A butterfly algorithm. for the geometric nonuniform FFT. John Strain Mathematics Department UC Berkeley February 2013

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1 A butterfly algorithm for the geometric nonuniform FFT John Strain Mathematics Department UC Berkeley February

2 FAST FOURIER TRANSFORMS Standard pointwise FFT evaluates f(k) = n j=1 e 2πikj/n f j k n/2 O(n d+1 ) O(n d log n) with rigid algebraic recursion for equidistant points Multidimensional pointwise nonuniform NUFFT evaluates f(t k ) = N j=1 e itt k s j f j 1 k N O(N 2 ) O(N log N) via low-rank expansion and butterfly recursion Geometric GNUFFT adds dimensional recursion for < g k χ Tk, f >= T k g k (t) N j=1 S j e itt s f j (s) ds dt with N polynomials g k, f j on simplices T k, S j in R D 2

3 OUTLINE Low-rank expansions separate variables enable fast local interactions Butterfly algorithm propagates information between scales simultaneously merge sources and focus targets New dimensional recursion simplifies matrix elements f kj exactly evaluated by fast low-rank expansion gives direct algorithms as well as fast algorithms 3

4 LOW-RANK EXPANSION Complex exponential Taylor series e z = m z α α=0 α! + E m, E m ( z e m ) m Approximates D-dimensional clustered nonuniform FFT f(t k ) = N j=1 e itt k s j f j = t α iα N k α! α m j=1 f j s α j + E m = α m C α t α k + E m E m F D ( ) Re m F = f j t T k m s j R j Fast algorithm for clustered interactions: form O(m D ) moments C α of N sources s j evaluate O(m D )-term series f(t k ) at N targets t k total cost O(Nm D ) = O(N log ɛ) for accuracy ɛ assuming all sources and targets have t T k s j R = O(1) 4

5 LOCALIZE Usually t T k s j = O(N) is not bounded by a constant R Instead s j = O(1) and t k = O(N) or vice versa so R = O(N) Shift to centers τ and σ of intervals T and S e itt k s j = e iτ T σ e i(t k τ) T σ e i(t k τ) T (s j σ) e iτ T (s j σ) = e iτ T σ e i(t k τ) T σ α m i α α! (t k τ) α (s j σ) α e iτ T (s j σ) + E m Accurate in Heisenberg pairs (T, S) where (t k T, s j S) (t k τ) T (s j σ) R E m ɛ 5

6 BUTTERFLY ALGORITHM Sort N sources into O(N) cells Build N partial expansions, each converging at all N targets Repeatedly split and merge shifted expansions split each target expansion into 2 D adjacent children merge 2 D adjacent source children into parent expansion until... Evaluate 1 total expansion at targets in each target cell 6

7 SHIFT SOURCE MOMENTS For targets in cell T near τ and sources in cell S near σ s j S e itt k s j f j = e i(t k τ)t σ α (t k τ) α C α (σ, τ) C α (σ, τ) = e iτ T σ iα α! s j S ( sj σ ) α e iτ T (s j σ) f j Exponential expansion shifts τ to target child cell centers {ξ} ( β + α C α (σ, ξ) = e i(ξ τ)t σ β β ) (ξ τ) β C β+α (σ, τ) Binomial theorem shifts {σ} to source parent cell center ρ C α (ρ, ξ) = σ β α i α β (α β)! (σ ρ)α β C β (σ, ξ) Step (S, T ) (parent of S, children of T ) preserves R 7

8 D-DIMENSIONAL POINTWISE NUFFT 0. Organize source and target points into D-dimensional L-level quadtrees with 2 L R S R T R so D(Re/m) m ɛ 1. Build coefficients C α (σ L, τ 0 ) for leaf source cells σ L and root target cells τ 0 2. For l = 1... L Recursively shift and merge coefficients to each child τ l of target cell τ l 1, yielding C α (σ L l+1, τ l ) parent σ L l of each source cell σ L l+1, summing to C α (σ L l, τ l ) 3. Evaluate expansion with coefficients C α (σ 0, τ L ) for root source cells σ 0 and leaf target cells τ L 8

9 POINTWISE COMPUTATIONAL KERNEL One computational kernel T α T α + iα α! (s σ)α e i(t τ)t (s σ) does all direct evaluation with n S = n T = 0 coefficient building with n S > n T = 0 expansion evaluation with n T > n S = 0 β n T (t τ) β S β α n S Key observation: either n S or n T is zero Generalize sums over points s or t to integrals over s or t 9

10 GEOMETRIC SOURCES AND TARGETS Points sources, targets and densities with geometry Points s j, t k d-dimensional simplices S j, T k in R D points, line segments, triangles, tetrahedra,... Densities f j polynomials f j (s), g k (t) on simplices Matrix elements e itt k s j integrals f kj = g k (t) e itt s f j (s) ds dt T k S j Fourier transform sum of integrals f(k) = N j=1 f kj = T k g k (t) N j=1 S j e itt s f j (s) ds dt 10

11 GNUFFT INITIALIZATION 0. Organize source and target simplices into D-dimensional L-level quadtrees where 2 L R S R T R and D(Re/m) m ɛ Approximate inclusion is enough But some simplices are left behind in non-leaf cells 11

12 GNUFFT COEFFICIENTS 1. Build geometric integrated coefficients in expansion S j σ S j e itt s f j (s) ds = e i(t τ)t σ α (t τ) α C α (σ, τ) C α (σ, τ) = e iτ T σ iα α! S j S S j (s σ) α e iτ T (s σ) f j (s) ds Oscillatory integrands challenge quadrature schemes Use dimensional recursion (later) to obtain exact coefficients 12

13 GNUFFT BUTTERFLY RECURSION Almost identical to pointwise NUFFT! 2. For l = 1... L a. Recursively shift and merge coefficients to each child τ l of target cell τ l 1, yielding C α (σ L l+1, τ l ) parent σ L l of each source cell σ L l+1, yielding C α (σ L l, τ l ) b. Add source simplices left behind on level L l 13

14 GNUFFT EXPANSION INTEGRATION 3. Integrate expansion (σ 0, τ L ) over target simplices in each level-l target cell τ L Dual to coefficient initialization Use same dimensional recursion to obtain exact integrals 14

15 DIMENSIONAL RECURSION On d-dimensional simplex S carrying polynomial p F (t, d, S, p, α, σ) = S eitt s (s σ) α p(s) ds Gauss theorem integrates by parts parallel to S S qt f(s) ds = S qt nf(σ) dσ Average over parallel target components t S S to get F (t, d, S, p, α, σ) = i t S 2 D j=1 d f=0 ( F (ts, d, S, t T S p, α, σ) α j F (t S, d, S, p, α e j, σ) t T S n ff (t S, d 1, f S, p, α, σ) 15

16 CONCLUSIONS Geometric extension of butterfly algorithm exact Fourier transform of simplicial polynomials arbitrary dimension and codimension O(N log N log ɛ) work with accuracy ɛ Simple speedups: Tabulate or approximate shift/merge operators Optimized compressed dimensional recursion Taylor expansion Chebyshev approximation Taylor expansion accurate on disk in complex plane GNUFFT evaluates geometric Laplace and Gauss transforms with single parametrized code 16