Long-Time Molecular Dynamics Simulations in Nano-Mechanics through Parallelization of the Time Domain Ashok Srinivasan Florida State University http://www.cs.fsu.edu/~asriniva Aim: Simulate for long time spans Solution features: Use data from prior simulations to parallelize the time domain Acknowledgements: NSF, ORNL, NERSC, NCSA Collaborators: Yanan Yu and Namas Chandra

Outline Background Other Time Parallelization Approaches Limitations of Conventional Parallelization Example Application: Carbon Nanotube Tensile Test Small Time Step Size in Molecular Dynamics Simulations Other Time Parallelization Approaches Data-Driven Time Parallelization Experimental Results Scaled efficiently to ~ 1000 processors, for a problem where conventional parallelization scales to just 2-3 processors Conclusions

Background Limitations of Conventional Parallelization Example Application: Carbon Nanotube Tensile Test Molecular Dynamics Simulations

Limitations of Conventional Parallelization Conventional parallelization decomposes the state space across processors It is effective for large state space It is not effective when computational effort arises from a large number of time steps … or when granularity becomes very fine due to a large number of processors

Example Application Carbon Nanotube Tensile Test Pull the CNT at a constant velocity Determine stress-strain response and yield strain (when CNT starts breaking) using MD Strain rate dependent

A Drawback of Molecular Dynamics In each time step, forces of atoms on each other modeled using some potential After force is computed, update positions Repeat for desired number of time steps Time steps size ~ 10 –15 seconds, due to physical and numerical considerations Desired time range is much larger A million time steps are required to reach 10-9 s Around a day of computing for a 3000-atom CNT MD uses unrealistically large strain-rates

Other Time Parallelization Approaches Waveform relaxation Repeatedly solve for the entire time domain Parallelizes well but convergence can be slow Several variants to improve convergence Parareal approach Features similar to ours and to waveform relaxation Precedes our approach Not data-driven Sequential phase for prediction Not very effective in practice so far Has much potential to be improved

Waveform Relaxation Special case: Picard iterations In general Ex: dy/dt = y, y(0) = 1 becomes dyn+1/dt = yn(t), y0(t) = 1 In general dy/dt = f(y,t), y(0) = y0 becomes dyn+1/dt = g(yn, yn+1, t), y0(t) = y0 g(u, u, t) = f(u, t) g(yn, yn+1, t) = f(yn, t): Picard g(yn, yn+1, t) = f(y, t): Converges in 1 iteration Jacobi, Gauss-Seidel, and SOR versions of g defined Many improvements Ex: DIRM combines above with reduced order modeling Exact N = 1 N = 2 N = 3 N = 4

Parareal approach Based on an “approximate-verify-correct” sequence An example of shooting methods for time-parallelization Not shown to be effective in realistic situations Second prediction Initial computed result Correction Initial prediction

Data-Driven Time Parallelization Use Prior Data

Time Parallelization Each processor simulates a different time interval Initial state is obtained by prediction, except for processor 0 Verify if prediction for end state is close to that computed by MD Prediction is based on dynamically determining a relationship between the current simulation and those in a database of prior results If time interval is sufficiently large, then communication overhead is small

Problems with multiple time-scales Fine-scale computations (such as MD) are more accurate, but more time consuming Much of the details at the finer scale are unimportant, but some are A simple schematic of multiple time scales

Use Prior Data Results for identical simulation exists Retrieve the results Results for slightly different parameter, with the same coarse-scale response exists Verify closeness, or pre-determine acceptable parameter range Current simulation behaves like different prior ones at different times Identify similar prior results, learn relationship, verify prediction Not similar to prior results Try to identify coarse-scale behavior, apply dynamic iterations to improve on predictions

Experimental Results CNT tensile test CNT identical to prior results, but different strain-rate 1000-atoms CNT, 300 K Static and dynamic prediction CNT identical to prior results, but different strain-rate and temperature CNT differs in size from prior result, and simulated with a different strain-rate

Dimensionality Reduction Movement of atoms in a 1000-atom CNT can be considered the motion of a point in 3000-dimensional space Find a lower dimensional subspace close to which the points lie We use principal orthogonal decomposition Find a low dimensional affine subspace Motion may, however, be complex in this subspace Use results for different strain rates Velocity = 10m/s, 5m/s, and 1 m/s At five different time points [U, S, V] = svd(Shifted Data) Shifted Data = U*S*VT States of CNT expressed as m + c1 u1 + c2 u2 u u m

Basis Vectors from POD CNT of ~ 100 A with 1000 atoms at 300 K Blue: z Green, Red: x, y u1 (blue) and u2 (red) for z u1 (green) for x is not “significant”

Relate strain rate and time Coefficients of u1 Blue: 1m/s Red: 5 m/s Green: 10m/s Dotted line: same strain Suggests that behavior is similar at similar strains In general, clustering similar coefficients can give parameter-time relationships

Prediction When v is the only parameter Static Predictor Independently predict change in each coordinate Use precomputed results for 40 different time points each for three different velocities To predict for (t; v) not in the database Determine coefficients for nearby v at nearby strains Fit a linear surface and interpolate/extrapolate to get coefficients c1 and c2 for (t; v) Get state as m + c1 u1 + c2 u2 Green: 10 m/s, Red: 5 m/s, Blue: 1 m/s, Magenta: 0.1 m/s, Black: 0.1m/s through direct prediction Dynamic Prediction Correct the above coefficients, by determining the error between the previously predicted and computed states

Verification of prediction Definition of equivalence of two states Atoms vibrate around their mean position Consider states equivalent if difference in position, potential energy, and temperature are within the normal range of fluctuations Max displacement ~= 0.2 A Mean displacement ~= 0.08 A Potential energy fluctuation = 0.35% Temperature fluctuation = 12.5 K Displacement (from mean) Mean position

Stress-strain response at 0.1 m/s Blue: Exact result Green: Direct prediction with interpolation / extrapolation Points close to yield involve extrapolation in velocity and strain Red: Time parallel results

Speedup Red line: Ideal speedup Blue: v = 0.1m/s Green: A different predictor v = 1m/s, using v = 10m/s CNT with 1000 atoms Xeon/ Myrinet cluster

Temperature and velocity vary Use 1000-atom CNT results Temperatures: 300K, 600K, 900K, 1200K Velocities: 1m/s, 5m/s, 10m/s Dynamically choose closest simulation for prediction Speedup __ 450K, 2m/s … Linear Stress-strain Blue: Exact 450K Red: 200 processors

CNTs of varying sizes Use a 1000-atom CNT, 10 m/s, 300K result Parallelize 1200, 1600, 2000-atom CNT runs Observe that the dominant mode is approximately a linear function of the initial z-coordinate Normalize coordinates to be in [0,1] z t+Dt = z t+ z’ t+Dt Dt, predict z’ Speedup - 2000 atoms .- 1600 atoms __ 1200 atoms … Linear Stress-strain Blue: Exact 2000 atoms, 1m/s Red: 200 processors

Predict change in coordinates Express x’ in terms of basis functions Example: x’ t+Dt = a0, t+Dt + a1, t+Dt x t a0, t+Dt, a1, t+Dt are unknown Express changes, y, for the base (old) simulation similarly, in terms of coefficients b and perform least squares fit Predict ai, t+Dt as bi, t+Dt + R t+Dt R t+Dt = (1-b) R t + b(ai, t- bi, t) Intuitively, the difference between the base coefficient and the current coefficient is predicted as a weighted combination of previous weights We use b = 0.5 Gives more weight to latest results Does not let random fluctuations affect the predictor too much Velocity estimated as latest accurate results known

Conclusions Data-driven time parallelization shows significant improvement in speed, without sacrificing accuracy significantly, in the CNT tensile test The 980-processor simulation attained a flop rate of ~ 420 Gflops Its flops per atom rate of 420 Mflops/atom is likely the largest flop per atom rate in classical MD simulations References See http://www.cs.fsu.edu/~asriniva/research.html

Future Work More complex problems Better prediction Better learning POD is good for representing data, but not necessarily for identifying patterns Use better dimensionality reduction / reduced order modeling techniques Better learning Better verification