An FPGA Implementation of Reciprocal Sums for SPME

Transcription

1 An FPGA Implementation of Reciprocal Sums for SPME Sam Lee and Paul Chow Edward S. Rogers Sr. Department of Electrical and Computer Engineering University of Toronto

2 Objectives Accelerate part of Molecular Dynamics Simulation Smooth Particle Mesh Ewald Implementation FPGA based Try it and learn Investigation Acceleration bottleneck Precision requirement Parallelization strategy

3 Presentation Outline Molecular Dynamics SPME The Reciprocal Sum Compute Engine Speedup and Parallelization Precision Future work

4 Molecular Dynamics Simulation 4

5 Molecular Dynamics Combines empirical force calculations with Newton s equations of motion. Predict the time trajectory of small atomic systems. Computationally demanding.. Calculate interatomic forces.. Calculate the net force.. Integrate Newton s equations of motion. a r = F m ( t + δt ) = r () t + δt v () t + 0.5δt a () t v ( t + δt ) = v () t δt a () t + a ( t + δt ) F 5

6 Molecular Dynamics k Θ All Angles + + k ( l b l o All Bonds ( Θ Θ ) o ) U = A [ + cos( nτ + φ)] All Torsions + + All Pairs All Pairs q q r σ 4ε r 6 σ r + δ δ 6

7 MD Simulation Problem scientists are facing: SLOW! O(N ) complexity. 0 CPU Years 7

8 Solutions Parallelize to more compute engines Accelerate with FPGA Especially: The non-bonded calculations To be more specific, this paper addresses: Electrostatic interaction (Reciprocal space) Smooth Particle Mesh Ewald algorithm. 8

9 Previous Work Software SPME Implementations: Original PME Package written by Toukmaji. Used in NAMD. Hardware Implementations: No previous hardware implementation of reciprocal sums calculation. MD-Grape & MD-Engine uses Ewald Summation. Ewald Summation is O(N ); SPME is O(NLogN)! 9

10 Smooth Particle Mesh Ewald 0

11 Electrostatic Interaction Coulombic equation: qq = 4πε r v coulomb 0 Under the Periodic Boundary Condition, the summation to calculate Electrostatic energy is only Conditionally Convergent. U = ' n N N q q i= j= ij, n i r j

12 Periodic Boundary Condition A 4 5 B 4 5 C 4 5 D 4 5 E 4 5 F 4 5 G 4 5 H 4 5 I 4 5 To combat Surface Effect 4 5 Replication

13 Ewald Summation Used For PBC To calculate the Coulombic Interactions O(N ) Direct Sum + O(N ) Reciprocal Sum q r Direct Sum q Reciprocal Sum q r r

14 Smooth Particle Mesh Ewald Shift the workload to the Reciprocal Sum. Use Fast Fourier Transform. O(N) Real + O(NLogN) Reciprocal. RSCE calculates the Reciprocal Sums using the SPME algorithm. 4

15 5 SPME Reciprocal Contribution ),m,m m Q)( (θ ),m,m (m r Q r E F K m K m rec K m αi αi rec ~ = = = = = ) (m b ) (m b ) b (m ),m,m B(m = 0 exp exp = + = n k i i n i i i i ) K πim k ( ) (k M ) K )m πi(n ( ) b (m exp m ) /β m π ( πv ),m,m C(m = = ),,,c( m ) m, m, m )F(Q)(,m,m F(Q)(m ),m,m B(m m ) /β m π ( πv E m ~ 0 exp = FFT FFT Energy: Force: ),m,m m Q)( (θ ),m,m Q(m E K m K m rec K m ~ = = = =

16 Charge Interpolation F D A C B E 6

17 Reciprocal Sum Compute Engine 7

18 RSCE Architecture 8

19 RSCE Verification Testbench 9

20 RSCE Validation Environment 0

21 Speedup Estimate RSCE vs. Software Implementation

22 RSCE Speedup 00MHz vs. P4 Speedup: x to 4x Why so insignificant? Reciprocal Sums calculations not easily parallelizable. QMM memory bandwidth limitation. Improvement: Using more QMM memories can improve the speedup. Slight design modifications are required.

23 Parallelization Strategy Multiple RSCE

24 RSCE Parallelization Strategy Assume a -D simulation system. Assume P=, K=8, N=6. Assume NumP = 4. An 8x8x8 mesh Four 4x4x4 Mini Meshes 4

25 RSCE Parallelization Strategy Mini-mesh composed -> D-IFFT D-IFFT = two passes of D-FFT (X and Y). X Direction FFT Y Direction FFT Ky Ky P P P P P P P4 D FFT Y direction P4 0 Kx D FFT X direction 0 Kx 5

26 Parallelization Strategy D-IFFT -> Energy Calculation -> D-FFT D-FFT -> Force Calculation Energy Calculation Force Calculation E Total = E P P= 0 D-FFT 6

27 MD Simulations RSCE + NAMD 7

28 RSCE Precision Precision goal: Relative error bound < 0-5. Two major calculation steps: B-Spline Calculation. D-FFT/IFFT Calculation. Due to the limited logic resource & limited precision FFT LogiCore. => Precision goal cannot be achieved. 8

29 RSCE Precision To achieve the relative error bound of < 0-5. Minimum calculation precision: FFT {4.0}, B-Spline {.7} 9

30 MD Simulation with RSCE RMS Energy Error Fluctuation: RMS Energy Fluctuatio n = E E E 0

31 FFT Precision Vs. Energy Fluctuation

32 Summary Implementation of FPGA-based Reciprocal Sums Compute Engine and its SystemC model. Integration of the RSCE into a widely used Molecular Dynamics program called NAMD for verification RSCE Speedup Estimate x to 4x Precision Requirement B-Spline: {.7} & FFT: {4:0} => 0-5 rel. error Parallelization Strategy

33 Future Work More in-depth precision analysis. Investigation on how to further speedup the SPME algorithm with FPGA.

34 Questions 4