( � x i − x j � ) .

Variably scaled kernels Properties Background Native Spaces Interpolation Scale parameter Variably scaled kernels Introduction Definition ◮ Numerical Examples Improving the stability P f = argmin {� s � H : s ∈ H , s ( x i ) = f ( x i ) , i = 1 , . . . , N } Reproduction quality ◮ � f − P f � H ≤ � f − s � H , N � s ∈ H X = { α i φ ( � x − x i � ) , x i ∈ X } i =1

Variably scaled kernels Scale parameter Background Native Spaces Interpolation Scale parameter Variably scaled kernels Fixed a positive number c Introduction Definition Numerical Examples x, y ∈ R d . K ( x, y ; c ) := K ( x/c, y/c ) Improving the stability Reproduction quality [Franke 82] √ c = 0 . 8 N , D D is the diameter of the smallest circle containing all data points. [Rippa 1999 ],[Fasshauer et al. 2007]

Variably scaled kernels Background Native Spaces Interpolation Scale parameter ◮ Variably scaled kernels K ( � x − x j � Introduction ) Definition c j Numerical Examples Improving the stability [Hardy 71], [Kansa 1990, 1992, 2000, 2006] Reproduction quality [B., Lenarduzzi, Schaback 2002], [B., Lenarduzzi, Rossini, Schaback 2004] [Fornberg and Zuev 2007] ◮ � x 1 − y 1 � , . . . , x d − y d K c 1 c d [B., Lenarduzzi 2005], [Fasshauer 2012]

Variably scaled kernels Introduction Background Native Spaces Interpolation Scale parameter Variably scaled kernels Given a domain Ω ⊂ R d , we consider a bijective map Introduction Definition Numerical Examples C : Ω �→ C (Ω) . Improving the stability Reproduction quality Given a kernel K : Ω × Ω → R and the map C, the kernel K C ( C ( x ) , C ( y )) := K ( x, y ) for all x, y ∈ Ω acts on C (Ω) and inherits the definiteness properties of K .

Variably scaled kernels Background Native Spaces Interpolation This gives rise to two native spaces and their properties, i.e. Scale parameter ◮ the space H ( K, Ω) Variably scaled kernels Introduction Definition f ( x ) = ( f, K ( x, · )) Numerical Examples Improving the stability K ( x, y ) = ( K ( x, · ) , K ( y, · )) Reproduction quality for all x, y ∈ Ω . ◮ The space H C ( K C , C (Ω)) g ( u ) = ( g, K C ( u, · )) H C K C ( u, v ) = ( K C ( u, · ) , K C ( v, · )) H C for all u, v ∈ C (Ω) .

We indicate by C the map Variably scaled kernels C : f on Ω → g on C (Ω) Background Native Spaces Interpolation such that Scale parameter g ( C ( x )) = ( C f )( C ( x )) := f ( x ) . Variably scaled kernels Introduction Definition Furthermore, Numerical Examples Improving the stability Reproduction quality C ( K ( · , y ))( C ( x )) := K ( x, y ) = K C ( C ( x ) , C ( y )) . The map C is linear. ◮ The two native spaces H and H C are isometric: ( K ( x, · ) , K ( y, · )) H = K ( x, y ) = K C ( C ( x ) , C ( y )) = ( K C ( C ( x ) , · ) , K C ( C ( y ) , · )) H C It follows that ( f, g ) H = ( C f, C g ) H C .

Variably scaled kernels Summarizing Background Native Spaces Interpolation Scale parameter Let Variably scaled kernels Introduction Definition C : Ω �→ C (Ω) . Numerical Examples Improving the stability Reproduction quality be a bijective map and K : Ω × Ω → R a positive definite kernel. ◮ We introduce the transformed kernel K C ( C ( x ) , C ( y )) which ineherits the definitness property of K . ◮ The native spaces H ( K, Ω) , H C ( K C , C (Ω)) are isometric.

Variably scaled kernels Definition and Properties Background Native Spaces Interpolation Scale parameter Let C : x ∈ Ω ⊂ R d �→ ( x, c ( x )) ∈ C (Ω) ⊂ R d +1 Variably scaled kernels Introduction Definition where Numerical Examples c : R d �→ (0 , ∞ ) . Improving the stability Reproduction quality ◮ Let K a positive definite kernel on R d +1 x, y ∈ R d . K c ( x, y ) := K (( x, c ( x )) , ( y, c ( y ))) K c is the Variably scaled kernel ◮ Since K is positive definite on the submanifold, so is K c . ◮ If K and c are continuous, so is K c .

Variably scaled kernels Background Therefore, given the set X := { x 1 , . . . , x N } on R d , the matrix Native Spaces Interpolation Scale parameter Variably scaled kernels A c,X := ( K c ( x i , x j )) 1 ≤ i,j ≤ N Introduction Definition is non singular and the interpolant is Numerical Examples Improving the stability Reproduction quality N N � � P f ( x ) := a j K c ( x, x j ) = a j K (( x, c ( x )) , ( x j , c ( x j ))) . j =1 j =1 If the kernel is radial, i.e. K ( x, y ) = φ ( � x − y � 2 2 ) , the interpolant is N � a j φ ( � x − x j � 2 2 + ( c ( x ) − c ( x j )) 2 ) . P f := j =1

Variably scaled kernels Examples If K is a power kernel φ ( r ) = r β , the interpolants take the form Background Native Spaces Interpolation N Scale parameter 2 + ( c ( x j ) − c ( x )) 2 � β/ 2 � � � x j − x � 2 P f ( x ) := a j Variably scaled kernels Introduction j =1 Definition Numerical Examples Improving the stability and are identical to power interpolants if the scale function c ( x ) is Reproduction quality constant, otherwise similar to scaled multiquadrics. If K is the Gaussian. N � a j exp( −� x j − x � 2 2 − ( c ( x j ) − c ( x )) 2 ) P f ( x ) = j =1 N � a j exp( −� x j − x � 2 2 ) exp ( − ( c ( x j ) − c ( x )) 2 ) = j =1 which can be seen as a superposition of Gaussians of the same scale but with varying amplitudes for evaluation.

Variably scaled kernels Background We observe that Native Spaces Interpolation ◮ the analysis of error and stability of the varying–scale problem in Scale parameter R d coincides with the analysis of a fixed–scale problem on a Variably scaled kernels Introduction submanifold in R d +1 . Definition Numerical Examples ◮ In particular, let Ω be a compact set and C be a diffeomorphism Improving the stability Reproduction quality between Ω and C (Ω) , then C (Ω) is compact. As usual, we consider the fill distance h ( X, Ω) := sup min x ∈ X � x − y � 2 y ∈ Ω and the separation distance q ( X ) := X ∋ x � = y ∈ X � x − y � 2 min

Variably scaled kernels Background Native Spaces Interpolation Scale parameter Variably scaled kernels Then Introduction q ( C (Ω)) = min � C ( x ) − C ( y ) � 2 Definition Numerical Examples Improving the stability and Reproduction quality � C ( x ) − C ( y ) � 2 � x − y � 2 2 + ( c ( x ) − c ( y )) 2 = 2 � x − y � 2 2 (1 + L ) 2 ≤ � C ( x ) − C ( y ) � 2 � x − y � 2 ≥ 2 2 L is a constant related to the norm of the gradient of c. It follows that the separation distance never decreases.

Variably scaled kernels Numerical examples Background Native Spaces Interpolation Scale parameter Variably scaled kernels Introduction Definition We now provide some examples that show the different roles of the Numerical Examples variable scale parameter: it may affect both the stability and the Improving the stability Reproduction quality accuracy. ◮ Its appropriate choice enhances stability, ◮ one can significantly improve the recovery quality, in particular by preserving shape properties in a much better way than for interpolation with constant scale.