MS 15: Data-Aware Parallel Computing Data-Driven Parallelization in Multi-Scale Applications – Ashok Srinivasan, Florida State University Dynamic Data.

MS 15: Data-Aware Parallel Computing

• Data-Driven Parallelization in Multi-Scale Applications

– Ashok Srinivasan, Florida State University

• Dynamic Data Driven Finite Element Modeling of Brain Shape Deformation During Neurosurgery

– Amitava Majumdar, San Diego Supercomputer Center

• Dynamic Computations in Large-Scale Graphs

– David Bader, Georgia Tech

• Tackling Obesity in Children

– Radha Nandkumar, NCSA

www.cs.fsu.edu/~asriniva/presentations/siampp06

Data-Driven Parallelization in Multi-Scale Applications

Ashok Srinivasan

Computer Science, 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, NCSACollaborators: Yanan Yu and Namas Chandra

Outline

• Background– Limitations of Conventional Parallelization

– Example Application: Carbon Nanotube Tensile Test

• Small Time Step Size in Molecular Dynamics Simulations

• Data-Driven Time Parallelization

• Experimental Results– Scaled efficiently to ~ 1000 processors, for a problem where

conventional parallelization scales to just 2-3 processors

• Other time parallelization approaches

• Conclusions

Background

• Limitations of Conventional Parallelization

• Example Application: Carbon Nanotube

Tensile Test

– Molecular Dynamics Simulations

• Problems with Multiple Time-Scales

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

• 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

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

Data-Driven Time Parallelization

• Time parallelization

• Data Driven Prediction

– Dimensionality Reduction

– Relate Simulation Parameters

– Static Prediction

– Dynamic Prediction

• Verification

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

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 • + c1 u1 + c2 u2

u

u

Basis Vectors from POD• CNT of ~ 100 A with 1000 atoms at 300 K

u1 (blue) and u2 (red) for z

u1 (green) for x is not “significant”

Blue: z Green, Red: x, y

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

• Dynamic Prediction– Correct the above coefficients, by determining the error between the

previously predicted and computed states

• Direct 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 + 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

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

Mean positionDisplacement (from mean)

Experimental Results

• Relate simulations with different strain rates

– Use the above strategy directly

• Relate simulations with different strain rates

and different CNT sizes

– Express basis vectors in a different functional form

• Relate simulations with different temperatures

and strain rates

– Dynamically identify different simulations that are

similar in current behavior

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: The next predictor

• v = 1m/s, using v = 10m/s

• CNT with 1000 atoms

• Xeon/ Myrinet cluster

CNTs of varying sizes

• Use a 1000-atom CNT 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+t = z t+ z’ t+t t, predict z’

Speedup

- - 2000 atoms

-.- 1600 atoms

__ 1200 atoms

… Linear

Stress-strain

Blue: Exact 2000 atoms

Red: 200 processors

Predict change in coordinates

• Express x’ in terms of basis functions– Example:

• x’ t+t = a0, t+t + a1, t+t x t

– a0, t+t, a1, t+t are unknown– Express changes, y, for the base (old) simulation similarly, in

terms of coefficients b and perform least squares fit

• Predict ai, t+t as bi, t+t + R t+t

• R t+t = (1-) R t + (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 = 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

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

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

Conclusions

• Data-driven time parallelization shows significant improvement in speed, without sacrificing accuracy significantly

• Direct prediction is very effective when applicable

• 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

Future Work

• More complex problems– Better prediction

• POD is good for representing data, but not necessarily for identifying patterns

• Use better dimensionality reduction / reduced order modeling techniques

• Use experimental data for prediction

– Better learning– Better verification– In CP8: Application of Dimensionality Reduction

Techniques to Time Parallelization, Yanan Yu• Tomorrow, 2:30 – 3:00 pm