the short-range part

Volltext

(1)Faculty of Mathematics. Silica melt. N -body problem. Cloud-wall. 3d-periodic. Task. N X X. φSp (r j ) := N = 12960 charged particles. N = 300 charged particles. 0. the short-range part. de. cos.. 1d-periodic. 3dp:. N X X. 2α 0q erfc(αkr ij + Lnk) − √ qj i kr ij + Lnk π n∈Sp i=1. Θ3d(k). N X X erf(αkr ij + Lnk) L qi φSp (r j ) = . kr ij + Lnk. a. 0d-periodic. Computing the Fourier series of the long-range part φLSp (r j ) along periodic dimensions converts x → kx, y → ky , z → kz and yields the well known Ewald formulas. N 2πi q X X 3dp: φLS3 (r j ) = qi Θ3d kx2 + ky2 + kz2 e L (kxxij +ky yij +kz zij ). 0.2. α = 2.0 α = 1.5 α = 1.0 α = 0.5. 0.1 0 r. 5. Long-range evaluation. 0. where the analytically known, continuous Fourier transform of 2 1 2d 2d −v b Θ (k, ·) fulfills Θ (k, v) ∼ v2 e for v → ∞.. C. type A. h ≥ 2L. But the kink at r = ±h implies a rather slow 2nd order convergence in Fourier space.. type B. N X. Sk := φLSp (r j ) ≈. qi e2πik·ri. i=1 X. pd ck Sk e−2πik·rj k∈M. 3h L2 h h 2. −L −h 2. L h h−L 2. 3h 2. Type B2: Instead, we construct an interpolating polynomial within [L, h − L] that fits the first m derivatives of Θpd(k, r) at r = ±L. Then, the convergence rate will be m + 2. Final approximation. φLSp (r j ) ≈ 0 r. 5. 10. N X i=1. X pd qi ck e2πik·rij , k∈M. where k = (kx, ky , kz ) runs in a finite 3d mesh M.. Modularized algorithm structure. Fast particle-mesh algorithms. 3dp 2dp 1dp 0dp. P3M [Hockney, Eastwood 1988] PME [Darden et al. 1993] SPME [Essmann et al. 1995] Type B1 approximated fast Ewald [Martyna et al. 2002] Gaussian split Ewald [Shan et. al 2004] NFFT based fast summation [Nieslony, Potts, Steidl 2004] NFFT based fast Ewald [Hedman, Laaksonen 2006] Spectrally accurate Ewald [Lindbo, Tornberg 2011, 2012] P2NFFT [Nestler, Pippig, Potts 2013, 2015]. ! ! ! % ! % ! ! !. % % % ! % % % ! !. % % % ! % % % % !. % % % ! % ! % % !. One of the benefits of the P2NFFT is its highly modularized structure. All the performance critical steps are encapsulated and can be implemented by existing software libraries. Algorithmic modules P2NFFT. ⊃. NFFT. ⊃. FFT. ScaFaCoS. ⊃. PNFFT. ⊃. PFFT. Parallel software libraries. Massive parallelism Mz - mesh size along non-periodic dimensions. rms force error. 10−1. Mz = 40. 2dp. 10−3 Mz = 80 10−5 Mz = 158 10−7 Mz = 300 10−9 N = 300 10−11 L = 10 Mz = 588 h ≈ 25 10−13 0.2 0.5 0.8 1.1 1.4 1.7 2 splitting parameter α. Visit our software page or join us at Github! www.tu-chemnitz.de/~mpip/software.php.en www.github.com/mpip. Mz - mesh size along non-periodic dimensions. 10−1 rms force error. 3dp. 10−3 Mz = 32 10−5 Mz = 64 10−7 Mz = 128 10−9 10−11 N = 300 Mz = 256 L = 10 10−13 0.2 0.5 0.8 1.1 1.4 1.7 2 splitting parameter α. 3h 2. In summary, we can write the truncated series as. 0 −10 −5. We demonstrate the high accuracy of P2NFFT at the example of a cloud-wall test system with N = 300 particles. Mz = 16. B2. Cm. L h h−L −L h −22 L h h−L 2. 0.2. High accuracy independent of periodicity Mx - mesh size along periodic dimensions. B1. C0. −L −h 2. 0.4. 10. m. But what if Θpd(k, r) does not decay fast enough? Type B1: A first attempt is to repeat Θpd(k, r) with period. type B. 0.6. B1. Type B Fourier approximations. type B. Mz = 58. 1dp. 10−3 Mz = 112 10−5 Mz = 224 10−7 Mz = 424 10−9 N = 300 Mz = 832 10−11 L = 10 h ≈ 40 10−13 0.2 0.5 0.8 1.1 1.4 1.7 2 splitting parameter α. The P2NFFT software library was designed for massive parallelism with distributed memory. We demonstrate the scalabilty on a BlueGene/P up to 16384 cores. wall clock time in [s]. 2. The approximated long-range part is evaluated by non-equispaced fast Fourier transforms (NFFT) within O(N log N ) operations.. rms force error. type A. Historical context. If the particles are sufficiently homogeneously distributed, the short-range part φSSp (r j ) can be computed directly within O(N ) operations.. S0 = {0}3. kz ∈Z. n∈Z. −L h −2. type A. 0 −10 −5. Short-range evaluation. 10−1. type B. The decay of the smooth functions Θpd(k, r) for r → ∞ falls into two categories. Type A functions decay very fast, while type B functions do not decay at all or not fast enough.. How to convert the non-periodic dimensions to Fourier space?. 3d-NFFT:. 2 2 2 2 −π k /(α L) e. Decay of type A and B. 0.3. S1 = Z × {0}2. If Θpd(k, r) is neglible for |r| ≥ h ≥ 2L, we can use its h-periodization instead and apply the Poisson summation formula. E.g., in the 2d-periodic case we have 2πi X X 1 b 2d k, kz e h kz z , Θ2d(k, z) ≈ Θ2d(k, z + hn) = Θ h h. All of these functions asymptotically tend to zero as k12 e−k for k → ∞, which justifies truncation of the Fourier series along periodic dimensions.. Remaining problem:. adjoint 3d-NFFT:. S2 = Z2 × {0}. 2. Θ1d(1, r). 0dp:. i=1 kx,ky ,kz N q 2πi X X φLS2 (r j ) = qi Θ2d kx2 + ky2 , |zij | e L (kxxij +ky yij ) i=1 kx,ky N q 2πi X X 2 + z2 L (kx xij ) φLS1 (r j ) = qi Θ1d |kx|, yij e ij i=1 kx N q X 2 + z2 φLS0 (r j ) = qi Θ0d x2ij + yij ij i=1. 0dp:. L. Type A Fourier approximation. C. 1dp:. Fourier series along periodic dimensions. 1dp:. S3 = Z3. 2 πLk √ 2 π 1 −α2r2 √ 2d Θ (0, r) := − 2 e + πz erf(αr) α L h 1 πk 2d 2πkr/L e erfc + αr Θ (k, r) := 2Lk αL i πk −2πkr/L +e erfc − αr αL h i 1 γ + Γ(0, α2r2) + ln(α2r2) Θ1d(0, r) := − L 2 2 1 π k Θ1d(k, r) := K0 α2Lx2 , α2r2 L erf(αr) 0d Θ (r) := r. 2dp:. n∈Sp i=1. :=. L. L. The Fourier coefficients and their type of decay are given as follows.. and the long-range part. 2dp:. L. L. Θ1d(0, r). φSSp (r j ) =. qi , kr ij + Lnk. Analytically known Fourier coefficients. erfc(αr) erf(αr) 1 The identity r = r + r splits φSp (r j ) = φSSp (r j ) + φLSp (r j ) with. Fourier Approximations. c s . w ww. L. where r ij = (xij , yij , zij ) := r i − r j ∈ [−L, L)3.. Ewald splitting. P NFFT. . For N particles r j = (xj , yj , zj ) ∈ [−L/2, L/2)3 with charges qj compute the Coulomb potential under periodic boundary conditions along the first p dimensions. n∈Sp i=1. Numerical Results. 2d-periodic. icle Part - Par. 2. fa. P NFFT -. A versatile framework for computing NFFT based fast Ewald summation. - NFFT e l tic. P2 NF FT. Applied Functional Analysis Franziska Nestler and Michael Pippig. 101 100. 0dp. P2NFFT ideal scaling long-range part short-range part. Computing potential and field of a 0d-periodic silica melt −1 10 with N = 829440 particles. The rms potential error was tuned to 10−5. 6 8 10 12 14 2 2 2 2 2 number of BlueGene/P cores. Franziska Nestler franziska.nestler@mathematik.tu-chemnitz.de www.tu-chemnitz.de/~nesfr. Michael Pippig michael.pippig@mathematik.tu-chemnitz.de www.tu-chemnitz.de/~mpip.

(2)