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'''
Ex2 from PETSc example files implemented for PETSc4py.
https://petsc.org/release/src/ksp/ksp/tutorials/ex2.c.html
By: Miguel Arriaga
Solves a linear system in parallel with KSP.
Input parameters include:
-view_exact_sol : write exact solution vector to stdout
-m <mesh_x> : number of mesh points in x-direction
-n <mesh_n> : number of mesh points in y-direction
Vec x,b,u; # approx solution, RHS, exact solution
Mat A; # linear system matrix
KSP ksp; # linear solver context
PetscReal norm; # norm of solution error
'''
import sys
import petsc4py
petsc4py.init(sys.argv)
from petsc4py import PETSc
import numpy as np
comm = PETSc.COMM_WORLD
size = comm.getSize()
rank = comm.getRank()
OptDB = PETSc.Options()
m = OptDB.getInt('m', 8)
n = OptDB.getInt('n', 7)
''' - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Compute the matrix and right-hand-side vector that define
the linear system, Ax = b.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - '''
'''
Create parallel matrix, specifying only its global dimensions.
When using MatCreate(), the matrix format can be specified at
runtime. Also, the parallel partitioning of the matrix is
determined by PETSc at runtime.
Performance tuning note: For problems of substantial size,
preallocation of matrix memory is crucial for attaining good
performance. See the matrix chapter of the users manual for details.
'''
A = PETSc.Mat().create(comm=comm)
A.setSizes((m*n,m*n))
A.setFromOptions()
A.setPreallocationNNZ((5,5))
'''
Currently, all PETSc parallel matrix formats are partitioned by
contiguous chunks of rows across the processors. Determine which
rows of the matrix are locally owned.
'''
Istart,Iend = A.getOwnershipRange()
'''
Set matrix elements for the 2-D, five-point stencil in parallel.
- Each processor needs to insert only elements that it owns
locally (but any non-local elements will be sent to the
appropriate processor during matrix assembly).
- Always specify global rows and columns of matrix entries.
Note: this uses the less common natural ordering that orders first
all the unknowns for x = h then for x = 2h etc; Hence you see J = Ii +- n
instead of J = I +- m as you might expect. The more standard ordering
would first do all variables for y = h, then y = 2h etc.
'''
for Ii in range(Istart,Iend):
v = -1.0
i = int(Ii/n)
j = int(Ii - i*n)
if (i>0):
J = Ii - n
A.setValues(Ii,J,v,addv=True)
if (i<m-1):
J = Ii + n
A.setValues(Ii,J,v,addv=True)
if (j>0):
J = Ii - 1
A.setValues(Ii,J,v,addv=True)
if (j<n-1):
J = Ii + 1
A.setValues(Ii,J,v,addv=True)
v = 4.0
A.setValues(Ii,Ii,v,addv=True)
'''
Assemble matrix, using the 2-step process:
MatAssemblyBegin(), MatAssemblyEnd()
Computations can be done while messages are in transition
by placing code between these two statements.
'''
A.assemblyBegin(A.AssemblyType.FINAL)
A.assemblyEnd(A.AssemblyType.FINAL)
''' A is symmetric. Set symmetric flag to enable ICC/Cholesky preconditioner '''
A.setOption(A.Option.SYMMETRIC,True)
'''
Create parallel vectors.
- We form 1 vector from scratch and then duplicate as needed.
- When using VecCreate(), VecSetSizes and VecSetFromOptions()
in this example, we specify only the
vector's global dimension; the parallel partitioning is determined
at runtime.
- When solving a linear system, the vectors and matrices MUST
be partitioned accordingly. PETSc automatically generates
appropriately partitioned matrices and vectors when MatCreate()
and VecCreate() are used with the same communicator.
- The user can alternatively specify the local vector and matrix
dimensions when more sophisticated partitioning is needed
(replacing the PETSC_DECIDE argument in the VecSetSizes() statement
below).
'''
u = PETSc.Vec().create(comm=comm)
u.setSizes(m*n)
u.setFromOptions()
b = u.duplicate()
x = b.duplicate()
'''
Set exact solution; then compute right-hand-side vector.
By default we use an exact solution of a vector with all
elements of 1.0;
'''
u.set(1.0)
b = A(u)
'''
View the exact solution vector if desired
'''
flg = OptDB.getBool('view_exact_sol', False)
if flg:
u.view()
''' - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Create the linear solver and set various options
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - '''
ksp = PETSc.KSP().create(comm=comm)
'''
Set operators. Here the matrix that defines the linear system
also serves as the preconditioning matrix.
'''
ksp.setOperators(A,A)
'''
Set linear solver defaults for this problem (optional).
- By extracting the KSP and PC contexts from the KSP context,
we can then directly call any KSP and PC routines to set
various options.
- The following two statements are optional; all of these
parameters could alternatively be specified at runtime via
KSPSetFromOptions(). All of these defaults can be
overridden at runtime, as indicated below.
'''
rtol=1.e-2/((m+1)*(n+1))
ksp.setTolerances(rtol=rtol,atol=1.e-50)
'''
Set runtime options, e.g.,
-ksp_type <type> -pc_type <type> -ksp_monitor -ksp_rtol <rtol>
These options will override those specified above as long as
KSPSetFromOptions() is called _after_ any other customization
routines.
'''
ksp.setFromOptions()
''' - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Solve the linear system
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - '''
ksp.solve(b,x)
''' - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Check the solution and clean up
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - '''
x = x - u # x.axpy(-1.0,u)
norm = x.norm(PETSc.NormType.NORM_2)
its = ksp.getIterationNumber()
'''
Print convergence information. PetscPrintf() produces a single
print statement from all processes that share a communicator.
An alternative is PetscFPrintf(), which prints to a file.
'''
if norm > rtol*10:
PETSc.Sys.Print("Norm of error {}, Iterations {}".format(norm,its),comm=comm)
else:
if size==1:
PETSc.Sys.Print("- Serial OK",comm=comm)
else:
PETSc.Sys.Print("- Parallel OK",comm=comm)
|
