Building More Efficient Time Parallel Solvers with Applications to Multiscale Modeling and Gyrokinetics

Transcription

1 Building More Efficient Time Parallel Solvers with Applications to Multiscale Modeling and Gyrokinetics January 2015 Carl D Lederman AFTC/PA clearance No , 16 January

2 The Differential Equation The general form of the differential equation (DE) of interest is: = ff xx xx 0 = xx 0 ff: R DD R DD xx tt : R R DD where the dimension, DD, may be any size and ff may be derived from a spatial discretization of a PDE. The differential equations are assumed computationally solvable by existing serial means. The solution is represented in a compact form using propagator FF. xx tt = FF(xx 0, tt) The letter CC represents a coarse propagator, which solves the same DE with less accuracy and with more computational speed. 2

3 Serial Solvers The serial solution to the described DE is summarized as: Serial Computations xx tt3 = FF(xx tt2, ) xx tt2 = FF(xx tt1, ) xx tt1 = FF(xx tt0, ) The propagator, FF, encapsulates any necessary processes required to advance the solution such as a Newton non-linear solver or Fourier transform. This approach is standard for solving DEs of the type described and generally works well. The only drawback is that computing on modern machines works fastest when tasks can be parallelized as much as possible and this approach restricts the possible parallelization that can occur. 3

4 TP Solvers For time parallel (TP) modeling, the solution, xx vv uu, depends on both a value vv [0, VV] related to physical time as well as uu, which serves as an iteration number in the simplest case. The TP method alternates between parallel and serial steps. Serial Computations xx 3 0 = CC(xx 2 0 ) xx 2 0 = CC(xx 1 0 ) xx 1 0 = CC(xx 0 0 ) Parallel Computations 0 FF xx VV 1, 0 FF xx VV 2, FF xx 1 0, FF xx 0 0, Serial Computations xx VV = gg(xx VV 1, FF xx VV 1, ) xx 2 1 = gg(xx 1 1, FF xx 1 0, ) xx 1 1 = gg(xx 0 1, FF xx 0 0, ) As much computation as possible should be performed in parallel. A simple means of achieving this is to make FF a composition propagators with smaller time steps. 4

5 Propagator Derivatives In the TP approach, the fine propagator is run independently on each time block. Thus, in general, the fine propagator FF(xx, tt) will be starting from an xx that disagrees with the desired final solution To account for the inaccurate propagator starting location, some measure of the variation in xx, or FF is needed. xx Initial Coarse Solution Fine Propagator Difference in Fine Propagator Starting Location 5

6 TP PDE A general approach yields an equation that can be discretized in a few different ways to produce TP schemes. A generic functional for the energy of the DE is defined. φφ xx(vv) = VV ddvv ff xx ff xx ddvv ddvv The functional is minimized by gradient descent with a carefully chosen functional gradient. = DDDD(φφ xx ) The resulting PDE is: 2 xx vvvvvv xx + ff = 0 ff: R DD R DD xx uu, vv : R R R DD 6

7 Direct TP PDE Approximations The simplest approach is to simply integrate in vv using known serial propagators: dd xx uu, vv + xx uu,vv, dd xx uu, vv + xx uu, vv + FF xx uu, vv, = 0 Which can be further simplified to: mm vv+1 uu mm vv + xx vv+1 FF xx uu vv = 0 xxuu vv mm vv = xx vv uu+1 xx vv uu = xx vv uu+1 xx vv uu Additionally the propagator derivative can be approximated with the substitution: xxuu vv ddcc xx vv uu An alternative substitution yields the Parareal Method : xx vv uu mm vv CC xx vv uu + mm vv CC xx vv uu = CC xx vv uu+1 CC xx vv uu These schemes are implemented efficiently when each of the fine propagators is precomputed in parallel. 7

8 Additional TP PDE Approximations Higher Accuracy Schemes This approach is based on substituting a better derivative formula into the TP PDE. ddff ddxx 1 ddff xxuu 2 ddxx + 1 ddff vv xxuu+1 2 vv xxuu vv The discretization relies on a hybrid approach in which the required computations are made based a few solution values at uu + 1 and many at uu. Much of the computational work that relies only on uu can be done in parallel. Multi-Propagator Schemes Approximates using ddcc. Computing the derivative of the fine propagator can be the computationally expensive for some DEs. The fine propagator derivative, computed over shorter intervals, can be combined with the coarse propagator derivative, computed over longer intervals, to produce an approximation that is close to the accuracy of the pure fine propagator derivative while remaining less computation expensive. 8

9 Sample Results Coarsest Initial Fine The TP schemes can vary considerably in their convergence to the desired solution 9

10 Potential Applications to Multiscale Problems For multiscale (MS) applications, the goal is not just to perform fewer computations in serial, but fewer computations. To accomplish this, as much useful information as possible needs to be taken from only a small sampling of the fine scale features. One of the TP methods discussed offers an efficient means to approximate the derivative (in xx) of a fine propagator based on only a sampling of values as well a coarse propagator derivative. The fine propagator itself can be then be approximated from the following relation: (xx, tt) tt = xx, tt ff(xx) 10

11 Gyrokinetic Particle Trajectories (Preliminary) A particle rapidly orbits around a slowly varying magnetic field (cyan). A known analytical approximation is the helix trajectory (blue). Standard numerical schemes are accurate with small time steps, but numerical stability is limited based upon the gyrofrequency. Fine Coarse Multiscale B Field The coarse and multiscale solutions are implemented with a single step (but interpolated to show the full trajectory) The multiscale approximation may allow for larger time step sizes than the standard numerical schemes and better accuracy than the analytical approximation. 11

12 Extensions Solve larger scale problems and PDEs with Time Parallel methods Fully develop the multiscale gyrokinetic solver Further analyze the causes of simulation failure for multiscale and time parallel methods Thank you! 12