A Fast N-Body Solver for the Poisson(-Boltzmann) Equation

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1 A Fast N-Body Solver for the Poisson(-Boltzmann) Equation Robert D. Skeel Departments of Computer Science (and Mathematics) Purdue University 1

2 Thesis Poisson(-Boltzmann) equation can be solved much faster. Using a fast N-body solver as preconditioner is key. Especially suited for modern computers. Work with D. Yershov and S. Bond. Multilevel summation is the method of choice for molecular biophysics / structural biology. Work with D. Hardy. 2

3 Poisson(-Boltzmann) equation Generalized Poisson equation with explicit ions.: (ε ( r) Φ ( r)) = 4π N i=1 q i δ ( r r i ). Extension to other equations possible. 3

4 Outline I. Multilevel summation: an approximation II. Multilevel summation: an algorithm III. Types of N-body solvers IV. Comparison of N-body solvers 4

5 Creating a fast N-body solver derive an approximation construct an algorithm Please defer any thoughts about an algorithm until later. 5

6 Interactions in 1 dimension An N-body solver calculates O(N 2 ) 2-body nonbonded interactions, U el = 1 2 N i=1 N j=1,j i q i q j k(x i, x j ), k(x, x ) = 1 x x where x i are particle positions and q i are partial charges. 6

7 Matrix form where q = charges, K ij = U el = 1 2 qt Kq, e = Kq = potentials { k(xi, x j ), j i 0, j = i Matrix form best reveals structure of algorithms. 7

8 Kernel splitting Separate length scales by splitting k = k 0 + k k ν where (i) the range of k µ increases with µ, but also (ii) the smoothness of k µ increases with µ. (k 1,..., k ν are approximated on grids of increasing coarseness.) A corresponding matrix splitting: K = K 0 + K K ν 8

9 Distant-dependent kernel If k(x, x ) = g( x x ), define k µ (x, x ) = g µ ( x x ) where g(r) = g 0 (r) + g 1 (r) + + g ν (r) e.g., 1/r (softened 1/r) softened 1/r 1 r = r }{{} g 0 (r) 0 0 r }{{} g 1 (r) 9

10 Banded matrices Also, if 0 x 1 < x 2 < < x n L, the values k µ (x i, x j ) = 0 for large enough i j, so the matrices K µ are banded with increasing bandwidth. The slowly varying parts k 1 (x, x ),..., k ν (x, x ) are approximated... 10

11 Digression: approximation in 1D Consider f(x) = k µ (x, x ) with x an arbitrary parameter. Goal is to approximate f(x) in terms of its values (and derivs.) on a uniform grid 0 = x h 0 < x h 1 < < x h M = L: f(x) φ 0 (x)f(x h 0) + φ 1 (x)f(x h 1) + + φ M (x)f(x h M) where the φ m (x) are basis functions having local support, e.g., piecewise polynomials: 11

12 Matrix form of approximation f(x) φ(x) T f h where φ(x) = [φ 0 (x), φ 1 (x),..., φ M (x)] T and f h = [ f(x h 0), f(x h 1),..., f(x h M) ] T. At particle positions f(x 1 ) φ(x 1 ) T f h. f(x N ) φ(x N ) T f h 12

13 Approximation operator f(x 1 ). f(x N ) φ(x 1 ) T. φ(x N ) T f h def = I h f h The operator I h maps gridded values to particle values. Only a fixed number, e.g., 4, of nonzeros for each row of I h. 13

14 Approximation in 2D Analogous to f(x) φ(x) T f h, it can be shown that with K h µ = we have k µ (x h 0, x h 0) k µ (x h 0, x h 1) k µ (x h 0, x h M ) k µ (x h 1, x h 0) k µ (x h 1, x h 1) k µ (x h 1, x h M ) k µ (x h M, xh 0) k µ (x h M, xh 1) k µ (x h M, xh M ) a separable approximation. k µ (x, x ) φ(x) T K h µφ(x ), 14

15 Therefore, tabulation of k µ (x, x ) tabulation of φ(x) T K h µφ(x ) and K µ = k µ (x 1, x 1 ).... k µ (x 1, x N ). k µ (x N, x 1 ) k µ (x N, x N ) φ(x 1 ) T. φ(x N ) T K h µ [ φ(x1 ) φ(x N ) ] = I h K h µi T h. 15

16 Multilevel summation approximation A 4-level summation method splits the kernel as k = k 0 + k 1 + k 2 + k 3, where each term has twice the range of its predecessor. With interpolation from successively coarser grids, the corresponding matrix decomposition is K K 0 + I h K h 1 I T h + I 2h K 2h 2 I T 2h + I 4h K 4h 3 I T 4h. 16

17 Outline I. Multilevel summation: an approximation II. Multilevel summation: an algorithm III. Types of N-body solvers IV. Comparison of N-body solvers 17

18 An O(N log N) method anterpolation: q h = I T hq, q 2h = I T 2hq, q 4h = I T 4hq U 1 ( q T K 0 q + (q h ) T K1 h q h + (q 2h ) T K2 2h q 2h + (q 4h ) T K3 4h q 4h). 2 With O(log N) terms the number of operations per term is only O(N). 18

19 An O(N) method For finite element-style interpolation I 2h = I h I h 2h and I 4h = I h I h 2hI 2h 4h. Not quite true for finite difference-style interpolation, e.g., for nodal basis functions Substituting and factoring out common factors gives... 19

20 Nested multilevel summation method K K 0 + I h ( K h 1 + I h 2h ( K 2h 2 + I 2h 4hK 4h 3 (I 2h 4h) T) (I h 2h) T) I T h. e Kq is computed as follows for a 5-level method: 20

21 q h = I T hq O(N) ops q 2h = (I h 2h) T q h O(M) ops q 4h = (I 2h 4h) T q 2h O(M/2) ops q 8h = (I 4h 8h) T q 4h O(M/4) ops e 8h = K 8h 4 q 8h O((M/8) 2 ) ops e 4h = K 4h 3 q 4h + I 4h 8he 8h e 2h = K 2h 2 q 2h + I 2h 4he 4h O(M/4) ops O(M/2) ops e h = K h 1 q h + I h 2he 2h O(M) ops e = K 0 q + I h e h O(N) ops 21

22 Nested multilevel summation method, diagram 22

23 Basis for all fast N-body solvers separable approximation for kernel k(x, x ), associativity (and distributivity) of matrix operations. 23

24 Outline I. Multilevel summation: an approximation II. Multilevel summation: an algorithm III. Types of N-body solvers IV. Comparison of N-body solvers 24

25 More generally, Nonbonded 2-body interactions U nb = U nb ( r 1, r 2,..., r N ) = 1 2 where r i are particle positions, the primed sum omits excluded pairs, kernels other than r r 1 possible, periodicity in any dimension possible, extension to dipoles, etc. possible. N i=1 N q i q j k( r i, r j ) j=1 25

26 Forces F nb i = i U nb ( r 1, r 2,..., r N ) are best obtained by differentiating the approximation to U nb ( r 1, r 2,..., r N ). 26

27 Types of N-body solvers I. kernel splitting method (KSM): multilevel summation, PME, P 3 M II. hierarchical clustering method (HCM): FMM, tree methods Another classification: 2-level: uses FFT for gridded values multilevel: uses O(log N) levels of grids/cells nested multilevel: reduces cost to O(N) 27

28 History of multilevel summation integral transforms, Brandt & Lubrecht (1990) particle monopoles and dipoles in 2D, Sandak (2001) C 1 potentials for particles in 3D, Skeel, Tezcan & Hardy (2002) generalized Born potentials, Lee, Salsbury & Olson (2004) higher order accuracy, analysis, periodicity, Hardy (2006) 28

29 Two-level method w/ real space FFT In K 1 I h K1 h Ih T k 1 (0 k 1 ( h) k 1 ( Mh) K1 h k = 1 (h) k 1 (0) k 1 (h Mh) k 1 (Mh) k 1 (Mh h) k 1 (0), is a nonperiodic convolution computable by an FFT if the dimension of the matrix is doubled. 29

30 Two-level method w/ reciprocal space FFT Modify g 1 (r) to be constant for r L + L and let κ 1 (r) be a periodic extension of g 1 (r) with period 2(L + L). Truncate Fourier series: k 1 (x, x ) = κ 1 (x x ) ψ(x ) T Dψ(x) where D is diagonal matrix and ψ(x) is Fourier basis. 30

31 Interpolating from gridded values F of the Fourier basis hence Tabulating at particle positions ψ(x) T φ(x) T F κ 1 (x x ) φ(x ) T F T DF φ(x) K 1 I h F T DF I T h. This is particle mesh Ewald (PME), a variant of an older approach, viz., particle particle particle mesh Ewald (P 3 M). 31

32 Hierarchical clustering methods employ an oct-tree decomposition of space to partition the interactions into different levels: K = K 0 + K 1 + K 2 + K 3, with higher level interactions for more separated particles. Short range directly. Longer range are partitioned into pairs of particle clusters, each using a different localized (polynomial) approximation for the kernel k( r, r ). 32

33 Truncated Taylor expansions are used in practice. For r r 1, each polynomial is harmonic and can expressed in terms of p 2 spherical harmonics, where p 1 is the degree of the polynomial, instead of 1 6 p p p monomials. The entries of K h 1, K 2h 2,... are then multipoles. 33

34 Outline I. Multilevel summation: an approximation II. Multilevel summation: an algorithm III. Types of N-body solvers IV. Comparison of N-body solvers 34

35 KSMs vs. HCMs Advantages of KSMs over HCMs: HCMs yield a U that is a discontinuous function of r, r ; KSMs can attain any degree of continuity. KSMs are simpler they do not need interaction lists. KSMs are faster in comparisons I have seen. Advantage of HCMs over KSMs: HCMs can exploit special properties of kernel. However, this depends on the kernel, and it can greatly complicate the algorithm. 35

36 Nonissue: Only HCMs honor Newton s 3rd Law. Unknown: Is adaptivity as efficient for KSMs as it is for HCMs? 36

37 Energy drift for HCMs HCMs compute discontinuous forces, so are not usable unless high accuracy is requested terms in FMA 12 terms in FMA 16 terms in FMA total energy (kcal/mol) time (fs) Our multisummation method is 11 times faster than a fast multipole method for MD, partly because the latter needs 12 terms to avoid rapid energy growth. 37

38 7 6 Multisummation vs. multipole CPU time vs. error for 10,002 waters DPMTA (FMM), theta=0.50 DPMTA (FMM), theta=0.75 C1 nonic, C5 Taylor, h=3 C1 septic, C4 Taylor, h=3 C1 quintic, C3 Taylor, h=3 cubic, sigma[2,6], h=3 Hermite, C3 Taylor, h=6 5 time in seconds e percent relative error in average force 38

39 Conservation of linear momentum HCM approximations yield forces that obey Newton s 3rd Law. KSM approximations lack this property. (And produce self forces!) The usual remedy F i (1/N) j F j yields nonconservative forces. However, the mass-weighted correction F i (...) m i m tot j F j (...) is the gradient of a potential. Skeel, Hardy & Phillips (2007) 39

40 Multilevel vs. 2-level Advantages of multilevel over 2-level methods: FFT-based methods require doubling the dimension in each direction that is nonperiodic. FFT-based methods are very difficult to parallelize. adaptivity not possible for 2-level methods multilevel methods are faster in comparisons we have done. Our multisummation method is twice as fast as PME. 40

41 20 15 Multisummation vs. particle mesh Ewald CPU time vs. error for 21,950 waters PME, quintic, 64 divisions PME, cubic, 64 divisions C1 nonic, C5 Taylor, 32 divisions C1 septic, C4 Taylor, 32 divisions C1 quintic, C3 Taylor, 32 divisions cubic, sigma[2,6], 32 divisions Hermite, C3 Taylor, 16 divisions time in seconds e percent relative error in average force 41

Laplacian-Centered Poisson Solvers and Multilevel Summation Algorithms

Laplacian-Centered Poisson Solvers and Multilevel Summation Algorithms Dmitry Yershov 1 Stephen Bond 1 Robert Skeel 2 1 University of Illinois at Urbana-Champaign 2 Purdue University 2009 SIAM Annual Meeting