Cauchy Fast Multipole Method for Analytic Kernel

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1 Cauchy Fast Multipole Method for Analytic Kernel Pierre-David Létourneau 1 Cristopher Cecka 2 Eric Darve 1 1 Stanford University 2 Harvard University ICIAM July 2011 Pierre-David Létourneau Cristopher Cecka Eric Darve Cauchy FMM 1 / 19

2 Outline 1 Overview 2 Algorithm 3 Results Pierre-David Létourneau Cristopher Cecka Eric Darve Cauchy FMM 2 / 19

3 Outline 1 Overview 2 Algorithm 3 Results Pierre-David Létourneau Cristopher Cecka Eric Darve Cauchy FMM 3 / 19

4 Overview Perform matrix-vector product of the form N σ i K (x i, y j ) ; j = 1..N i=1 e.g. electrostatic kernel : K (x i, y j ) = 1 x i y j Naive implementation : O(N 2 ) Pierre-David Létourneau Cristopher Cecka Eric Darve Cauchy FMM 4 / 19

5 Overview One way to speed up the computation is to group together particles and use a low-rank approximation of the kernel. N σ i K (x i, y j ) P Q K (x, y) a pq u p (x)v q (y) p=1 q=1 ( Q P )) N a pq σ i u p (x i v q (y j ) i=1 q=1 p=1 i=1 Pierre-David Létourneau Cristopher Cecka Eric Darve Cauchy FMM 5 / 19

6 Tree decomposition Usually, it is not possible to get an efficient global separable approximation. Instead, use approximations valid only in certain region (far apart) and use a tree decomposition. 1 Use different low-rank approximation at each scale/tree level 2 Compute close-term interactions at lowest level e.g far-neighbors (top) and close-terms (bottom) Pierre-David Létourneau Cristopher Cecka Eric Darve Cauchy FMM 6 / 19

7 Goal We want to develop an algorithm with the following properties : 1 The algorithmic complexity is O(N) (up to log-factor) 2 The translation and transfer operators are diagonal P Q P K (x, y) a pqu p(x)v q(y) vs a pu p(x)v p(y) p=1 q=1 p=1 3 The algorithm is applicable to any analytic kernel Pierre-David Létourneau Cristopher Cecka Eric Darve Cauchy FMM 7 / 19

8 Outline 1 Overview 2 Algorithm 3 Results Pierre-David Létourneau Cristopher Cecka Eric Darve Cauchy FMM 8 / 19

9 Algorithm The algorithm is based on Cauchy Integral formula i.e. given K (z), z C, complex differentiable z Ω (holomorphic), then K (x) = 1 K (z) dz (1) 2πı Γ z x Pierre-David Létourneau Cristopher Cecka Eric Darve Cauchy FMM 9 / 19

10 Algorithm (Formal derivation) Assume that R(z (x y)) > 0 z Γ. Then, by the Laplace transform K (x y) = 1 K (z) 2πı Γ z (x y) dz = 1 K (z) e s(z (x y)) ds dz 2πı Γ 0 which upon discretization gives, K (x y) 1 K (z n ) e sp(zn (x y)) 2πı n p = ( ) 1 K (z n )e spzn e spx e spy 2πı p n This expansion possesses the desired form Pierre-David Létourneau Cristopher Cecka Eric Darve Cauchy FMM 10 / 19

11 Algorithm (Formal derivation) Current issues with the separable expansion : 1 It is not true that R(z (x y)) > 0 z Γ. 2 The use of exponentials e s k z n, e s k x and e s k y is numerically unstable Both problems can however be solved. Pierre-David Létourneau Cristopher Cecka Eric Darve Cauchy FMM 11 / 19

12 Algorithm (Formal derivation) To solve the first problem, we break the contour into paths and rotate each path by multiplying by {λ k } {1, ı, 1, ı}. Γ 3 Im Γ 2 x y Γ 1 Re Im Re x Γ2 = ıγ 2 Γ 4 1 λ k K (z) 2πı Γ k λ k z (x y) dz = λ k K (z) 2πı Γ k 0 e sλ k (z (x y)) ds dz Pierre-David Létourneau Cristopher Cecka Eric Darve Cauchy FMM 12 / 19

13 Algorithm (Formal derivation) To solve the stability problem, we introduce parameters {l k } such that R(λ k z 2l k ) > 0 (2) R(l k λ k x) > 0 (3) R(l k + λ k y) > 0 (4) Thanks to the well-separated condition, the tree decomposition and the choice of paths, it is always possible to find such parameters. Then, 0 e sλ k (z (x y)) ds = 0 e s(λ k z 2l k ) e s(l k λ k x) e s(l k +λ k y) ds which can be discretized efficiently to produce a stable approximation. Pierre-David Létourneau Cristopher Cecka Eric Darve Cauchy FMM 13 / 19

14 Outline 1 Overview 2 Algorithm 3 Results Pierre-David Létourneau Cristopher Cecka Eric Darve Cauchy FMM 14 / 19

15 Results Kernel studied, Algebraic growth (Multiquadric) : K (x y) = 1 + (x y) 2 Algebraic decay (Inverse Multiquadric) : K (x y) = Oscillatory (Helmholtz) : K (x y) = eık x y k x y Error considered, Maximum absolute error : max n φ FMM Relative L 2 error : Particle distribution, n (φfmm n φ Direct n ) 2 n (φdirect n ) 2 n φ Direct n N locations uniformly distributed in an interval Various interval size : 0.1, 1, 10, (x y) 2 Pierre-David Létourneau Cristopher Cecka Eric Darve Cauchy FMM 15 / 19

16 Results (Single level - Maximum absolute error vs # Quadrature points) Pierre-David Létourneau Cristopher Cecka Eric Darve Cauchy FMM 16 / 19

17 Results (Multiple levels - Relative L 2 error vs # Quadrature points) Number of particles : Number of levels : 8 Pierre-David Létourneau Cristopher Cecka Eric Darve Cauchy FMM 17 / 19

18 Results Multiple levels - Timing vs Problem Size Interval length : 100 Relative L 2 error : Multiquadric( 10 6 ), Inverse Multiquadric( 10 3 ), Helmholtz( 10 5 ) Computer : Dell 3GHz CPU, 8GBytes RAM Pierre-David Létourneau Cristopher Cecka Eric Darve Cauchy FMM 18 / 19

19 Conclusion 1 The algorithmic complexity is O(N log(n)) 2 The translation and transfer operators are diagonal 3 The algorithm is applicable to any analytic kernel 4 Extension to n-d is trivial K (x, y, z) = 1 K (z 1, z 2, z 3 ) (2πı) 3 Γ 1 Γ 2 Γ 3 (z 1 x)(z 2 y)(z 3 z) dz 1 dz 2 dz 3 Pierre-David Létourneau Cristopher Cecka Eric Darve Cauchy FMM 19 / 19

Kernel Independent FMM

Kernel Independent FMM FMM Issues FMM requires analytical work to generate S expansions, R expansions, S S (M2M) translations S R (M2L) translations R R (L2L) translations Such analytical work leads to