Using petsc

The chunk of f90 code at the bottom of this page shows one way to solve a sparse linear system, on a single processor - parallel version at using petsc (parallel) - using PETSc. I'll try to solve the a discrete version of the elliptic problem

$$ \nabla. (a(x,y) \nabla u) + b(x,y)u = c(x,y)$$

$$ u(0) = 0$$

$$ u'(L) = d$$

on a 256x256 grid, with a,b and c set so 1 on the left half of the domain, and h, r and p on the right - a bit like a grounding line problem

If you're new to PETSc, you might find all of the MatSetValues statements a bit odd. It *is* possible to full up normal Fortran arrays, and then insert them into PETSc structures, but the methods I've used below are (a bit) easier to adapt to multiple processors. The constants in the code

!x > 1/2 diffusion = h u' h = 1e-2 !x > 1/2 dissipation = r u, set small r to have problems inverting the matrix !so that direct solvers win r = 1e-6 !x > 1/2 source = p p = 0.5 ! u' at x = 1 d = 100

were picked to give the solvers a hard time - if they were all 1, the iterative methods below would beat the direct solver decisively in CPU time. I think the strength of PETSc is the number of solvers and preconditioners you get to try without too much fiddling

To run this, first you need PETSc, I configured it like this (stuff I compile myself ends up in /local)

./configure --prefix=/local/petsc/opt --download-parmetis=yes --download-mumps=yes --download-f-blas-lapack=yes --download-blacs=yes --download-scalapack=yes --enable-debugging=0 --with-mpi-dir=/local then followed the install instructions. Most of the options are to do with obtaining and compiling mumps, and PETSc has interfaces to dozens of third party solvers and preconditioners

Then compiled like this (with correct filenames and include / lib dirs)

mpif90 -I /local/petsc/opt/include/ -I /local/include/ -x f95-cpp-input scalar.f90 -L /local/petsc/opt/lib/ -L/local/lib -lpetscksp -lpetscdm -lpetscmat -lpetscvec -lpetsc -ldmumps -lmumps_common -lparmetis -lmetis -lpord -lscalapack -lblacs -lflapack -lfblas -lX11

and run >time ./a.out -ksp_monitor

0 KSP Residual norm 1.559571605727e+04 1 KSP Residual norm 1.529424636367e+04 2 KSP Residual norm 1.515888043725e+04 3 KSP Residual norm 1.497463214399e+04 4 KSP Residual norm 1.480442438141e+04 . . . 541 KSP Residual norm 1.587899253129e-01 542 KSP Residual norm 1.570278720318e-01 543 KSP Residual norm 1.555037845050e-01 ||Ax - b||/||b|| =  6.395110096044148E-006

real   0m3.702s user   0m3.651s sys    0m0.031s These default args give you GMRES(30) precondined with ilu(0) - much the same as the SLAP DSLUGM subroutine does. 543 is a lot of GMRES iterations - anything more than 30 and GMRES restarts, by default. It's possible to change this restart number, but instead I'll try a direct solver, MUMPS

> time ./a.out -ksp_monitor -pc_type lu -pc_factor_mat_solver_package mumps

0 KSP Residual norm 3.232253436059e+07 1 KSP Residual norm 1.077336304460e-06

real   0m1.377s user   0m1.263s sys    0m0.097s
 * Ax - b||/||b|| =  1.173428394516308E-012

much better. But in this case, a different choice of iterative solver, biconjugate gradients, works well too (and will use less memory as the problem size increases) time ./a.out -ksp_monitor -ksp_type bcgs

0 KSP Residual norm 1.559571605727e+04 1 KSP Residual norm 2.654841192325e+04 2 KSP Residual norm 3.504875048005e+04 3 KSP Residual norm 3.727376385776e+04 . . . 107 KSP Residual norm 2.332690535574e-01 108 KSP Residual norm 1.976699334772e-01 109 KSP Residual norm 1.560968894387e-01 110 KSP Residual norm 1.366366438749e-01 ||Ax - b||/||b|| =  3.980990389893450E-006
 * Ax - b||/||b|| =  1.84931214581757455E-005

real   0m0.892s user   0m0.857s sys    0m0.014s

or, with a different (more scalable) preconditioner, successive over relaxation time ./a.out -ksp_monitor -ksp_type bcgs -pc_type sor 0 KSP Residual norm 9.191868858477e+03 1 KSP Residual norm 1.399973304466e+04 2 KSP Residual norm 2.017715315232e+04 3 KSP Residual norm 2.256365643228e+04 4 KSP Residual norm 2.245000461834e+04 . . . 114 KSP Residual norm 2.371656459961e-01 115 KSP Residual norm 2.201423306979e-01 116 KSP Residual norm 4.348021435156e-02 ||Ax - b||/||b|| =  5.648286513658119E-006

real   0m0.913s user   0m0.884s sys    0m0.013s

contents of scalar.f90 !! solve a 2D n x n (a(x,y)u')' + b(x,y) u = c(x,y) !! with periodic boundary conditions in y, u(x=0,y) = 0 and u'(x=1,y)= d !! a(x,y) = 1 for x < 1/2, h for x > 1/2 !! b(x,y) = 1 for x < 1/2, r for x > 1/2 !! c(x,y) = 1 for x < 1/2, p for x > 1/2

program main

implicit none

Mat A Vec x,b, res KSP ksp PetscInt ierr, ndof, n, m, i, j PetscScalar dfluxe(0:1), dfluxw(0:1), dfluxn(0:1), dfluxs(0:1), su(0:0), sc, & h, r, p, d, rnorm, bnorm, one PetscInt row(0:0), fcol(0:1) n = 256 m = n/2 ndof = n * n
 * 1) include "finclude/petsc.h"
 * 2) include "finclude/petscvec.h"
 * 3) include "finclude/petscmat.h"
 * 4) include "finclude/petscksp.h"
 * 5) include "finclude/petscpc.h"

!x > 1/2 diffusion = h u' h = 1e-2 !x > 1/2 dissipation = r u, set small r to have problems inverting the matrix !so that direct solvers win r = 1e-6 !x > 1/2 source = p p = 0.5 ! u' at x = 1 d = 100

! this has to be the first petsc call call PetscInitialize(PETSC_NULL_CHARACTER,ierr)

! alocate space for rhs, residual and solution call VecCreateSeq(PETSC_COMM_SELF, ndof, x, ierr); call VecCreateSeq(PETSC_COMM_SELF, ndof, b, ierr); call VecCreateSeq(PETSC_COMM_SELF, ndof, res, ierr);

!Matrix assembly. I'm doing this in the church of PETSc style, which might !look a bit odd. If you already have arrays i,j,a in the CSR form, there is  !a single call, ! ! call MatCreateSeqAIJWithArrays(PETSC_COMM_SELF, ndof, ndof, i, j, a, A, ierr) ! which would do the trick ! ! m is the number of rows, n > m the number of nonzeros ! a(0:n-1) contains values and j(0:n-1) coulmn indices ! i(0:m) contains row pointers, e.g i(4) data for row 4 starts ! at a(i(4)) and finishes at a(i(5)-1) ! create matrix A, allocating 5 nonzeros per row, this doesn't have to be exactly ! right, but if it is to small, there will be costly mallocs, and if too ! large, additional operations. There is a variant which allows a per-row sparisity ! pattern to be set. call MatCreateSeqAIJ(PETSC_COMM_SELF, ndof, ndof, 5, PETSC_NULL_INTEGER,A,ierr) ! bulk dfluxe(0) = 1.0 dfluxe(1) = -1.0 su(0) = 1.0 sc = 1.0

do i = 1, n-2

dfluxw(0) = dfluxe(0) dfluxw(1) = dfluxe(1)

dfluxn(0) = 1.0 dfluxn(1) = -1.0

if (i > m) then su(0) = r       sc = p        dfluxe(0) = h        dfluxe(1) = -h dfluxn(0) = h       dfluxn(1) = -h endif dfluxs(0) = dfluxn(0) dfluxs(1) = dfluxn(1) do j = 1, n-2 row(0) = j*n + i       ! source term call MatSetValues(A, 1 ,row, 1, row, su, ADD_VALUES,ierr) call VecSetValue( b, row(0), sc , INSERT_VALUES, ierr)

! fluxes fcol(0) = row(0)

fcol(1) = row(0) + 1 call MatSetValues(A, 1 ,row, 2, fcol, dfluxe, ADD_VALUES,ierr) fcol(1) = row(0) - 1 call MatSetValues(A, 1 ,row, 2, fcol, dfluxw, ADD_VALUES,ierr) fcol(1) = row(0) + n       call MatSetValues(A, 1 ,row, 2, fcol, dfluxn, ADD_VALUES,ierr) fcol(1) = row(0) - n       call MatSetValues(A, 1 ,row, 2, fcol, dfluxs, ADD_VALUES,ierr)

end do

! periodic boundaries j = 0 row(0) = j*n + i    ! source term call MatSetValues(A, 1 ,row, 1, row, su, ADD_VALUES,ierr) call VecSetValue( b, row(0), sc , INSERT_VALUES, ierr)

! fluxes fcol(0) = row(0)

fcol(1) = row(0) + 1 call MatSetValues(A, 1 ,row, 2, fcol, dfluxe, ADD_VALUES,ierr) fcol(1) = row(0) - 1 call MatSetValues(A, 1 ,row, 2, fcol, dfluxw, ADD_VALUES,ierr) fcol(1) = row(0) + n    call MatSetValues(A, 1 ,row, 2, fcol, dfluxn, ADD_VALUES,ierr) fcol(1) = (n-1)*n + i    call MatSetValues(A, 1 ,row, 2, fcol, dfluxs, ADD_VALUES,ierr)

j = n-1 row(0) = j*n + i    ! source term call MatSetValues(A, 1 ,row, 1, row, su, ADD_VALUES,ierr) call VecSetValue( b, row(0), sc , INSERT_VALUES, ierr)

! fluxes fcol(0) = row(0)

fcol(1) = row(0) + 1 call MatSetValues(A, 1 ,row, 2, fcol, dfluxe, ADD_VALUES,ierr) fcol(1) = row(0) - 1 call MatSetValues(A, 1 ,row, 2, fcol, dfluxw, ADD_VALUES,ierr) fcol(1) = row(0) - n    call MatSetValues(A, 1 ,row, 2, fcol, dfluxn, ADD_VALUES,ierr) fcol(1) = i    call MatSetValues(A, 1 ,row, 2, fcol, dfluxs, ADD_VALUES,ierr)

end do !x-boundaries

dfluxw(0) = h dfluxw(1) = -h

do j = 0, n - 1 i = 0 row(0) = j*n + i    call MatSetValues(A, 1 ,row, 1, row, su, ADD_VALUES,ierr)

i = n - 1 row(0) = j*n + i    su = r     sc = h*d call MatSetValues(A, 1 ,row, 1, row, su, ADD_VALUES,ierr) call VecSetValue( b, row(0), sc , INSERT_VALUES, ierr) fcol(0) = row(0) fcol(1) = row(0) - 1 call MatSetValues(A, 1 ,row, 2, fcol, dfluxw, ADD_VALUES,ierr)

end do

!needs to be done before A can be used call MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY,ierr) call MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY,ierr)

!KSP context: KSP is petsc's interface to linear solvers call KSPCreate(PETSC_COMM_SELF,ksp,ierr)

!A is defines both the operator and preconditioner (but might not) call KSPSetOperators(ksp,A,A,DIFFERENT_NONZERO_PATTERN,ierr)

!grab all the options from the command line call KSPSetFromOptions(ksp,ierr)

! solve call KSPSolve(ksp,b,x,ierr)

!check results call MatMult(A, x, res, ierr) one = -1.0 call VecAXPY(res, one , b, ierr) call VecNorm(res, NORM_2, rnorm, ierr) call VecNorm(b, NORM_2, bnorm, ierr) write(*,*) "||Ax - b||/||b|| = ", rnorm/bnorm

!clean up  call VecDestroy(x,ierr) call VecDestroy(b,ierr) call MatDestroy(A,ierr)

!last petsc call call PetscFinalize(ierr)

end program main