A fast Chebyshev Legendre transform using an asymptotic formula

Transcription

1 A fast Chebyshev Legendre transform using an asymptotic formula Alex Townsend MIT (Joint work with ick Hale) SIAM OPSFA, 5th June 2015

2 Introduction What is the Chebyshev Legendre transform? Suppose we have a degree 1 polynomial. It can be expressed in the Chebyshev or Legendre polynomial basis: p 1 pxq 1 ÿ k 0 The Chebyshev Legendre transform: c leg 0,..., cleg Applications in: Convolution [Hale & T., 14] Legendre-tau spectral methods QR of a quasimatrix [Trefethen, 08] best least squares approximation c leg 1 ÿ P k k pxq ðñ p 1 pxq forward transform ÝÝÝÝÝÝÝÝÝÝÝÝÝá k 0 c cheb k T k pxq Opplog q 2 { log log q c cheb,..., c cheb 1 0 âýýýýýýýýýýýýý inverse transform 1. Alex MIT 1/18

3 Introduction What is the Chebyshev Legendre transform? Suppose we have a degree 1 polynomial. It can be expressed in the Chebyshev or Legendre polynomial basis: p 1 pxq 1 ÿ k 0 The Chebyshev Legendre transform: c leg 0,..., cleg Applications in: Convolution [Hale & T., 14] Legendre-tau spectral methods QR of a quasimatrix [Trefethen, 08] best least squares approximation c leg 1 ÿ P k k pxq ðñ p 1 pxq forward transform ÝÝÝÝÝÝÝÝÝÝÝÝÝá k 0 c cheb k T k pxq Opplog q 2 { log log q c cheb,..., c cheb 1 0 âýýýýýýýýýýýýý inverse transform 1. Alex MIT 1/18

4 Introduction What is the Chebyshev Legendre transform? Suppose we have a degree 1 polynomial. It can be expressed in the Chebyshev or Legendre polynomial basis: p 1 pxq 1 ÿ k 0 The Chebyshev Legendre transform: c leg 0,..., cleg Applications in: Convolution [Hale & T., 14] Legendre-tau spectral methods QR of a quasimatrix [Trefethen, 08] best least squares approximation c leg 1 ÿ P k k pxq ðñ p 1 pxq forward transform ÝÝÝÝÝÝÝÝÝÝÝÝÝá k 0 c cheb k T k pxq Opplog q 2 { log log q c cheb,..., c cheb 1 0 âýýýýýýýýýýýýý inverse transform 1. Alex MIT 1/18

5 Introduction Related work Values at Chebyshev points Potts et al. O `plog q 2 Mori et al. O p log q discrete cosine transform O p log q Legendre coefficients Alpert & Rokhlin O pq Chebyshev coefficients Values in the complex plane Iserles O p log q Tygert O `plog q 2 Values at Legendre points ew features of our algorithms FFT-based approach Analysis-based o precomputation Alex MIT 2/18

6 Introduction The fast Fourier transform The FFT computes the DFT in Op log q operations: X k ř 1 n 0 x n e 2πink{, 0 ď k ď 1. Top 10 algorithm e 2πit sinp 2πtq Imag axis ; James Cooley t cosp 2πtq Real axis ; John Tukey Alex MIT 3/18

7 Introduction Many special functions are trigonometric-like Trigonometric functions cospωxq, sinpωxq Chebyshev polynomials T n pxq Legendre polynomials P n pxq Bessel functions J ν pzq Airy functions Aipxq Also, Jacobi polynomials, Hermite polynomials, cylinder functions, etc. Alex MIT 4/18

8 Introduction Many special functions are trigonometric-like Trigonometric functions cospωxq, sinpωxq Chebyshev polynomials T n pxq Legendre polynomials P n pxq Bessel functions J ν pzq Airy functions Aipxq Also, Jacobi polynomials, Hermite polynomials, cylinder functions, etc. Alex MIT 4/18

9 Introduction Many special functions are trigonometric-like Trigonometric functions cospωxq, sinpωxq Chebyshev polynomials T n pxq Legendre polynomials P n pxq Bessel functions J ν pzq Airy functions Aipxq Also, Jacobi polynomials, Hermite polynomials, cylinder functions, etc. Alex MIT 4/18

10 Introduction Many special functions are trigonometric-like Trigonometric functions cospωxq, sinpωxq Chebyshev polynomials T n pxq Legendre polynomials P n pxq Bessel functions J ν pzq Airy functions Aipxq Also, Jacobi polynomials, Hermite polynomials, cylinder functions, etc. Alex MIT 4/18

11 Introduction Many special functions are trigonometric-like Trigonometric functions cospωxq, sinpωxq Chebyshev polynomials T n pxq Legendre polynomials P n pxq Bessel functions J ν pzq Airy functions Aipxq Also, Jacobi polynomials, Hermite polynomials, cylinder functions, etc. Alex MIT 4/18

12 Introduction Many special functions are trigonometric-like Trigonometric functions cospωxq, sinpωxq Chebyshev polynomials T n pxq Legendre polynomials P n pxq Bessel functions J ν pzq Airy functions Aipxq Also, Jacobi polynomials, Hermite polynomials, cylinder functions, etc. Alex MIT 4/18

13 Introduction An asymptotic expansion of Legendre polynomials Legendre polynomials M 1 ÿ P n pcos θq C n C n m 0 cos`pm ` n ` 1 1 2qθ pm ` 2 h q π 2 m,n p2 sin θq m`1{2 c # 4 Γpn ` 1q π Γpn ` 3{2q, h m,n ś m j 1 ` R n,m pθq 1, m 0, pj 1{2q 2 jpn`j`1{2q, m ą 0. Thomas Stieltjes Boundary Region: Bessel like Interior Region: Trig like Alex MIT 5/18

14 Introduction umerical pitfalls of an asymptotic expansionist M 1 ÿ P n pcos θq C n m 0 cos`pm ` n ` 1 1 2qθ pm ` 2 h q π 2 m,n ` R p2 sin θq m`1{2 n,m pθq M 1 Rn,Mpθq M 3 M 5, 7 n 1,000 x Fix M. Where is the asymptotic expansion accurate? Alex MIT 6/18

15 Computing the Chebyshev Legendre transform The transform comes in two parts The Chebyshev Legendre transform naturally splits into two parts: This bit ÝÝÝÝÝÝÝá c leg 0,..., cleg p 1 1 px cheb q,..., p 0 1 px cheb âýýýýýýý IDCT ÝÝÝÝÝÝá 1 q ccheb DCT âýýýýýý 0,..., c cheb 1 Task: Compute the following matrix-vector product in quasilinear time? P 0 pcos θ 0 q... P 1 pcos θ 0 q c leg P c , θ k kπ 1 P 0 pcos θ 1 q... P 1 pcos θ 1 q c leg 1 Alex MIT 7/18

16 Computing the Chebyshev Legendre transform The transform comes in two parts The Chebyshev Legendre transform naturally splits into two parts: This bit ÝÝÝÝÝÝÝá c leg 0,..., cleg p 1 1 px cheb q,..., p 0 1 px cheb âýýýýýýý IDCT ÝÝÝÝÝÝá 1 q ccheb DCT âýýýýýý 0,..., c cheb 1 Task: Compute the following matrix-vector product in quasilinear time? P 0 pcos θ 0 q... P 1 pcos θ 0 q c leg P c , θ k kπ 1 P 0 pcos θ 1 q... P 1 pcos θ 1 q c leg 1 Alex MIT 7/18

17 Computing the Chebyshev Legendre transform Asymptotic expansions as a matrix decomposition The asymptotic expansion M 1 ÿ P n pcos θ k q C n m 0 cos`pm ` n ` 1 2 h qθ k pm ` 1 2 q π 2 m,n ` R p2 sin θ k q m`1{2 n,m pθ k q gives a matrix decomposition (sum of diagonally scaled DCTs and DSTs):» fi M 1 ÿ P D um C D Chm ` D vm 0 S 2 0fl D Chm ` R,M. m P ` P ASY R,M Alex MIT 8/18

18 Computing the Chebyshev Legendre transform Asymptotic expansions as a matrix decomposition The asymptotic expansion M 1 ÿ P n pcos θ k q C n m 0 cos`pm ` n ` 1 2 h qθ k pm ` 1 2 q π 2 m,n ` R p2 sin θ k q m`1{2 n,m pθ k q gives a matrix decomposition (sum of diagonally scaled DCTs and DSTs):» fi M 1 ÿ P D um C D Chm ` D vm 0 S 2 0fl D Chm ` R,M. m P ` P ASY R,M Alex MIT 8/18

19 Computing the Chebyshev Legendre transform Asymptotic expansions as a matrix decomposition The asymptotic expansion M 1 ÿ P n pcos θ k q C n m 0 cos`pm ` n ` 1 2 h qθ k pm ` 1 2 q π 2 m,n ` R p2 sin θ k q m`1{2 n,m pθ k q gives a matrix decomposition (sum of diagonally scaled DCTs and DSTs):» fi M 1 ÿ P D um C D Chm ` D vm 0 S 2 0fl D Chm ` R,M. m P ` P ASY R,M Alex MIT 8/18

20 Computing the Chebyshev Legendre transform Be careful and stay safe Error curve: R n,m pθ k q ɛ Fix M. Where is the asymptotic expansion accurate? R,M M 5 M 7 M 6 M 15 R n,m pθ k q ď 2C n h M,n p2 sin θ k q M`1{2 Alex MIT 9/18

21 Computing the Chebyshev Legendre transform Partitioning and balancing competing costs α 3 α 2 α P EVAL P P (3) P (2) P (1) Theorem The matrix-vector product f P c can be computed in Opplog q 2 { loglog q operations. α too small P EVAL c Op 2 q P (j) c Op log q Alex MIT 10/18

22 Computing the Chebyshev Legendre transform Partitioning and balancing competing costs α 3 α 2 α P EVAL P P (3) P (2) P (1) Theorem The matrix-vector product f P c can be computed in Opplog q 2 { loglog q operations. α too small P EVAL c Op 2 q P (j) c Op log q Alex MIT 10/18

23 Computing the Chebyshev Legendre transform Partitioning and balancing competing costs α 3 α 2 α P EVAL P P (3) P (2) P (1) Theorem The matrix-vector product f P c can be computed in Opplog q 2 { loglog q operations. α too small P EVAL c Op 2 q P (j) c Op log q Alex MIT 10/18

24 Computing the Chebyshev Legendre transform Partitioning and balancing competing costs α 3 α 2 α P EVAL P P (3) P (2) P (1) Theorem The matrix-vector product f P c can be computed in Opplog q 2 { loglog q operations. P (j) c α too large Op 2 log q P EVAL c Op log q Alex MIT 11/18

25 Computing the Chebyshev Legendre transform Partitioning and balancing competing costs α 3 α 2 α P EVAL P P (3) P (2) P (1) Theorem The matrix-vector product f P c can be computed in Opplog q 2 { loglog q operations. α Op1{ log q P (j) c Opplog q 2 { loglog q P EVAL c Opplog q 2 { loglog q Alex MIT 12/18

26 Computing the Chebyshev Legendre transform umerical results Mollification of rough signals Op 2 q Opplog q 2 { loglog q o precomputation. ż 1 1 d P pxqe iωx n dx i m 2π ω J m`1{2p ωq Alex MIT 13/18

27 Computing the Chebyshev Legendre transform The inverse Chebyshev Legendre transform The integral formula for Legendre coefficients gives the following relation: c leg j I `1 0 Ds2 P 2 px cheb 2 qt D w2 T 2 px cheb 2 q I`1 c 0 cheb, 2α 3 2α 2 2α i 1 1 i1 2 i 1 3 P EVAL T 2 P (3) T 2 P (2) T 2 P (1) T 2 Execution time Op 2 q Opplog q 2 { loglog q 2 P 2 px cheb 2 qt Alex MIT 14/18

28 Future work Fast spherical harmonic transform Spherical harmonic transform: fpθ, φq 1 ÿ lÿ l 0 m l α m l P m pcos θqe imφ l [Mohlenkamp, 1999], [Rokhlin & Tygert, 2006], [Tygert, 2008] A new generation of fast transforms with no precomputation. Alex MIT 15/18

29 Thank you Thank you Alex MIT 16/18

30 References D. H. Bailey, K. Jeyabalan, and X. S. Li, A comparison of three high-precision quadrature schemes, P. Baratella and I. Gatteschi, The bounds for the error term of an asymptotic approximation of Jacobi polynomials, Lecture notes in Mathematics, Z. Battels and L.. Trefethen, An extension of Matlab to continuous functions and operators, SISC, Iteration-free computation of Gauss Legendre quadrature nodes and weights, SISC, G. H. Golub and J. H. Welsch, Calculation of Gauss quadrature rules, Math. Comput., Hale and A. Townsend, Fast and accurate computation of Gauss Legendre and Gauss Jacobi quadrature nodes and weights, SISC, Hale and A. Townsend, A fast, simple, and stable Chebyshev Legendre transform using an asymptotic formula, SISC, Y. akatsukasa, V. oferini, and A. Townsend, Computing the common zeros of two bivariate functions via Bezout resultants, umerische Mathematik, Alex MIT 17/18

31 References F. W. J. Olver, Asymptotics and Special Functions, Academic Press, ew York, A. Townsend and L.. Trefethen, An extension of Chebfun to two dimensions, SISC, A. Townsend, T. Trogdon, and S. Olver, Fast computation of Gauss quadrature nodes and weights on the whole real line, to appear in IMA umer. Anal., A. Townsend, The race to compute high-order Gauss quadrature, SIAM ews, A. Townsend, A fast analysis-based discrete Hankel transform using asymptotics expansions, submitted, F. G. Tricomi, Sugli zeri dei polinomi sferici ed ultrasferici, Ann. Mat. Pura. Appl., Alex MIT 18/18