Towards a Faster Spherical Harmonic transform

Size: px

Start display at page:

Download "Towards a Faster Spherical Harmonic transform"

Miranda McKinney
5 years ago
Views:

1 Towards a Faster Spherical Harmonic transform Nathanaël Schaeffer LGIT / CNRS / Université Joseph Fourier IUGG CMG, Pisa, 7 May 2010 N. Schaeffer (LGIT/CNRS/UJF) Towards a faster SHT IUGG CMG, Pisa, 7 May / 13

2 1 Introduction 2 Taking advantage of the DCT 3 The SHTns library N. Schaeffer (LGIT/CNRS/UJF) Towards a faster SHT IUGG CMG, Pisa, 7 May / 13

3 Spherical Harmonic Transform (SHT) Definition Spherical Harmonic of degree l and order m, defined on the sphere : Y m l (θ, φ) Eigenfunction of the Laplace operator on the sphere : Y m l = l(l + 1)Y m l N. Schaeffer (LGIT/CNRS/UJF) Towards a faster SHT IUGG CMG, Pisa, 7 May / 13

4 Spherical Harmonic Transform (SHT) Definition Spherical Harmonic of degree l and order m, defined on the sphere : Y m l (θ, φ) Eigenfunction of the Laplace operator on the sphere : Y m l They form an orthonormal basis f (θ, φ) = l,m = l(l + 1)Y m l Q m l Y m l (θ, φ) Q m l = f (θ, φ) Y m l (θ, φ) sin(θ)dθdφ N. Schaeffer (LGIT/CNRS/UJF) Towards a faster SHT IUGG CMG, Pisa, 7 May / 13

5 Application for numerical simulations Advantages Spectral convergence Exact derivatives Boundary conditions for a magnetic field are straightforward. N. Schaeffer (LGIT/CNRS/UJF) Towards a faster SHT IUGG CMG, Pisa, 7 May / 13

6 Application for numerical simulations Advantages Spectral convergence Exact derivatives Boundary conditions for a magnetic field are straightforward. Drawback As of today, Gauss-Legendre algorithm is the best choice (complexity N θ (N lm + N φ log N φ )), which means that for high resolution you ll spend most of your time performing SHT! N. Schaeffer (LGIT/CNRS/UJF) Towards a faster SHT IUGG CMG, Pisa, 7 May / 13

7 Application for numerical simulations Advantages Spectral convergence Exact derivatives Boundary conditions for a magnetic field are straightforward. Drawback As of today, Gauss-Legendre algorithm is the best choice (complexity N θ (N lm + N φ log N φ )), which means that for high resolution you ll spend most of your time performing SHT! A fast algorithm would not harm! N. Schaeffer (LGIT/CNRS/UJF) Towards a faster SHT IUGG CMG, Pisa, 7 May / 13

8 Fast algorithms There are a few fast algorithm : Healy (1996,2003) : FPM (source available) Mohlenkamp (1997,1999) : Wavelet approach (approximate) Reiji Suda & Masayasu Takami (2001) : FMM (approximate) Pots (2003, 2006) : DCT, FPM (source available) Tygert (2008) : FMM (approximate) N. Schaeffer (LGIT/CNRS/UJF) Towards a faster SHT IUGG CMG, Pisa, 7 May / 13

9 Fast algorithms There are a few fast algorithm : Healy (1996,2003) : FPM (source available) Mohlenkamp (1997,1999) : Wavelet approach (approximate) Reiji Suda & Masayasu Takami (2001) : FMM (approximate) Pots (2003, 2006) : DCT, FPM (source available) Tygert (2008) : FMM (approximate) They all claim to be faster for N 512. But... N. Schaeffer (LGIT/CNRS/UJF) Towards a faster SHT IUGG CMG, Pisa, 7 May / 13

10 Fast algorithms There are a few fast algorithm : Healy (1996,2003) : FPM (source available) Mohlenkamp (1997,1999) : Wavelet approach (approximate) Reiji Suda & Masayasu Takami (2001) : FMM (approximate) Pots (2003, 2006) : DCT, FPM (source available) Tygert (2008) : FMM (approximate) They all claim to be faster for N 512. But... Timing for 1 scalar SHT l max Pots Healy Gauss Healy/Gauss s 5.5 ms 0.69 ms s 27 ms 6.5 ms 4.2 N. Schaeffer (LGIT/CNRS/UJF) Towards a faster SHT IUGG CMG, Pisa, 7 May / 13

11 Fast algorithms There are a few fast algorithm : Healy (1996,2003) : FPM (source available) Mohlenkamp (1997,1999) : Wavelet approach (approximate) Reiji Suda & Masayasu Takami (2001) : FMM (approximate) Pots (2003, 2006) : DCT, FPM (source available) Tygert (2008) : FMM (approximate) They all claim to be faster for N 512. But... Timing for 1 scalar SHT l max Pots Healy Gauss Healy/Gauss s 5.5 ms 0.69 ms s 27 ms 6.5 ms s 143 ms 49 ms 2.9 N. Schaeffer (LGIT/CNRS/UJF) Towards a faster SHT IUGG CMG, Pisa, 7 May / 13

12 Fast algorithms There are a few fast algorithm : Healy (1996,2003) : FPM (source available) Mohlenkamp (1997,1999) : Wavelet approach (approximate) Reiji Suda & Masayasu Takami (2001) : FMM (approximate) Pots (2003, 2006) : DCT, FPM (source available) Tygert (2008) : FMM (approximate) They all claim to be faster for N 512. But... Timing for 1 scalar SHT l max Pots Healy Gauss Healy/Gauss s 5.5 ms 0.69 ms s 27 ms 6.5 ms s 143 ms 49 ms s 860 ms 297 ms 2.9 N. Schaeffer (LGIT/CNRS/UJF) Towards a faster SHT IUGG CMG, Pisa, 7 May / 13

13 Gauss beats them all! Gauss-Legendre Quadrature P k (x)dx N P k (x j )w j For polynomials of degree l, the quadrature is exact if 2N > k. j=1 N. Schaeffer (LGIT/CNRS/UJF) Towards a faster SHT IUGG CMG, Pisa, 7 May / 13

14 Gauss beats them all! Gauss-Legendre Quadrature P k (x)dx N P k (x j )w j For polynomials of degree l, the quadrature is exact if 2N > k. j=1 This means that our integral f m (θ) l (θ) sin θdθ }{{}}{{} l l Y m is exactly computed by the quadrature if we use 2N θ > 2l max Gauss points. N. Schaeffer (LGIT/CNRS/UJF) Towards a faster SHT IUGG CMG, Pisa, 7 May / 13

15 Gauss beats them all! Gauss-Legendre Quadrature P k (x)dx N P k (x j )w j For polynomials of degree l, the quadrature is exact if 2N > k. j=1 This means that our integral f m (θ) l (θ) sin θdθ }{{}}{{} l l Y m is exactly computed by the quadrature if we use 2N θ > 2l max Gauss points. The Gauss-Legendre quadrature rule keeps the number of colocation points low N. Schaeffer (LGIT/CNRS/UJF) Towards a faster SHT IUGG CMG, Pisa, 7 May / 13

16 Gauss on steroids θ Parity separation : all Y m l have defined parity (x2 speedup) N. Schaeffer (LGIT/CNRS/UJF) Towards a faster SHT IUGG CMG, Pisa, 7 May / 13

17 Gauss on steroids θ Parity separation : all Yl m have defined parity (x2 speedup) Polar optimization : for large m, Yl m (θ) is almost zero near the poles ( 15% speed-up) N. Schaeffer (LGIT/CNRS/UJF) Towards a faster SHT IUGG CMG, Pisa, 7 May / 13

18 Go to DCT space For m even with Y m l (θ, φ) = P m l (cos θ)e imφ P m l (cos θ) = l k=0 ak lm cos(kθ) N. Schaeffer (LGIT/CNRS/UJF) Towards a faster SHT IUGG CMG, Pisa, 7 May / 13

19 Go to DCT space For m even with Y m l (θ, φ) = P m l (cos θ)e imφ P m l (cos θ) = l k=0 ak lm cos(kθ) + Less terms in the sum (independant of N θ ) : Efficient for small m only. l max fk m = Ql m ak lm l=k + Fast DCT : f m (θ i ) = DCT [f m k ] + Fast DCT requires N θ > l max equally spaced nodes. We can no longer use the Gauss quadrature. N. Schaeffer (LGIT/CNRS/UJF) Towards a faster SHT IUGG CMG, Pisa, 7 May / 13

20 Fejér (or Clenshaw-Curtis) quadrature Quadrature in DCT space l max Ql m = f (θ)yl m (θ) = f k k=0 l max T k Yl m = k=0 f k W lm k With T k (cos θ) = cos(kθ) the Tchebychev polynomial. This is te Fejér quadrature with weight Y m l, which is exact for N θ > l max. As the Wk lm matrix is dense, there is no advantage in computing the quadrature in the cosine domain, so we go back to real domain : Q m l = k f m (θ k ) IDCT [W lm k ] which is still exact for N θ > l max as the Gauss quadrature We can of course precompute the W lm k coefficients. N. Schaeffer (LGIT/CNRS/UJF) Towards a faster SHT IUGG CMG, Pisa, 7 May / 13

21 For non-linear simulations? Aliasing For an equation with quadratic non-linear terms : f m (θ) Yl m (θ) sin θdθ }{{}}{{} 2l l is exactly computed by a Gauss quadrature if we use 2N θ > 3l max Gauss points. a Fejér quadrature if we use N θ > 2l max regular grid points. Aliasing for non-linear problems penalizes the Fejér quadrature. N. Schaeffer (LGIT/CNRS/UJF) Towards a faster SHT IUGG CMG, Pisa, 7 May / 13

22 a library for Spherical Harmonic Transform Aimed at Numerical Simulations, it focuses on speed (but with high double precision accuracy). both scalar and vector transforms flexible truncation latitude-major or longitude-major spatial data ordering various conventions (normalizations, Condon-Shortley phase) can be called from fortran or c/c++ programs Highly optimized Gauss-Legendre algorithm DCT algorithm for regular nodes (generalized Fejér quadrature) support for SSE2 & SSE3 vectorization with gcc N. Schaeffer (LGIT/CNRS/UJF) Towards a faster SHT IUGG CMG, Pisa, 7 May / 13

23 Timing comparison The inverse SHT Ql m f (θ, φ) is much faster with the DCT : l max m max perf. gain /Gauss x4 x1.9 x1.15 N. Schaeffer (LGIT/CNRS/UJF) Towards a faster SHT IUGG CMG, Pisa, 7 May / 13

24 Timing comparison The inverse SHT Ql m f (θ, φ) is much faster with the DCT : l max m max perf. gain /Gauss x4 x1.9 x1.15 However, the analysis step involving quadrature is just the same. N. Schaeffer (LGIT/CNRS/UJF) Towards a faster SHT IUGG CMG, Pisa, 7 May / 13

25 Timing comparison The inverse SHT Ql m f (θ, φ) is much faster with the DCT : l max m max perf. gain /Gauss x4 x1.9 x1.15 However, the analysis step involving quadrature is just the same. Hence the real-life results are not as good : l max = 1021, m max = 10, NL order=2, 2x isht, 1x SHT DCT is % faster N. Schaeffer (LGIT/CNRS/UJF) Towards a faster SHT IUGG CMG, Pisa, 7 May / 13

26 Conclusions & future developments Conclusions Asymptotically fast algorithms do not appear fast yet! A DCT based SHT is more efficient for anisotropic truncations. A highly optimized Gauss SHT is still the overall winner! Anyone producing a fast algorithm should compare with it. Freely available SHTns library that picks the fastest for you N. Schaeffer (LGIT/CNRS/UJF) Towards a faster SHT IUGG CMG, Pisa, 7 May / 13

27 Conclusions & future developments Conclusions Asymptotically fast algorithms do not appear fast yet! A DCT based SHT is more efficient for anisotropic truncations. A highly optimized Gauss SHT is still the overall winner! Anyone producing a fast algorithm should compare with it. Freely available SHTns library that picks the fastest for you Future work SHTns will use the tremendous computing power of GPUs. Maybe parallelization. Perhaps somebody needs to writes a good implementation of a fast algorithm... N. Schaeffer (LGIT/CNRS/UJF) Towards a faster SHT IUGG CMG, Pisa, 7 May / 13

Angular Momentum. Classically the orbital angular momentum with respect to a fixed origin is. L = r p. = yp z. L x. zp y L y. = zp x. xpz L z.

Angular Momentum. Classically the orbital angular momentum with respect to a fixed origin is. L = r p. = yp z. L x. zp y L y. = zp x. xpz L z. Angular momentum is an important concept in quantum theory, necessary for analyzing motion in 3D as well as intrinsic properties such as spin Classically the orbital angular momentum with respect to a