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From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Interpolation points and interpolation formulae on the square Stefano De Marchi Department of Computer

  1. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Padua points The Padua points (PD) can be defined as follows (C.DeM.V. AMC 05): � (2 k − 1) π �  cos if m odd  � ( m − 1) π �  n + 1  x PD y PD = cos , = m k n � 2( k − 1) π �  cos if m even   n + 1 1 ≤ m ≤ n + 1, 1 ≤ k ≤ n / 2 + 1, N = ( n + 2)( n + 1) / 2. The PD points are equispaced w.r.t. Dubiner metric on [ − 1 , 1] 2 . They are modified Morrow-Patterson points discovered in Padua in 2003 by B.DeM.V.&W. Moreover, there are four families which correspond to successive rotations of 90 degrees, clockwise for even degrees and counterclockwise for odd degrees. Stefano De Marchi Interpolation points and interpolation formulae on the square

  2. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Graphs of MP, EMP, PD pts and their Lebesgue constants 1 MP 1000 EMP PD 0.8 0.6 Lebesgue constants MP 0.4 2 (0.7·n+1.0) 100 EMP 0.2 2 (0.4·n+0.9) PD 0 2 (2/ π ·log(n+1)+1.1) −0.2 −0.4 10 −0.6 −0.8 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 −1 degree n −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Left : the graphs of MP, EMP, PD for n = 8. Right : the growth of the corresponding Lebesgue constants. Stefano De Marchi Interpolation points and interpolation formulae on the square

  3. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature An interpolation formula for MP points L. Bos in a note described an interpolation formula for MP points. Letting, n � P T j ( x MP m , y MP L m , k ( x , y ) := ) P j ( x , y ) , 1 ≤ m ≤ n +1 , 1 ≤ k ≤ n / 2+1 , (1) k j =0 where   U 0 ( x ) U j ( y ) U 1 ( x ) U j − 1 ( y )   P j ( x , y ) =  .  .   .   U j ( x ) U 0 ( y ) U s being the Chebyshev polynomials of second type, so that L m , k ( x s , y r ) = 0 , ( s , r ) � = ( m , k ) . Stefano De Marchi Interpolation points and interpolation formulae on the square

  4. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature An interpolation formula for MP points 1 ℓ m , k ( x , y ) := ) L m , k ( x , y ) , (2) � n j =0 P T j ( x MP m , y MP ) P j ( x MP m , y MP k k m = 1 , . . . , n + 1 ; k = 1 , . . . , n / 2 + 1, are the fundamental Lagrange polynomials; � n j =0 P T j ( x MP m , y MP ) P j ( x MP m , y MP ) ≥ 1 . He k k also gave a naive upper bound (overestimate) for the Lebesgue constant n +1 n / 2+1 � � Λ MP | ℓ m , k ( x , y ) | ≤ c 1 n 6 , := n m =1 k =1 for appropriate c 1 > 0, while from our numerical results the growth is O ( n 2 ) ( ≈ (0 . 7 n + 1) 2 ). The computational cost for evaluating the interpolant at any ( x , y ) is O ( N 2 ). Stefano De Marchi Interpolation points and interpolation formulae on the square

  5. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature An interpolation formula for MP points 1 ℓ m , k ( x , y ) := ) L m , k ( x , y ) , (2) � n j =0 P T j ( x MP m , y MP ) P j ( x MP m , y MP k k m = 1 , . . . , n + 1 ; k = 1 , . . . , n / 2 + 1, are the fundamental Lagrange polynomials; � n j =0 P T j ( x MP m , y MP ) P j ( x MP m , y MP ) ≥ 1 . He k k also gave a naive upper bound (overestimate) for the Lebesgue constant n +1 n / 2+1 � � Λ MP | ℓ m , k ( x , y ) | ≤ c 1 n 6 , := n m =1 k =1 for appropriate c 1 > 0, while from our numerical results the growth is O ( n 2 ) ( ≈ (0 . 7 n + 1) 2 ). The computational cost for evaluating the interpolant at any ( x , y ) is O ( N 2 ). Stefano De Marchi Interpolation points and interpolation formulae on the square

  6. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Generating curves For MP points � t π � � � ( n + 3 − t ) π �� γ MP n ( t ) = cos , cos , 0 ≤ t ≤ ( n +2)( n +3) ; n + 2 n + 3 (3) For the PD points of the first family � � t π � t π �� � γ PD n ( t ) = − cos , − cos π − , 0 ≤ t ≤ n ( n +1) . n n + 1 (4) Stefano De Marchi Interpolation points and interpolation formulae on the square

  7. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature The interpolant of the PD pts Let A j = ( x j , y j ) ∈ [ − 1 , 1] 2 be a PD, it belongs to the curve, x = − cos( n + 1) t , y = − cos nt , 0 ≤ t ≤ π . K n ( x , y ) = D n ( θ 1 + φ 1 , θ 2 + φ 2 ) + D n ( θ 1 + φ 1 , θ 2 − φ 2 ) + (5) + D n ( θ 1 − φ 1 , θ 2 + φ 2 ) + D n ( θ 1 − φ 1 , θ 2 − φ 2 ) , x = (cos θ 1 , cos θ 2 ) , y = (cos φ 1 , cos φ 2 ) , where the function D n is defined by D n ( α, β ) = 1 cos(( n + 1 / 2) α ) cos( α/ 2) − cos(( n + 1 / 2) β ) cos( β/ 2) . 2 cos α − cos β (6) The Lagrange polynomials are l j ( x , y ) = w j { K n (( x , y ) , A j ) − T n ( x j ) T n ( x ) } and the coefficients w j are the corresponding cubature weights. Stefano De Marchi Interpolation points and interpolation formulae on the square

  8. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature The interpolant of the PD pts  1 / 2 vertex pts 1  w j = 2 interior pts n ( n + 1) 1 boundary pts  K n is the reproducing kernel of P n ([ − 1 , 1] 2 ) (Xu, JAT 95) equipped with the scalar product < f , g > = 1 � dx dy √ [ − 1 , 1] 2 f ( x , y ) g ( x , y ) π 2 � 1 − x 2 1 − y 2 holds the reproducing property ∀ p ∈ P n ( R 2 ) < p ( x , y ) , K n (( x , y ) , A ) > = p ( A ) , and A = ( a , b ) ∈ [ − 1 , 1] 2 . Stefano De Marchi Interpolation points and interpolation formulae on the square

  9. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature The Xu points and the Xu interpolant Given the Chebyshev-Lobatto points on the interval [ − 1 , 1] (cf. Xu, JAT 95) ξ k = ξ k , n = cos k π n , k = 0 , . . . , n , n = 2 m , the Xu interpolation points on the square Q = [ − 1 , 1] 2 are the two-dimensional Chebyshev array X N = { z r , s } of dimension N = n ( n + 2) / 2 z 2 i , 2 j +1 = ( ξ 2 i , ξ 2 j +1 ) , 0 ≤ i ≤ m , 0 ≤ j ≤ m − 1 , z 2 i +1 , 2 j = ( ξ 2 i +1 , ξ 2 j ) , 0 ≤ i ≤ m − 1 , 0 ≤ j ≤ m . Note: the Xu points are exactly equally spaced w.r.t. the Dubiner metric. Stefano De Marchi Interpolation points and interpolation formulae on the square

  10. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature The Xu points and the Xu interpolant Given the Chebyshev-Lobatto points on the interval [ − 1 , 1] (cf. Xu, JAT 95) ξ k = ξ k , n = cos k π n , k = 0 , . . . , n , n = 2 m , the Xu interpolation points on the square Q = [ − 1 , 1] 2 are the two-dimensional Chebyshev array X N = { z r , s } of dimension N = n ( n + 2) / 2 z 2 i , 2 j +1 = ( ξ 2 i , ξ 2 j +1 ) , 0 ≤ i ≤ m , 0 ≤ j ≤ m − 1 , z 2 i +1 , 2 j = ( ξ 2 i +1 , ξ 2 j ) , 0 ≤ i ≤ m − 1 , 0 ≤ j ≤ m . Note: the Xu points are exactly equally spaced w.r.t. the Dubiner metric. Stefano De Marchi Interpolation points and interpolation formulae on the square

  11. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature The Xu points and the Xu interpolant The Xu interpolant of degree n in Lagrange form of a function f on Q is K ∗ n ( x , z r , s ) � L n f ( x ) = Xu f ( z r , s ) ℓ n ( x , z r , s ) , ℓ n ( x , z r , s ) := n ( z r , s , z r , s ) , (7) K ∗ z r , s ∈ X N n ( z r , s , z r , s ) = 1 K ∗ 2 ( K n − 1 ( z r , s , z r , s ) + K n ( z r , s , z r , s )) − 1 , (8) with K n as defined for the PD points in (5). 1 XU 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Stefano De Marchi Interpolation points and interpolation formulae on the square Xu points for m = 8 .

  12. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature The Xu points and the Xu interpolant Remarks In the Xu interpolation formula, the ℓ n in (7) are based on the K ∗ n (cf. (8)), that make use of K n − 1 and K n : the interpolant based on the PD points only need the K n (i.e no K ∗ n ). On the other hand, the dimension of the corresponding polynomial space, V n , is dim( P n − 1 ( R 2 )) < dim( V n ) := n ( n + 2) / 2 < dim( P n ( R 2 )). Drawback: numerical instability in computing D n ( α, β ) when cos α ≈ cos β ! Stefano De Marchi Interpolation points and interpolation formulae on the square

  13. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature The Xu points and the Xu interpolant Remarks In the Xu interpolation formula, the ℓ n in (7) are based on the K ∗ n (cf. (8)), that make use of K n − 1 and K n : the interpolant based on the PD points only need the K n (i.e no K ∗ n ). On the other hand, the dimension of the corresponding polynomial space, V n , is dim( P n − 1 ( R 2 )) < dim( V n ) := n ( n + 2) / 2 < dim( P n ( R 2 )). Drawback: numerical instability in computing D n ( α, β ) when cos α ≈ cos β ! Stefano De Marchi Interpolation points and interpolation formulae on the square

  14. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature The Xu points and the Xu interpolant Remarks In the Xu interpolation formula, the ℓ n in (7) are based on the K ∗ n (cf. (8)), that make use of K n − 1 and K n : the interpolant based on the PD points only need the K n (i.e no K ∗ n ). On the other hand, the dimension of the corresponding polynomial space, V n , is dim( P n − 1 ( R 2 )) < dim( V n ) := n ( n + 2) / 2 < dim( P n ( R 2 )). Drawback: numerical instability in computing D n ( α, β ) when cos α ≈ cos β ! Stefano De Marchi Interpolation points and interpolation formulae on the square

  15. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature The Xu points and the Xu interpolant Stabilization D n ( α, β ) = 1 4 [ U n − 1 (cos φ ) U n − 1 (cos ψ ) + U n − 2 (cos φ ) U n − 2 (cos ψ )] , where φ = ( α − β ) / 2, ψ = ( α + β ) / 2, U n Chebyshev polynomial of the second kind computed by the three-term recurrence, with overall computational cost ≈ 8 nN ≈ 11 N 3 / 2 flops. Hybrid stable formula for U n (cos θ ): three-term recurrence whenever | θ − k π | ≤ ε , otherwise U n (cos θ ) = sin ( n + 1) θ/ sin θ . For ε = 0 . 01, L Xu n f ( x ) is computed at machine precision, the recurrence relation is used globally less than 1%, and for degrees n up to the hundreds, overall computational cost ≈ 32 c sin N flops, c sin being the average evaluation cost of the sine function. In practical applications the computational cost becomes linear in the number N of Xu points We made a Fortran implementation of the Xu and PD interpolation formula: http://profs.sci.univr.it/ ∼ demarchi/software.htm . Stefano De Marchi Interpolation points and interpolation formulae on the square

  16. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature The Xu points and the Xu interpolant Stabilization D n ( α, β ) = 1 4 [ U n − 1 (cos φ ) U n − 1 (cos ψ ) + U n − 2 (cos φ ) U n − 2 (cos ψ )] , where φ = ( α − β ) / 2, ψ = ( α + β ) / 2, U n Chebyshev polynomial of the second kind computed by the three-term recurrence, with overall computational cost ≈ 8 nN ≈ 11 N 3 / 2 flops. Hybrid stable formula for U n (cos θ ): three-term recurrence whenever | θ − k π | ≤ ε , otherwise U n (cos θ ) = sin ( n + 1) θ/ sin θ . For ε = 0 . 01, L Xu n f ( x ) is computed at machine precision, the recurrence relation is used globally less than 1%, and for degrees n up to the hundreds, overall computational cost ≈ 32 c sin N flops, c sin being the average evaluation cost of the sine function. In practical applications the computational cost becomes linear in the number N of Xu points We made a Fortran implementation of the Xu and PD interpolation formula: http://profs.sci.univr.it/ ∼ demarchi/software.htm . Stefano De Marchi Interpolation points and interpolation formulae on the square

  17. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature The Xu points and the Xu interpolant Stabilization D n ( α, β ) = 1 4 [ U n − 1 (cos φ ) U n − 1 (cos ψ ) + U n − 2 (cos φ ) U n − 2 (cos ψ )] , where φ = ( α − β ) / 2, ψ = ( α + β ) / 2, U n Chebyshev polynomial of the second kind computed by the three-term recurrence, with overall computational cost ≈ 8 nN ≈ 11 N 3 / 2 flops. Hybrid stable formula for U n (cos θ ): three-term recurrence whenever | θ − k π | ≤ ε , otherwise U n (cos θ ) = sin ( n + 1) θ/ sin θ . For ε = 0 . 01, L Xu n f ( x ) is computed at machine precision, the recurrence relation is used globally less than 1%, and for degrees n up to the hundreds, overall computational cost ≈ 32 c sin N flops, c sin being the average evaluation cost of the sine function. In practical applications the computational cost becomes linear in the number N of Xu points We made a Fortran implementation of the Xu and PD interpolation formula: http://profs.sci.univr.it/ ∼ demarchi/software.htm . Stefano De Marchi Interpolation points and interpolation formulae on the square

  18. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature The Xu points and the Xu interpolant Implementation details and performance (cf. B.C.DeM.V. ’05): Comparison with the MPI software by T. Sauer (cf. S. AiCM 95, S. Xu Math.Comp.95). The MPI software is one of the most efficient and robust implementations of multivariate interpolation by polynomials. We compared the CPU times necessary to build the interpolant and the interpolation errors for both XU and MPI on many tests functions. Stefano De Marchi Interpolation points and interpolation formulae on the square

  19. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature The Xu points and the Xu interpolant Table 1: CPU times (secs.) and interpolation errors on [0 , 1] 2 for the Franke funct. n 20 30 40 50 60 XU 2 . 1 5 . 2 10 . 3 17 . 8 28 . 4 7.3E-03 3.6E-04 3.1E-06 1.8E-08 2.5E-11 MPI 0 . 6 Unsolv. Unsolv. Unsolv. Unsolv. 3.8E-02 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Table 2: CPU times and interpolation errors of MPI for the Franke function on different domains by a change of variables and reordering the points as Leja sequences. n 20 30 40 50 60 MPI 0 . 6 4 . 3 21 . 0 75 . 6 Unsolv. [ − 1 , 1] 2 6.3E-03 3.5E-04 2.0E-01 3.8E-02 ∗ ∗ ∗ MPI 0 . 5 3 . 7 17 . 4 62 . 3 183 . 4 [ − 2 , 2] 2 6.4E-03 1.0E-02 2.7E+02 1.3E+14 1.9E+35 MPI-Leja 0 . 6 4 . 3 21 . 0 75 . 6 Unsolv. [ − 1 , 1] 2 6.4E-03 3.5E-04 1.1E-04 2.0E-03 ∗ ∗ ∗ Stefano De Marchi Interpolation points and interpolation formulae on the square

  20. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature The Lebesgue constant of the Xu points Since, the maximum is attained at the four vertices of the square, the computation became ”easy” Table 1. Lebesgue constants size of different nodal sets on Q : Morrow-Patterson (MP), Extended Morrow-Patterson (EMP), Padua points (PD), Xu points (XU). interp. pts. Λ 34 Λ 48 Λ 62 Λ 76 MP 649 1264 2082 3102 EMP 237 456 746 1106 PD 11 13 14 15 XU 10 12 13 14 Stefano De Marchi Interpolation points and interpolation formulae on the square

  21. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature The Lebesgue constant of the Xu points � 2 � � 2 � Λ Xu ≤ 2 2 + 4 π log n + 5 n � 2 � 2 = 8 π log n + 5 + 4 . Stefano De Marchi Interpolation points and interpolation formulae on the square

  22. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature The Lebesgue constant of the Xu points Figure 1. Left: the distribution of 144 Xu points on Q (i.e n = 16). Right: the behavior of the Lebesgue constant up to degree n = 100. 1 16 14 0,5 12 10 0 8 Lebesgue constant of Xu points (0.95+2/ π log(n+1)) 2 6 (1+2/ π *log(n+1)) 2 -0,5 4 2 -1 0 -1 -0,5 0 0,5 1 0 10 20 30 40 50 60 70 80 90 100 Stefano De Marchi Interpolation points and interpolation formulae on the square

  23. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Applications of the Xu interpolant We studied two main applications 1. We compressed a surface given as a large set of scattered data, i.e. by “interpolated interpolations”. 2. Compression of a Finite Element PDE solution. Concerning 1. we adopted for sufficiently regular surfaces Xu-like interpolation of a cubic Shepard-like interpolant (cf. Renka TOMS99). The compression ratio obtained is compr. ratio = 3 × numb. of scatt. pts. numb. of Xu nodes ≈ 6 × numb. of scatt. pts. , n 2 where n is the polynomial degree. Concerning 2. Given a FEM discretization (ex. Delaunay mesh) we used Xu-like interpolation of the Finite Element solution. Stefano De Marchi Interpolation points and interpolation formulae on the square

  24. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Applications of the Xu interpolant Table 3. Compression errors (in the max-norm) for the Finite Element solution of the Poisson equation ∆ f ( x ) = − 10 , x ∈ Ω ; f ( x ) = 0 , x ∈ ∂ Ω, where Ω is the “lynx-eye” shaped domain in the Fig. below. mesh size n = 8 n = 12 n = 16 n = 20 n = 24 n = 28 n = 32 41402 1E-1 3E-2 1E-2 5E-3 2E-3 1E-3 1E-3 0,5 0,25 0 -0,25 -0,5 -1 -0,5 0 0,5 1 Left: the distribution of N = 312 Xu-like points (deg n = 24) in the “lynx-eye” shaped domain (generalized sector). Right: Plot of the Xu-like interpolated solution (deg n = 24: compression ratio ≈ 400:1, compression error ≈ 2 · 10 − 3 ). Stefano De Marchi Interpolation points and interpolation formulae on the square

  25. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Applications of the Xu interpolant Recently we applied the Xu interpolation to functions in parametric form f ( x ( u , v ) , y ( u , v ) , z ( u , v )), where u ∈ [ a , b ] , v ∈ [ c , d ]. Here some interesting pictures Left: Xu points over the cilinder and the function to be interpolated f ( x , y , z ) = y ( x 2 + z 2 ). Right: The same for the sphere. Stefano De Marchi Interpolation points and interpolation formulae on the square

  26. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Applications of the Xu interpolant Stefano De Marchi Interpolation points and interpolation formulae on the square

  27. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Bivariate interpolation problem and Padua Pts Let P 2 n be the space of bivariate polynomials of total degree ≤ n . Question: is there a set Ξ ⊂ [ − 1 , 1] 2 of points such that: n ) = ( n +1)( n +2) card (Ξ) = dim ( P 2 ; 2 the problem of finding the interpolation polynomial on Ξ of degree n is unisolvent; the Lebesgue constant Λ n behaves like log 2 n for n → ∞ . Answer: yes, it is the set Ξ = Pad n of Padua points. Stefano De Marchi Interpolation points and interpolation formulae on the square

  28. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Bivariate interpolation problem and Padua Pts Let P 2 n be the space of bivariate polynomials of total degree ≤ n . Question: is there a set Ξ ⊂ [ − 1 , 1] 2 of points such that: n ) = ( n +1)( n +2) card (Ξ) = dim ( P 2 ; 2 the problem of finding the interpolation polynomial on Ξ of degree n is unisolvent; the Lebesgue constant Λ n behaves like log 2 n for n → ∞ . Answer: yes, it is the set Ξ = Pad n of Padua points. Stefano De Marchi Interpolation points and interpolation formulae on the square

  29. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Bivariate interpolation problem and Padua Pts Let P 2 n be the space of bivariate polynomials of total degree ≤ n . Question: is there a set Ξ ⊂ [ − 1 , 1] 2 of points such that: n ) = ( n +1)( n +2) card (Ξ) = dim ( P 2 ; 2 the problem of finding the interpolation polynomial on Ξ of degree n is unisolvent; the Lebesgue constant Λ n behaves like log 2 n for n → ∞ . Answer: yes, it is the set Ξ = Pad n of Padua points. Stefano De Marchi Interpolation points and interpolation formulae on the square

  30. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Bivariate interpolation problem and Padua Pts Let P 2 n be the space of bivariate polynomials of total degree ≤ n . Question: is there a set Ξ ⊂ [ − 1 , 1] 2 of points such that: n ) = ( n +1)( n +2) card (Ξ) = dim ( P 2 ; 2 the problem of finding the interpolation polynomial on Ξ of degree n is unisolvent; the Lebesgue constant Λ n behaves like log 2 n for n → ∞ . Answer: yes, it is the set Ξ = Pad n of Padua points. Stefano De Marchi Interpolation points and interpolation formulae on the square

  31. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Bivariate interpolation problem and Padua Pts Let P 2 n be the space of bivariate polynomials of total degree ≤ n . Question: is there a set Ξ ⊂ [ − 1 , 1] 2 of points such that: n ) = ( n +1)( n +2) card (Ξ) = dim ( P 2 ; 2 the problem of finding the interpolation polynomial on Ξ of degree n is unisolvent; the Lebesgue constant Λ n behaves like log 2 n for n → ∞ . Answer: yes, it is the set Ξ = Pad n of Padua points. Stefano De Marchi Interpolation points and interpolation formulae on the square

  32. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Bivariate interpolation problem and Padua Pts Let P 2 n be the space of bivariate polynomials of total degree ≤ n . Question: is there a set Ξ ⊂ [ − 1 , 1] 2 of points such that: n ) = ( n +1)( n +2) card (Ξ) = dim ( P 2 ; 2 the problem of finding the interpolation polynomial on Ξ of degree n is unisolvent; the Lebesgue constant Λ n behaves like log 2 n for n → ∞ . Answer: yes, it is the set Ξ = Pad n of Padua points. Stefano De Marchi Interpolation points and interpolation formulae on the square

  33. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Padua points Let us consider n + 1 Chebyshev–Lobatto points on [ − 1 , 1] � � ( j − 1) π � � z n C n +1 = j = cos , j = 1 , . . . , n + 1 n and the two subsets of points with odd or even indexes C odd � z n � n +1 = j , j = 1 , . . . , n + 1 , j odd z n C even � � n +1 = j , j = 1 , . . . , n + 1 , j even Then, the Padua points are the set Pad n = C odd n +1 × C even n +2 ∪ C even n +1 × C odd n +2 ⊂ C n +1 × C n +2 Stefano De Marchi Interpolation points and interpolation formulae on the square

  34. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Padua points Let us consider n + 1 Chebyshev–Lobatto points on [ − 1 , 1] � � ( j − 1) π � � z n C n +1 = j = cos , j = 1 , . . . , n + 1 n and the two subsets of points with odd or even indexes C odd � z n � n +1 = j , j = 1 , . . . , n + 1 , j odd z n C even � � n +1 = j , j = 1 , . . . , n + 1 , j even Then, the Padua points are the set Pad n = C odd n +1 × C even n +2 ∪ C even n +1 × C odd n +2 ⊂ C n +1 × C n +2 Stefano De Marchi Interpolation points and interpolation formulae on the square

  35. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Padua points Let us consider n + 1 Chebyshev–Lobatto points on [ − 1 , 1] � � ( j − 1) π � � z n C n +1 = j = cos , j = 1 , . . . , n + 1 n and the two subsets of points with odd or even indexes C odd � z n � n +1 = j , j = 1 , . . . , n + 1 , j odd z n C even � � n +1 = j , j = 1 , . . . , n + 1 , j even Then, the Padua points are the set Pad n = C odd n +1 × C even n +2 ∪ C even n +1 × C odd n +2 ⊂ C n +1 × C n +2 Stefano De Marchi Interpolation points and interpolation formulae on the square

  36. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature The generating curve There exists an alternative representation as self-intersections and boundary contacts of the generating curve γ ( t ) = ( − cos(( n + 1) t ) , − cos( nt )) , t ∈ [0 , π ] Stefano De Marchi Interpolation points and interpolation formulae on the square

  37. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature The generating curve γ ( t ) ( n = 4) 1 0.5 0 -0.5 -1 -1 -0.5 0 0.5 1 t = 0 Stefano De Marchi Interpolation points and interpolation formulae on the square

  38. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature The generating curve γ ( t ) ( n = 4) 1 0.5 0 -0.5 -1 -1 -0.5 0 0.5 1 � � 4 π t ∈ 0 , ( n ( n +1)) Stefano De Marchi Interpolation points and interpolation formulae on the square

  39. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature The generating curve γ ( t ) ( n = 4) 1 0.5 0 -0.5 -1 -1 -0.5 0 0.5 1 � � 4 π 5 π t ∈ ( n ( n +1)) , ( n ( n +1)) Stefano De Marchi Interpolation points and interpolation formulae on the square

  40. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature The generating curve γ ( t ) ( n = 4) 1 0.5 0 -0.5 -1 -1 -0.5 0 0.5 1 � � 5 π 8 π t ∈ ( n ( n +1)) , ( n ( n +1)) Stefano De Marchi Interpolation points and interpolation formulae on the square

  41. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature The generating curve γ ( t ) ( n = 4) 1 0.5 0 -0.5 -1 -1 -0.5 0 0.5 1 � � 8 π 9 π t ∈ ( n ( n +1)) , ( n ( n +1)) Stefano De Marchi Interpolation points and interpolation formulae on the square

  42. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature The generating curve γ ( t ) ( n = 4) 1 0.5 0 -0.5 -1 -1 -0.5 0 0.5 1 � � 9 π 10 π t ∈ ( n ( n +1)) , ( n ( n +1)) Stefano De Marchi Interpolation points and interpolation formulae on the square

  43. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature The generating curve γ ( t ) ( n = 4) 1 0.5 0 -0.5 -1 -1 -0.5 0 0.5 1 � � 10 π 12 π t ∈ ( n ( n +1)) , ( n ( n +1)) Stefano De Marchi Interpolation points and interpolation formulae on the square

  44. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature The generating curve γ ( t ) ( n = 4) 1 0.5 0 -0.5 -1 -1 -0.5 0 0.5 1 � � 12 π 13 π t ∈ ( n ( n +1)) , ( n ( n +1)) Stefano De Marchi Interpolation points and interpolation formulae on the square

  45. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature The generating curve γ ( t ) ( n = 4) 1 0.5 0 -0.5 -1 -1 -0.5 0 0.5 1 � � 13 π 14 π t ∈ ( n ( n +1)) , ( n ( n +1)) Stefano De Marchi Interpolation points and interpolation formulae on the square

  46. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature The generating curve γ ( t ) ( n = 4) 1 0.5 0 -0.5 -1 -1 -0.5 0 0.5 1 � � 14 π 15 π t ∈ ( n ( n +1)) , ( n ( n +1)) Stefano De Marchi Interpolation points and interpolation formulae on the square

  47. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature The generating curve γ ( t ) ( n = 4) 1 0.5 0 -0.5 -1 -1 -0.5 0 0.5 1 � � 15 π 16 π t ∈ ( n ( n +1)) , ( n ( n +1)) Stefano De Marchi Interpolation points and interpolation formulae on the square

  48. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature The generating curve γ ( t ) ( n = 4) 1 0.5 0 -0.5 -1 -1 -0.5 0 0.5 1 � � 16 π 17 π t ∈ ( n ( n +1)) , ( n ( n +1)) Stefano De Marchi Interpolation points and interpolation formulae on the square

  49. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature The generating curve γ ( t ) ( n = 4) 1 0.5 0 -0.5 -1 -1 -0.5 0 0.5 1 � � 17 π 18 π t ∈ ( n ( n +1)) , ( n ( n +1)) Stefano De Marchi Interpolation points and interpolation formulae on the square

  50. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature The generating curve γ ( t ) ( n = 4) 1 0.5 0 -0.5 -1 -1 -0.5 0 0.5 1 � � 18 π 19 π t ∈ ( n ( n +1)) , ( n ( n +1)) Stefano De Marchi Interpolation points and interpolation formulae on the square

  51. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature The generating curve γ ( t ) ( n = 4) 1 0.5 0 -0.5 -1 -1 -0.5 0 0.5 1 � � 19 π 20 π t ∈ ( n ( n +1)) , ( n ( n +1)) Stefano De Marchi Interpolation points and interpolation formulae on the square

  52. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature The generating curve γ ( t ) ( n = 4) 1 0.5 0 -0.5 -1 -1 -0.5 0 0.5 1 C odd n +1 × C even n +2 Stefano De Marchi Interpolation points and interpolation formulae on the square

  53. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature The generating curve γ ( t ) ( n = 4), is a Lissajous curve 1 0.5 0 -0.5 -1 -1 -0.5 0 0.5 1 Pad n = C odd n +1 × C even n +2 ∪ C even n +1 × C odd n +2 ⊂ C n +1 × C n +2 Stefano De Marchi Interpolation points and interpolation formulae on the square

  54. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Lagrange polynomials The fundamental Lagrange polynomials of the Padua points are L ξ ( x ) = w ξ ( K n ( ξ , x ) − T n ( ξ 1 ) T n ( x 1 )) , L ξ ( η ) = δ ξη , ξ , η ∈ Pad n where 1  if ξ is a vertex point  2 1   w ξ = n ( n + 1) · 1 if ξ is an edge point    2 if ξ is an interior point (Note: { w ξ } are weights of cubature formula for the prod. Cheb. measure, exact ”on almost” Π n 2 n ([ − 1 , 1] 2 )), i.e. pol. orthogonal to T 2 n ( x 1 ) n k � � T j ( x 1 ) ˆ ˆ T k − j ( x 2 ) ˆ T j ( y 1 ) ˆ K n ( x , y ) = T k − j ( y 2 ) , (9) k =0 j =0 ˆ T j is the normalized Chebyshev polynomial of degree j . Stefano De Marchi Interpolation points and interpolation formulae on the square

  55. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Lagrange polynomials The fundamental Lagrange polynomials of the Padua points are L ξ ( x ) = w ξ ( K n ( ξ , x ) − T n ( ξ 1 ) T n ( x 1 )) , L ξ ( η ) = δ ξη , ξ , η ∈ Pad n where 1  if ξ is a vertex point  2 1   w ξ = n ( n + 1) · 1 if ξ is an edge point    2 if ξ is an interior point (Note: { w ξ } are weights of cubature formula for the prod. Cheb. measure, exact ”on almost” Π n 2 n ([ − 1 , 1] 2 )), i.e. pol. orthogonal to T 2 n ( x 1 ) n k � � T j ( x 1 ) ˆ ˆ T k − j ( x 2 ) ˆ T j ( y 1 ) ˆ K n ( x , y ) = T k − j ( y 2 ) , (9) k =0 j =0 ˆ T j is the normalized Chebyshev polynomial of degree j . Stefano De Marchi Interpolation points and interpolation formulae on the square

  56. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Lagrange polynomials The fundamental Lagrange polynomials of the Padua points are L ξ ( x ) = w ξ ( K n ( ξ , x ) − T n ( ξ 1 ) T n ( x 1 )) , L ξ ( η ) = δ ξη , ξ , η ∈ Pad n where 1  if ξ is a vertex point  2 1   w ξ = n ( n + 1) · 1 if ξ is an edge point    2 if ξ is an interior point (Note: { w ξ } are weights of cubature formula for the prod. Cheb. measure, exact ”on almost” Π n 2 n ([ − 1 , 1] 2 )), i.e. pol. orthogonal to T 2 n ( x 1 ) n k � � T j ( x 1 ) ˆ ˆ T k − j ( x 2 ) ˆ T j ( y 1 ) ˆ K n ( x , y ) = T k − j ( y 2 ) , (9) k =0 j =0 ˆ T j is the normalized Chebyshev polynomial of degree j . Stefano De Marchi Interpolation points and interpolation formulae on the square

  57. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Reproducing kernel K n ( x , y ) is the reproducing kernel of P 2 n ([ − 1 , 1] 2 ) equipped with the inner product � d x 1 d x 2 � f , g � = [ − 1 , 1] 2 f ( x 1 , x 2 ) g ( x 1 , x 2 ) , � 1 − x 2 � 1 − x 2 π π 1 2 with reproduction property � ∀ p n ∈ P 2 [ − 1 , 1] 2 K n ( x , y ) p n ( y ) w ( y ) d y = p n ( x ) , n 1 1 w ( x ) = w ( x 1 , x 2 ) = � 1 − x 2 � 1 − x 2 π π 1 2 Stefano De Marchi Interpolation points and interpolation formulae on the square

  58. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Reproducing kernel K n ( x , y ) is the reproducing kernel of P 2 n ([ − 1 , 1] 2 ) equipped with the inner product � d x 1 d x 2 � f , g � = [ − 1 , 1] 2 f ( x 1 , x 2 ) g ( x 1 , x 2 ) , � 1 − x 2 � 1 − x 2 π π 1 2 with reproduction property � ∀ p n ∈ P 2 [ − 1 , 1] 2 K n ( x , y ) p n ( y ) w ( y ) d y = p n ( x ) , n 1 1 w ( x ) = w ( x 1 , x 2 ) = � 1 − x 2 � 1 − x 2 π π 1 2 Stefano De Marchi Interpolation points and interpolation formulae on the square

  59. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Lebesgue constant The Lebesgue constant � Λ n = x ∈ [ − 1 , 1] 2 λ n ( x ) , max λ n ( x ) = | L ξ ( x ) | ξ ∈ Pad n is bounded by Λ n ≤ C log 2 n (optimal order of growth on a square). Stefano De Marchi Interpolation points and interpolation formulae on the square

  60. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Interpolant Given the representation (9) for the reproducing kernel, the interpolant of a function f : [ − 1 , 1] 2 → R is � L n f ( x ) = f ( ξ ) w ξ ( K n ( ξ , x ) − T n ( ξ 1 ) T n ( x 1 )) = ξ ∈ Pad n n k T k − j ( x 2 ) − c n , 0 � � c j , k − j ˆ T j ( x 1 ) ˆ T n ( x 1 ) ˆ ˆ = T 0 ( x 2 ) , 2 k =0 j =0 where the coefficients � f ( ξ ) w ξ ˆ T j ( ξ 1 ) ˆ c j , k − j = T k − j ( ξ 2 ) , 0 ≤ j ≤ k ≤ n ξ ∈ Pad n can be computed once and for all. Stefano De Marchi Interpolation points and interpolation formulae on the square

  61. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Interpolant Given the representation (9) for the reproducing kernel, the interpolant of a function f : [ − 1 , 1] 2 → R is � L n f ( x ) = f ( ξ ) w ξ ( K n ( ξ , x ) − T n ( ξ 1 ) T n ( x 1 )) = ξ ∈ Pad n n k T k − j ( x 2 ) − c n , 0 � � c j , k − j ˆ T j ( x 1 ) ˆ T n ( x 1 ) ˆ ˆ = T 0 ( x 2 ) , 2 k =0 j =0 where the coefficients � f ( ξ ) w ξ ˆ T j ( ξ 1 ) ˆ c j , k − j = T k − j ( ξ 2 ) , 0 ≤ j ≤ k ≤ n ξ ∈ Pad n can be computed once and for all. Stefano De Marchi Interpolation points and interpolation formulae on the square

  62. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Coefficient matrix Let us define the coefficient matrix   c 0 , 0 c 0 , 1 . . . . . . c 0 , n c 1 , 0 c 1 , 1 . . . c 1 , n − 1 0    . . .  ... ... . . . C 0 =   . . .     c n − 1 , 0 c n − 1 , 1 0 . . . 0   c n , 0 0 . . . 0 0 2 and for a vector S = ( s 1 , . . . , s m ) , S ∈ [ − 1 , 1] m , the ( n + 1) × m Chebyshev collocation matrix ˆ ˆ   T 0 ( s 1 ) . . . T 0 ( s m ) . . . . T ( S ) =   . . . . .   ˆ ˆ T n ( s 1 ) . . . T n ( s m ) Stefano De Marchi Interpolation points and interpolation formulae on the square

  63. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Coefficient matrix factorization Letting C n +1 the vector of the Chebyshev-Lobatto pts z n 1 , . . . , z n � � C n +1 = n +1 we construct the ( n + 1) × ( n + 2) matrix � w ξ f ( z n r , z n +1 if ξ = ( z n r , z n +1 ) ) ∈ Pad n s s G ( f ) = ( g r , s ) = . if ξ = ( z n r , z n +1 0 ) ∈ ( C n +1 × C n +2 ) \ Pad n s Then C 0 is essentially the upper-left triangular part of C ( f ) = P 1 G ( f ) P T 2 P 1 = T ( C n +1 ) ∈ R ( n +1) × ( n +1) and P 2 = T ( C n +2 ) ∈ R ( n +1) × ( n +2) . Stefano De Marchi Interpolation points and interpolation formulae on the square

  64. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Coefficient matrix factorization Letting C n +1 the vector of the Chebyshev-Lobatto pts z n 1 , . . . , z n � � C n +1 = n +1 we construct the ( n + 1) × ( n + 2) matrix � w ξ f ( z n r , z n +1 if ξ = ( z n r , z n +1 ) ) ∈ Pad n s s G ( f ) = ( g r , s ) = . if ξ = ( z n r , z n +1 0 ) ∈ ( C n +1 × C n +2 ) \ Pad n s Then C 0 is essentially the upper-left triangular part of C ( f ) = P 1 G ( f ) P T 2 P 1 = T ( C n +1 ) ∈ R ( n +1) × ( n +1) and P 2 = T ( C n +2 ) ∈ R ( n +1) × ( n +2) . Stefano De Marchi Interpolation points and interpolation formulae on the square

  65. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Coefficient matrix factorization Letting C n +1 the vector of the Chebyshev-Lobatto pts z n 1 , . . . , z n � � C n +1 = n +1 we construct the ( n + 1) × ( n + 2) matrix � w ξ f ( z n r , z n +1 if ξ = ( z n r , z n +1 ) ) ∈ Pad n s s G ( f ) = ( g r , s ) = . if ξ = ( z n r , z n +1 0 ) ∈ ( C n +1 × C n +2 ) \ Pad n s Then C 0 is essentially the upper-left triangular part of C ( f ) = P 1 G ( f ) P T 2 P 1 = T ( C n +1 ) ∈ R ( n +1) × ( n +1) and P 2 = T ( C n +2 ) ∈ R ( n +1) × ( n +2) . Stefano De Marchi Interpolation points and interpolation formulae on the square

  66. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Linear algebra approach The construction of the coefficients is performed by a matrix-matrix product. It can be easily (and efficiently) implemented in Fortran77 � (based on (by, eventually optimized, BLAS) and in Matlab R optimized BLAS). Stefano De Marchi Interpolation points and interpolation formulae on the square

  67. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Linear algebra approach The construction of the coefficients is performed by a matrix-matrix product. It can be easily (and efficiently) implemented in Fortran77 � (based on (by, eventually optimized, BLAS) and in Matlab R optimized BLAS). Stefano De Marchi Interpolation points and interpolation formulae on the square

  68. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature A new approach based on FFT Since the coefficients are approximated Fourier–Chebyshev coefficients, they can be computed also by FFT techniques. FFT is competitive and more stable than the matrix-matrix multiplication at high degree of interpolation. Stefano De Marchi Interpolation points and interpolation formulae on the square

  69. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature A new approach based on FFT Since the coefficients are approximated Fourier–Chebyshev coefficients, they can be computed also by FFT techniques. FFT is competitive and more stable than the matrix-matrix multiplication at high degree of interpolation. Stefano De Marchi Interpolation points and interpolation formulae on the square

  70. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature � code for the FFT approach R Matlab Input: G ↔ G ( f ) C0 = G; X = real(fft(C0,max(1,2*(size(G,2)-1)),2)); X = X(:,1:n+1); Y = real(fft(X,max(1,2*(size(G,1)-1)))); C0 = Y(1:n+1,:); C0 = 2*C0; C0(1,:) = C0(1,:)/sqrt(2); C0(:,1) = C0(:,1)/sqrt(2); C0 = fliplr(triu(fliplr(C0))); C0(n1,1) = C0(n1,1)/2; Output: C0 ↔ C 0 Stefano De Marchi Interpolation points and interpolation formulae on the square

  71. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Evaluation Given the point x = ( x 1 , x 2 ) and the coefficient matrix C 0 , the polynomial interpolation formula can be evaluated by a double matrix-vector product L n f ( x ) = T ( x 1 ) T C 0 ( f ) T ( x 2 ) Stefano De Marchi Interpolation points and interpolation formulae on the square

  72. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Franke’s function 1.2 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1 1 Stefano De Marchi Interpolation points and interpolation formulae on the square

  73. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Numerical results Interpolation on Pad n (total degree n ) vs. tensor-product interpolation on TCL n = C n +1 × C n +1 (maximum degree n 2 ) for the Franke’s function: TCL n Pad n TCL n Pad n degree n 25 34 35 48 points 625 630 1225 1225 1 . 2 · 10 − 3 4 . 3 · 10 − 5 2 . 3 · 10 − 6 3 . 3 · 10 − 8 error TCL n Pad n TCL n Pad n degree n 45 62 55 76 points 2025 2016 3015 3003 1 . 5 · 10 − 9 5 . 4 · 10 − 12 1 . 9 · 10 − 13 1 . 9 · 10 − 14 error (number of points = number of function evaluations) Stefano De Marchi Interpolation points and interpolation formulae on the square

  74. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Beyond the square The interpolation formula can be extended to other domains Ω ⊂ R 2 , by means of a suitable mapping of the square. Given σ : [ − 1 , 1] 2 → Ω t �→ x = σ ( t ) it is possible to construct the (in general nonpolynomial) interpolation formula L n f ( x ) = T ( σ ← 1 ( x )) T C 0 ( f ◦ σ ) T ( σ ← 2 ( x )) Stefano De Marchi Interpolation points and interpolation formulae on the square

  75. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Beyond the square The interpolation formula can be extended to other domains Ω ⊂ R 2 , by means of a suitable mapping of the square. Given σ : [ − 1 , 1] 2 → Ω t �→ x = σ ( t ) it is possible to construct the (in general nonpolynomial) interpolation formula L n f ( x ) = T ( σ ← 1 ( x )) T C 0 ( f ◦ σ ) T ( σ ← 2 ( x )) Stefano De Marchi Interpolation points and interpolation formulae on the square

  76. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Cubature Integration of the interpolant at the Padua points gives a nontensorial Clenshaw–Curtis cubature formula � � [ − 1 , 1] 2 f ( x ) d x ≈ [ − 1 , 1] 2 L n f ( x ) d x exact for f ∈ P 2 n . Stefano De Marchi Interpolation points and interpolation formulae on the square

  77. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Cubature Defining  1 2 c j , k − j = 1 � f ( ξ ) w ξ ˆ T j ( ξ 1 ) ˆ T k − j ( ξ 2 ) , j = k = n   2    ξ ∈ Pad n c ′ j , k − j = � f ( ξ ) w ξ ˆ T j ( ξ 1 ) ˆ c j , k − j = T k − j ( ξ 2 ) , otherwise      ξ ∈ Pad n then n k � � j , k − j ˆ T j ( x 1 ) ˆ c ′ L n f ( x ) = T k − j ( x 2 ) k =0 j =0 Stefano De Marchi Interpolation points and interpolation formulae on the square

  78. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Cubature Defining  1 2 c j , k − j = 1 � f ( ξ ) w ξ ˆ T j ( ξ 1 ) ˆ T k − j ( ξ 2 ) , j = k = n   2    ξ ∈ Pad n c ′ j , k − j = � f ( ξ ) w ξ ˆ T j ( ξ 1 ) ˆ c j , k − j = T k − j ( ξ 2 ) , otherwise      ξ ∈ Pad n then n k � � j , k − j ˆ T j ( x 1 ) ˆ c ′ L n f ( x ) = T k − j ( x 2 ) k =0 j =0 Stefano De Marchi Interpolation points and interpolation formulae on the square

  79. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Moments n k � � � c ′ [ − 1 , 1] 2 L n f ( x ) d x = j , k − j m j , k − j , k =0 j =0 �� 1 � �� 1 � ˆ ˆ m j , k − j = T j ( t ) d t T k − j ( t ) d t − 1 − 1 2 j = 0  � 1  ˆ 0 j odd T j ( t ) d t = √ − 1 2 2 j even  1 − j 2 Stefano De Marchi Interpolation points and interpolation formulae on the square

  80. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Cubature weights n k � � � � c ′ [ − 1 , 1] 2 L n f ( x ) d x = j , k − j m j , k − j = λ ξ f ( ξ ) k =0 j =0 ξ ∈ Pad n where 1  2 m j , k − j , j = k = n n k   � � m ′ j , k − j ˆ T j ( ξ 1 ) ˆ m ′ λ ξ = w ξ T k − j ( ξ 2 ) , j , k − j = m j , k − j , otherwise  k =0 j =0  The cubature weights λ ξ are not all positive. However they satisfy � lim | λ ξ | = 4 n →∞ ξ ∈ Pad n and � � [ − 1 , 1] 2 L n f ( x ) d x + o ( n − p ) , f ∈ C p ([ − 1 , 1] 2 ) [ − 1 , 1] 2 f ( x ) d x = Stefano De Marchi Interpolation points and interpolation formulae on the square

  81. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Cubature weights: stability n k � � � � c ′ [ − 1 , 1] 2 L n f ( x ) d x = j , k − j m j , k − j = λ ξ f ( ξ ) k =0 j =0 ξ ∈ Pad n where 1  2 m j , k − j , j = k = n n k   � � m ′ j , k − j ˆ T j ( ξ 1 ) ˆ m ′ λ ξ = w ξ T k − j ( ξ 2 ) , j , k − j = m j , k − j , otherwise  k =0 j =0  The cubature weights λ ξ are not all positive. However they satisfy � lim | λ ξ | = 4 n →∞ ξ ∈ Pad n and � � [ − 1 , 1] 2 L n f ( x ) d x + o ( n − p ) , f ∈ C p ([ − 1 , 1] 2 ) [ − 1 , 1] 2 f ( x ) d x = Stefano De Marchi Interpolation points and interpolation formulae on the square

  82. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Cubature weights: stability and convergence n k � � � � c ′ [ − 1 , 1] 2 L n f ( x ) d x = j , k − j m j , k − j = λ ξ f ( ξ ) k =0 j =0 ξ ∈ Pad n where 1  2 m j , k − j , j = k = n n k   � � m ′ j , k − j ˆ T j ( ξ 1 ) ˆ m ′ λ ξ = w ξ T k − j ( ξ 2 ) , j , k − j = m j , k − j , otherwise  k =0 j =0  The cubature weights λ ξ are not all positive. However they satisfy � lim | λ ξ | = 4 n →∞ ξ ∈ Pad n and � � [ − 1 , 1] 2 L n f ( x ) d x + o ( n − p ) , f ∈ C p ([ − 1 , 1] 2 ) [ − 1 , 1] 2 f ( x ) d x = Stefano De Marchi Interpolation points and interpolation formulae on the square

  83. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature � code for the cubature R Matlab Input C0 ↔ C 0 k = [0:2:n]; mom = 2*sqrt(2)./(1-k.^2); mom(1) = 2; Mmom = mom’*mom; CM = C0(1:2:n1,1:2:n1).*Mmom; Int = sum(sum(CM)); � Output: Int ↔ [ − 1 , 1] 2 L n f ( x ) d x Stefano De Marchi Interpolation points and interpolation formulae on the square

  84. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Numerical results Clenshaw–Curtis cubature on Pad n ( CCPad n ) vs. tensor-product Gauss–Legendre–Lobatto cubature ( TGLL n ) for the Franke’s function: TGLL n CCPad n TGLL n CCPad n degree n 6 7 8 10 points 36 36 64 66 4 . 2 · 10 − 3 3 . 8 · 10 − 4 1 . 3 · 10 − 4 1 . 3 · 10 − 5 error TGLL n CCPad n TGLL n CCPad n degree n 11 14 15 20 points 121 120 225 231 5 . 7 · 10 − 5 9 . 4 · 10 − 6 1 . 1 · 10 − 6 1 . 1 · 10 − 7 error (number of points = number of function evaluations) Stefano De Marchi Interpolation points and interpolation formulae on the square

  85. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Numerical results Clenshaw–Curtis cubature on Pad n ( CCPad n ) vs. tensor-product Gauss–Legendre–Lobatto cubature ( TGLL n ) for the function ( x 2 + y 2 ) 3 / 2 : TGLL n CCPad n TGLL n CCPad n degree n 6 7 8 10 points 36 36 64 66 1 . 8 · 10 − 3 3 . 8 · 10 − 4 3 . 1 · 10 − 4 1 . 4 · 10 − 7 error TGLL n CCPad n TGLL n CCPad n degree n 11 14 15 20 points 121 120 225 231 7 . 7 · 10 − 5 2 . 8 · 10 − 7 1 . 5 · 10 − 5 9 . 8 · 10 − 9 error (number of points = number of function evaluations) Stefano De Marchi Interpolation points and interpolation formulae on the square

  86. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Conclusions We studied different families of point sets for polynomial interpolation on the square. The most promising, from theoretical purposes and computational cost both of the interpolant and Lebesgue constant growth are the Padua points. More on Padua points (papers, software, links) at the CAA research group: http://www.math.unipd.it/ ∼ marcov/CAA.html http://en.wikipedia.org/wiki/Padua points. Stefano De Marchi Interpolation points and interpolation formulae on the square

  87. From Dubiner metric to Xu and Padua pts More on Padua pts Computational aspects of Padua pts Application of Padua pts to Cubature Conclusions We studied different families of point sets for polynomial interpolation on the square. The most promising, from theoretical purposes and computational cost both of the interpolant and Lebesgue constant growth are the Padua points. More on Padua points (papers, software, links) at the CAA research group: http://www.math.unipd.it/ ∼ marcov/CAA.html http://en.wikipedia.org/wiki/Padua points. Stefano De Marchi Interpolation points and interpolation formulae on the square

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