Scalable Parallel-in-Time Simulation of Extreme-Scale Chaotic Systems

Presentation at #ExaMath13

Transcript

1. 1 Scalable Parallel-in-Time Simulation of Extreme-Scale Chaotic Systems Qiqi Wang ExaMath Workshop 2013.8.21
2. 2 Why space-time parallel simulation? Science: High fidelity simulation of long time scale processes is infeasible Engineering: High fidelity simulation is too slow for design Today: Parallel only in space Future: Space- time Parallel Nsteps = T / Δt, Δt is set by the fastest resolved process, T is set by the slowest simulated process.
3. 3 Why space-time parallel simulation? Today: Parallel only in space Future: Space- time Parallel Science: High fidelity simulation of long time scale processes is infeasible Engineering: High fidelity simulation is too slow for design
4. Many extreme scale simulations are chaotic • Initial value problem is ill-conditioned. • Small update at earlier time cause large update at later time. • Slow convergence of time parallel solvers of initial value problem. Time steps Residual First 12 iterations Reynolds-Barredoa et al. J. Comp. Phys. 231-23, 2012 4 Coarse solver Fine solver Heat transfer rate, Lorenz-63 model of Rayleigh Benard convection
5. Redefine a simulation for time parallelism • Assumption: quantities of interest are insensitive to small scale uncertainties in the initial condition. • Replace the traditional initial value problem with the Least Squares Shadowing (LSS) problem • An initial value problem of chaos is ill-conditioned; A LSS problem of chaos can be well-conditioned. 5 Coarse solution Time dilation
6. 6 Phase space cartoon of a perturbed initial value problem
7. 7 Phase space cartoon of a perturbed least squares shadowing (LSS) problem A nearby trajectory exists by the shadowing lemma.
8. 8 Least squares shadowing Condition number O(T2) Initial value problem Condition number O(eλT) Time Time dependent output Time dependent output Coarse solver Fine solver 8 Coarse solver Fine solver
9. A new paradigm for simulating chaos 9 Initial Value Formulation Least Squares Formulation Ill-conditioned Well-conditioned Suitable for time advancing Storing only a few time steps Needs iterative solution Storing hundreds of time steps Sequential in nature, do not parallelize well in time Breaks causality, scalable parallelization in space and time demonstrated on Lorenz system and Kuramoto-Sivashinsky eqn Requires checkpoint-restart upon individual core failure Potentially resilient to individual core failure Suitable when computer size to problem size ratio is small Current computing paradigm Suitable when computer size to problem size ratio is large Potential “Exascale” computing paradigm