New developments on rational RBF Stefano De Marchi Department of Mathematics “Tullio Levi-Civita” University of Padova Montecatini, 14th February 2018
Outline From RBF to Rational RBF (RRBF) 1 2 Eigen-rational interpolant 3 Numerical experiments Lebesgue functions and constants Errors Landmark-based Image registration Progetto GNCS 2016-17 4 2 of 33
From RBF to Rational RBF (RRBF) work with A. Martinez and E. Perracchione 3 of 33
Notations Data: Ω ⊂ R d , X ⊂ Ω , test function f 1 X N = { x 1 , . . . , x N } ⊂ Ω , f = { f 1 , . . . , f N } , where f i = f ( x i ) 4 of 33
Notations Data: Ω ⊂ R d , X ⊂ Ω , test function f 1 X N = { x 1 , . . . , x N } ⊂ Ω , f = { f 1 , . . . , f N } , where f i = f ( x i ) Approximation setting: 2 φ ( ε · ) : Conditionally Positive Definite (CPD) of order ℓ or Strictly Positive Definite (SPD) and radial ( ε , shape parameter) name φ ℓ globally supported: e − ε 2 r 2 Gaussian C ∞ (GA) 0 Generalized Multiquadrics C ∞ (GM) ( 1 + r 2 /ε 2 ) 3 / 2 2 locally supported: name φ ℓ Wendland C 2 (W2) ( 1 − ε r ) 4 + ( 4 ε r + 1 ) 0 Buhmann C 2 (B2) 2 r 4 log r − 7 / 2 r 4 + 16 / 3 r 3 − 2 r 2 + 1 / 6 0 4 of 33
Notations Data: Ω ⊂ R d , X ⊂ Ω , test function f 1 X N = { x 1 , . . . , x N } ⊂ Ω , f = { f 1 , . . . , f N } , where f i = f ( x i ) Approximation setting: 2 φ ( ε · ) : Conditionally Positive Definite (CPD) of order ℓ or Strictly Positive Definite (SPD) and radial ( ε , shape parameter) name φ ℓ globally supported: e − ε 2 r 2 Gaussian C ∞ (GA) 0 Generalized Multiquadrics C ∞ (GM) ( 1 + r 2 /ε 2 ) 3 / 2 2 locally supported: name φ ℓ Wendland C 2 (W2) ( 1 − ε r ) 4 + ( 4 ε r + 1 ) 0 Buhmann C 2 (B2) 2 r 4 log r − 7 / 2 r 4 + 16 / 3 r 3 − 2 r 2 + 1 / 6 0 kernel notation K ε ( · , · ) native space N K (Ω) (where K is the reproducing kernel) finite subspace N K ( X N ) = span { K ( · , x ) : x ∈ X N } ⊂ N K (Ω) 4 of 33
RBF Interpolation Given Ω , X N , f , K Aim Find P f ∈ N K ( X N ) s.t. ( P f ) X N = f 5 of 33
RBF Interpolation Given Ω , X N , f , K Aim Find P f ∈ N K ( X N ) s.t. ( P f ) X N = f N � Classical interpolant: P f ( x ) = α k K ( x , x k ) , x ∈ Ω , x k ∈ X N . k = 1 [Hardy and Gofert 1975] used multiquadrics � K ( x , y ) = 1 + ǫ 2 � x − y � 2 . 5 of 33
RBF Interpolation Given Ω , X N , f , K Aim Find P f ∈ N K ( X N ) s.t. ( P f ) X N = f N � Classical interpolant: P f ( x ) = α k K ( x , x k ) , x ∈ Ω , x k ∈ X N . k = 1 [Hardy and Gofert 1975] used multiquadrics � K ( x , y ) = 1 + ǫ 2 � x − y � 2 . � N k = 1 α k K ( x , x k ) P f ( x ) = P f ( x ) Rescaled interpolant: ˆ P g ( x ) = where P g � N k = 1 β k K ( x , x k ) is the kernel interpolant of g ( x ) = 1 , ∀ x ∈ Ω . Localized Rescaled and exactness on constants in [Deparis et al 2014]. In [DeM et al 2017] it is shown that it is a Shepard’s PU method. Linear convergence of localized rescaled interpolants [DeM and Wendland, draft 2017] 5 of 33
Rational RBF definition � N R ( x ) = R ( 1 ) ( x ) k = 1 α k K ( x , x k ) R ( 2 ) ( x ) = � N k = 1 β k K ( x , x k ) [Jackbsson et al. 2009, Sarra and Bai 2017] = ⇒ RRBFs well approximate data with steep gradients or discontinuites [rational with PU+VSK in DeM et al. 2017]. 6 of 33
Rational RBF Learning from rational functions, d = 1 polynomial case. a m x m + · · · + a 0 x 0 r ( x ) = p 1 ( x ) p 2 ( x ) = . x n + b n − 1 x n − 1 · · · + b 0 M = m + n + 1 unknowns (Pad´ e approximation). If M < N to get the coefficients we may solve the LS problem N 2 � � � min � f ( x k ) − r ( x k ) . � � � p 1 ∈ Π 1 m , p 2 ∈ Π 1 n k = 1 7 of 33
Rational RBF Learning from rational functions, d = 1 polynomial case. a m x m + · · · + a 0 x 0 r ( x ) = p 1 ( x ) p 2 ( x ) = . x n + b n − 1 x n − 1 · · · + b 0 M = m + n + 1 unknowns (Pad´ e approximation). If M < N to get the coefficients we may solve the LS problem N 2 � � � min � f ( x k ) − r ( x k ) . � � � p 1 ∈ Π 1 m , p 2 ∈ Π 1 n k = 1 RBF case. Let X m = { x k , . . . , x k + m − 1 } , X n = { x j , . . . , x j + n − 1 } ⊂ X N be non empty, such that m + n ≤ N � k + m − 1 α i 1 K ( x , x i 1 ) R ( x ) = R ( 1 ) ( x ) i 1 = k R ( 2 ) ( x ) = (1) , � j + n − 1 β i 2 K ( x , x i 2 ) i 2 = j provided R ( 2 ) ( x ) � 0 , for all x ∈ Ω . 7 of 33
Rational RBF Find the coefficients: I [Jackobsson et al 2009] proved the well-posedness of the interpolation on X N via R ( x ) = R ( 1 ) ( x ) � N i = 1 α i K ( x , x i ) R ( 2 ) ( x ) = , (2) � N k = 1 β k K ( x , x k ) 8 of 33
Rational RBF Find the coefficients: I [Jackobsson et al 2009] proved the well-posedness of the interpolation on X N via R ( x ) = R ( 1 ) ( x ) � N i = 1 α i K ( x , x i ) R ( 2 ) ( x ) = , (2) � N k = 1 β k K ( x , x k ) Letting ξ = ( α T , β T ) ∈ R 2 N and B the N × 2 N matrix K ( x 1 , x 1 ) K ( x 1 , x N ) − f 1 K ( x 1 , x 1 ) − f 1 K ( x 1 , x N ) · · · · · · . . . B = . . . . . . . K ( x N , x 1 ) · · · K ( x N , x N ) − f N K ( x N , x 1 ) · · · − f N K ( x N , x N ) The system B ξ = 0 can be written as ( A − DA )( ξ ) = 0 with D = diag ( f 1 , . . . , f N ) , and A i , j = K ( x i , x j ) ... 8 of 33
Rational RBF Find the coefficients: I [Jackobsson et al 2009] proved the well-posedness of the interpolation on X N via R ( x ) = R ( 1 ) ( x ) � N i = 1 α i K ( x , x i ) R ( 2 ) ( x ) = , (2) � N k = 1 β k K ( x , x k ) Letting ξ = ( α T , β T ) ∈ R 2 N and B the N × 2 N matrix K ( x 1 , x 1 ) K ( x 1 , x N ) − f 1 K ( x 1 , x 1 ) − f 1 K ( x 1 , x N ) · · · · · · . . . B = . . . . . . . K ( x N , x 1 ) · · · K ( x N , x N ) − f N K ( x N , x 1 ) · · · − f N K ( x N , x N ) The system B ξ = 0 can be written as ( A − DA )( ξ ) = 0 with D = diag ( f 1 , . . . , f N ) , and A i , j = K ( x i , x j ) ... which is underdetermined! 8 of 33
Rational RBF Find the coefficients: I [Jackobsson et al 2009] proved the well-posedness of the interpolation on X N via R ( x ) = R ( 1 ) ( x ) � N i = 1 α i K ( x , x i ) R ( 2 ) ( x ) = , (2) � N k = 1 β k K ( x , x k ) Letting ξ = ( α T , β T ) ∈ R 2 N and B the N × 2 N matrix K ( x 1 , x 1 ) K ( x 1 , x N ) − f 1 K ( x 1 , x 1 ) − f 1 K ( x 1 , x N ) · · · · · · . . . B = . . . . . . . K ( x N , x 1 ) · · · K ( x N , x N ) − f N K ( x N , x 1 ) · · · − f N K ( x N , x N ) The system B ξ = 0 can be written as ( A − DA )( ξ ) = 0 with D = diag ( f 1 , . . . , f N ) , and A i , j = K ( x i , x j ) ... which is underdetermined! Non trivial solution Following [GolubReinsch1975] non-trivial solutions exist by asking || ξ || 2 = 1 i.e. solving the problem ξ ∈ R N , || ξ || 2 = 1 || B ξ || 2 . min 8 of 33
Rational RBF Find the coefficients: II Obs: (Since R ( 1 ) ( x i ) = f i R ( 2 ) ( x i ) , i = 1 , . . . , N ) find q = ( R ( 2 ) ( x 1 ) , . . . , R ( 2 ) ( x N )) T and, as p = D q , then p = ( R ( 1 ) ( x i ) , ..., R ( 1 ) ( x N )) T . 9 of 33
Rational RBF Find the coefficients: II Obs: (Since R ( 1 ) ( x i ) = f i R ( 2 ) ( x i ) , i = 1 , . . . , N ) find q = ( R ( 2 ) ( x 1 ) , . . . , R ( 2 ) ( x N )) T and, as p = D q , then p = ( R ( 1 ) ( x i ) , ..., R ( 1 ) ( x N )) T . If p, q are given then the rational interpolant is known by solving A α = p , A β = q . (3) 9 of 33
Rational RBF Find the coefficients: II Obs: (Since R ( 1 ) ( x i ) = f i R ( 2 ) ( x i ) , i = 1 , . . . , N ) find q = ( R ( 2 ) ( x 1 ) , . . . , R ( 2 ) ( x N )) T and, as p = D q , then p = ( R ( 1 ) ( x i ) , ..., R ( 1 ) ( x N )) T . If p, q are given then the rational interpolant is known by solving A α = p , A β = q . (3) Existence & Uniqueness of (3): K is SPD Using the native space norms the above problem is equivalent Problem 1 1 || R ( 1 ) || 2 N K + || R ( 2 ) || 2 min (4) . N K || f || 2 R ( 1 ) , R ( 2 ) ∈N K , 2 1 / || f || 2 2 || p || 2 2 + || q || 2 2 = 1 , R ( 1 ) ( x k )= f k R ( 2 ) ( x k ) . 9 of 33
Rational RBF Find the coefficients: III || R ( 1 ) || 2 N K = α T A α , || R ( 2 ) || 2 N K = β T A β . and Then, from (3) and symmetry of A || R ( 1 ) || 2 N K = p T A − 1 p , || R ( 2 ) || 2 N K = q T A − 1 q . and Therefore, the Problem 1 reduces to solve Problem 2 1 q T D T A − 1 D q + q T A − 1 q min . || f || 2 q ∈ R N , 2 1 / || f || 2 2 || D q || 2 2 + || q || 2 2 = 1 . 10 of 33
Rational RBF Find the coefficients: IV [Jackbsson 2009] show that this is equivalent to solve the following generalized eigenvalue problem Problem 3 Σ q = λ Θ q , with 1 1 D T A − 1 D + A − 1 , D T D + I N , Σ = and Θ = || f || 2 || f || 2 2 2 where I N is the identity matrix. 11 of 33
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