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5.1 Poisson Equation in Parallel

NGSolve can be executed on a cluster using the MPI message passing interface. You can download poisson_mpi.py and run it as

mpirun -np 4 python3 poisson_mpi.py

This computes and saves a solution which we can visualize it with drawsolution.py

netgen drawsolution.py

For proper parallel execution, Netgen/NGSolve must be configured with -DUSE_MPI=ON. Recent binaries for Linux and Mac are built with MPI support (?). If you are unsure, when your Netgen/NGSolve supports MPI, look for outputs like "Including MPI version 3.1" during Netgen startup.

MPI-parallel execution using ipyparallel

For the MPI jupyter-tutorials we use ipyparallel module. Please consult https://ipyparallel.readthedocs.io for installation.

We start an MPI server:

from ipyparallel import Cluster
c = await Cluster(engines="mpi").start_and_connect(n=4, activate=True)

Starting 4 engines with <class 'ipyparallel.cluster.launcher.MPIEngineSetLauncher'>

We use mpi4py https://mpi4py.readthedocs.io/ for issuing MPI calls from Python. The %%px syntax magic causes parallel execution of that cell:

%%px
from mpi4py.MPI import COMM_WORLD as comm
print (comm.rank, comm.size)

[stdout:1] 1 4

[stdout:3] 3 4

[stdout:0] 0 4

[stdout:2] 2 4

The master process (rank==0) generates the mesh, and distributes it within the group of processes defined by the communicator. All other ranks receive a part of the mesh. The function mesh.GetNE(VOL) returns the local number of elements:

%%px
from ngsolve import *
from netgen.occ import unit_square

if comm.rank == 0:
    mesh = Mesh(unit_square.GenerateMesh(maxh=0.1).Distribute(comm))
else:
    mesh = Mesh(netgen.meshing.Mesh.Receive(comm))
print (mesh.GetNE(VOL))

[stdout:2] 62

[stdout:0] 56

[stdout:1] 55

[stdout:3] 57

We can define spaces, bilinear / linear forms, and gridfunctions in the same way as in sequential mode. But now, the degrees of freedom are distributed on the cluster following the distribution of the mesh. The finite element spaces define how the dofs match together.

%%px
fes = H1(mesh, order=3, dirichlet=".*")
u,v = fes.TnT()

a = BilinearForm(grad(u)*grad(v)*dx)
pre = Preconditioner(a, "local")
a.Assemble()

f = LinearForm(1*v*dx).Assemble()
gfu = GridFunction(fes)

from ngsolve.krylovspace import CGSolver
inv = CGSolver(a.mat, pre.mat, printrates=comm.rank==0, maxiter=200, tol=1e-8)
gfu.vec.data = inv*f.vec

[stdout:0] CG iteration 1, residual = 0.05313636107212468
CG iteration 2, residual = 0.0732448385167346
CG iteration 3, residual = 0.06077337326925862
CG iteration 4, residual = 0.04965675927264466
CG iteration 5, residual = 0.046283992940999
CG iteration 6, residual = 0.0305156333154046
CG iteration 7, residual = 0.02589831756935775
CG iteration 8, residual = 0.016994550808865833
CG iteration 9, residual = 0.013010287656153193
CG iteration 10, residual = 0.011432088760998969
CG iteration 11, residual = 0.007816924495134898
CG iteration 12, residual = 0.0035098066609772533
CG iteration 13, residual = 0.0017387130849062828
CG iteration 14, residual = 0.0011502309396829063
CG iteration 15, residual = 0.0008184957095722692
CG iteration 16, residual = 0.0005419630762720773
CG iteration 17, residual = 0.00038670700524591505
CG iteration 18, residual = 0.00028367421577578963
CG iteration 19, residual = 0.00017773436149414444
CG iteration 20, residual = 0.00013341157334192957
CG iteration 21, residual = 9.379683067760127e-05
CG iteration 22, residual = 6.550835827505411e-05
CG iteration 23, residual = 5.1235792984120506e-05
CG iteration 24, residual = 3.536225147414022e-05
CG iteration 25, residual = 2.2782716580339387e-05
CG iteration 26, residual = 1.567745927863831e-05
CG iteration 27, residual = 1.1117149587134072e-05
CG iteration 28, residual = 7.709451159725054e-06
CG iteration 29, residual = 5.474421176424829e-06
CG iteration 30, residual = 3.6876796657406495e-06
CG iteration 31, residual = 2.083369575095351e-06
CG iteration 32, residual = 1.6564423874702285e-06
CG iteration 33, residual = 1.373699995999589e-06
CG iteration 34, residual = 8.539631243750763e-07
CG iteration 35, residual = 4.6174975273229637e-07
CG iteration 36, residual = 2.950720008145955e-07
CG iteration 37, residual = 1.9900657697860535e-07
CG iteration 38, residual = 1.4268161595525394e-07
CG iteration 39, residual = 9.868137345834071e-08
CG iteration 40, residual = 7.020839769082992e-08
CG iteration 41, residual = 4.57707147332425e-08
CG iteration 42, residual = 2.962279920593732e-08
CG iteration 43, residual = 1.9267023820022404e-08
CG iteration 44, residual = 1.3684980860636776e-08
CG iteration 45, residual = 9.117459310061326e-09
CG iteration 46, residual = 5.918193053332316e-09
CG iteration 47, residual = 3.739344480410759e-09
CG iteration 48, residual = 2.4133694037355697e-09
CG iteration 49, residual = 1.7270530805917795e-09
CG iteration 50, residual = 1.1180335861372133e-09
CG iteration 51, residual = 7.62860621780241e-10
CG iteration 52, residual = 5.294659183175954e-10

The solution is distributed over the processes. Gather() is a collective operation that assembles the whole mesh and solution on the process with rank=0, while pickling (what c[:][...] does) transfers the local parts:

%%px
gfu_global = gfu.Gather()   # collective: whole solution on rank 0, None elsewhere

We pull the gathered solution from rank 0 to the client and draw it:

from ngsolve.webgui import Draw
Draw (c[0]["gfu_global"]);

Pulling gfu from all engines gives the local parts; drawing gfu[n] shows the part that the process with rank=n possesses.

gfu = c[:]["gfu"]   # the local parts
Draw (gfu[3]);

We can also visualize the sub-domains obtained by the automatic partitioning, without using any computed solution, as follows.

%%px
fesL2 = L2(mesh, order=0)
gfL2 = GridFunction(fesL2)
gfL2.vec.local_vec[:] = comm.rank

%%px
gfL2_global = gfL2.Gather()

Draw (c[0]["gfL2_global"]);