!!$ !!$ !!$ MLD2P4 version 2.0 !!$ MultiLevel Domain Decomposition Parallel Preconditioners Package !!$ based on PSBLAS (Parallel Sparse BLAS version 3.0) !!$ !!$ (C) Copyright 2008,2009,2010 !!$ !!$ Salvatore Filippone University of Rome Tor Vergata !!$ Alfredo Buttari CNRS-IRIT, Toulouse !!$ Pasqua D'Ambra ICAR-CNR, Naples !!$ Daniela di Serafino Second University of Naples !!$ !!$ Redistribution and use in source and binary forms, with or without !!$ modification, are permitted provided that the following conditions !!$ are met: !!$ 1. Redistributions of source code must retain the above copyright !!$ notice, this list of conditions and the following disclaimer. !!$ 2. Redistributions in binary form must reproduce the above copyright !!$ notice, this list of conditions, and the following disclaimer in the !!$ documentation and/or other materials provided with the distribution. !!$ 3. 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File: mld_dexample_ml.f90 ! ! This sample program solves a linear system obtained by discretizing a ! PDE with Dirichlet BCs. The solver is BiCGStab coupled with one of the ! following multi-level preconditioner, as explained in Section 6.1 of ! the MLD2P4 User's and Reference Guide: ! - choice = 1, default multi-level Schwarz preconditioner (Sec. 6.1, Fig. 2) ! - choice = 2, hybrid three-level Schwarz preconditioner (Sec. 6.1, Fig. 3) ! - choice = 3, additive three-level Schwarz preconditioner (Sec. 6.1, Fig. 4) ! ! The PDE is a general second order equation in 3d ! ! b1 dd(u) b2 dd(u) b3 dd(u) a1 d(u) a2 d(u) a3 d(u) ! - ------ - ------ - ------ - ----- - ------ - ------ + a4 u = 0 ! dxdx dydy dzdz dx dy dz ! ! with Dirichlet boundary conditions, on the unit cube 0<=x,y,z<=1. ! ! Example taken from: ! C.T.Kelley ! Iterative Methods for Linear and Nonlinear Equations ! SIAM 1995 ! ! In this sample program the index space of the discretized ! computational domain is first numbered sequentially in a standard way, ! then the corresponding vector is distributed according to a BLOCK ! data distribution. ! ! Boundary conditions are set in a very simple way, by adding ! equations of the form ! ! u(x,y) = exp(-x^2-y^2-z^2) ! ! Note that if a1=a2=a3=a4=0., the PDE is the well-known Laplace equation. ! program mld_dexample_ml use psb_base_mod use mld_prec_mod use psb_krylov_mod use psb_util_mod use data_input implicit none ! input parameters ! sparse matrices type(psb_dspmat_type) :: A ! sparse matrices descriptor type(psb_desc_type):: desc_A ! preconditioner type(mld_dprec_type) :: P ! right-hand side, solution and residual vectors real(psb_dpk_), allocatable , save :: b(:), x(:), r(:) ! solver and preconditioner parameters real(psb_dpk_) :: tol, err integer :: itmax, iter, istop integer :: nlev ! parallel environment parameters integer :: ictxt, iam, np ! other variables integer :: choice integer :: i,info,j integer(psb_long_int_k_) :: amatsize, precsize, descsize integer :: idim, ierr, ircode real(psb_dpk_) :: t1, t2, tprec, resmx, resmxp character(len=20) :: name ! initialize the parallel environment call psb_init(ictxt) call psb_info(ictxt,iam,np) if (iam < 0) then ! This should not happen, but just in case call psb_exit(ictxt) stop endif name='mld_dexample_ml' if(psb_get_errstatus() /= 0) goto 9999 info=psb_success_ call psb_set_errverbosity(2) ! ! Hello world ! if (iam == psb_root_) then write(*,*) 'Welcome to MLD2P4 version: ',mld_version_string_ write(*,*) 'This is the ',trim(name),' sample program' end if ! get parameters call get_parms(ictxt,choice,idim,itmax,tol) ! allocate and fill in the coefficient matrix, rhs and initial guess call psb_barrier(ictxt) t1 = psb_wtime() call create_matrix(idim,a,b,x,desc_a,ictxt,info) call psb_barrier(ictxt) t2 = psb_wtime() - t1 if(info /= psb_success_) then info=psb_err_from_subroutine_ call psb_errpush(info,name) goto 9999 end if if (iam == psb_root_) write(*,'("Overall matrix creation time : ",es12.5)')t2 if (iam == psb_root_) write(*,'(" ")') select case(choice) case(1) ! initialize the default multi-level preconditioner, i.e. hybrid ! Schwarz, using RAS (with overlap 1 and ILU(0) on the blocks) ! as post-smoother and 4 block-Jacobi sweeps (with UMFPACK LU ! on the blocks) as distributed coarse-level solver call mld_precinit(P,'ML',info) case(2) ! set a three-level hybrid Schwarz preconditioner, which uses ! block Jacobi (with ILU(0) on the blocks) as post-smoother, ! a coarsest matrix replicated on the processors, and the ! LU factorization from UMFPACK as coarse-level solver call mld_precinit(P,'ML',info,nlev=3) call mld_precset(P,mld_smoother_type_,'BJAC',info) call mld_precset(P,mld_coarse_mat_,'REPL',info) call mld_precset(P,mld_coarse_solve_,'UMF',info) case(3) ! set a three-level additive Schwarz preconditioner, which uses ! RAS (with overlap 1 and ILU(0) on the blocks) as pre- and ! post-smoother, and 5 block-Jacobi sweeps (with UMFPACK LU ! on the blocks) as distributed coarsest-level solver call mld_precinit(P,'ML',info,nlev=3) call mld_precset(P,mld_ml_type_,'ADD',info) call mld_precset(P,mld_smoother_pos_,'TWOSIDE',info) call mld_precset(P,mld_coarse_sweeps_,5,info) end select ! build the preconditioner call psb_barrier(ictxt) t1 = psb_wtime() call mld_precbld(A,desc_A,P,info) tprec = psb_wtime()-t1 call psb_amx(ictxt, tprec) if (info /= psb_success_) then call psb_errpush(psb_err_from_subroutine_,name,a_err='psb_precbld') goto 9999 end if ! set the solver parameters and the initial guess call psb_geall(x,desc_A,info) x(:) =0.0 call psb_geasb(x,desc_A,info) ! solve Ax=b with preconditioned BiCGSTAB call psb_barrier(ictxt) t1 = psb_wtime() call psb_krylov('BICGSTAB',A,P,b,x,tol,desc_A,info,itmax,iter,err,itrace=1,istop=2) t2 = psb_wtime() - t1 call psb_amx(ictxt,t2) call psb_geall(r,desc_A,info) r(:) =0.0 call psb_geasb(r,desc_A,info) call psb_geaxpby(done,b,dzero,r,desc_A,info) call psb_spmm(-done,A,x,done,r,desc_A,info) call psb_genrm2s(resmx,r,desc_A,info) call psb_geamaxs(resmxp,r,desc_A,info) amatsize = psb_sizeof(A) descsize = psb_sizeof(desc_A) precsize = mld_sizeof(P) call psb_sum(ictxt,amatsize) call psb_sum(ictxt,descsize) call psb_sum(ictxt,precsize) call mld_precdescr(P,info) if (iam == psb_root_) then write(*,'(" ")') write(*,'("Matrix from PDE example")') write(*,'("Computed solution on ",i8," processors")')np write(*,'("Iterations to convergence : ",i6)')iter write(*,'("Error estimate on exit : ",es12.5)')err write(*,'("Time to build prec. : ",es12.5)')tprec write(*,'("Time to solve system : ",es12.5)')t2 write(*,'("Time per iteration : ",es12.5)')t2/(iter) write(*,'("Total time : ",es12.5)')t2+tprec write(*,'("Residual 2-norm : ",es12.5)')resmx write(*,'("Residual inf-norm : ",es12.5)')resmxp write(*,'("Total memory occupation for A : ",i12)')amatsize write(*,'("Total memory occupation for DESC_A : ",i12)')descsize write(*,'("Total memory occupation for PREC : ",i12)')precsize end if call psb_gefree(b, desc_A,info) call psb_gefree(x, desc_A,info) call psb_spfree(A, desc_A,info) call mld_precfree(P,info) call psb_cdfree(desc_A,info) 9999 continue if(info /= psb_success_) then call psb_error(ictxt) end if call psb_exit(ictxt) stop contains ! ! get parameters from standard input ! subroutine get_parms(ictxt,choice,idim,itmax,tol) use psb_base_mod implicit none integer :: choice, idim, ictxt, itmax real(psb_dpk_) :: tol integer :: iam, np call psb_info(ictxt,iam,np) if (iam == psb_root_) then ! read input parameters call read_data(choice,5) call read_data(idim,5) call read_data(itmax,5) call read_data(tol,5) end if call psb_bcast(ictxt,choice) call psb_bcast(ictxt,idim) call psb_bcast(ictxt,itmax) call psb_bcast(ictxt,tol) end subroutine get_parms ! ! subroutine to allocate and fill in the coefficient matrix and ! the rhs ! subroutine create_matrix(idim,a,b,xv,desc_a,ictxt,info) ! ! Discretize the partial diferential equation ! ! b1 dd(u) b2 dd(u) b3 dd(u) a1 d(u) a2 d(u) a3 d(u) ! - ------ - ------ - ------ - ----- - ------ - ------ + a4 u = 0 ! dxdx dydy dzdz dx dy dz ! ! with Dirichlet boundary conditions, on the unit cube 0<=x,y,z<=1. ! ! Boundary conditions are set in a very simple way, by adding ! equations of the form ! ! u(x,y) = exp(-x^2-y^2-z^2) ! ! Note that if a1=a2=a3=a4=0., the PDE is the well-known Laplace equation. ! use psb_base_mod implicit none integer :: idim integer, parameter :: nb=20 real(psb_dpk_), allocatable :: b(:),xv(:) type(psb_desc_type) :: desc_a integer :: ictxt, info character :: afmt*5 type(psb_dspmat_type) :: a real(psb_dpk_) :: zt(nb),x,y,z integer :: m,n,nnz,glob_row,nlr,i,ii,ib,k integer :: ix,iy,iz,ia,indx_owner, ipoints integer :: np, iam, nr, nt integer :: element integer, allocatable :: irow(:),icol(:),myidx(:) real(psb_dpk_), allocatable :: val(:) ! deltah dimension of each grid cell ! deltat discretization time real(psb_dpk_) :: deltah, deltah2 real(psb_dpk_),parameter :: rhs=0.d0,one=1.d0,zero=0.d0 real(psb_dpk_) :: t0, t1, t2, t3, tasb, talc, ttot, tgen real(psb_dpk_) :: a1, a2, a3, a4, b1, b2, b3 external :: a1, a2, a3, a4, b1, b2, b3 integer :: err_act character(len=20) :: name, ch_err info = psb_success_ name = 'create_matrix' call psb_erractionsave(err_act) call psb_info(ictxt, iam, np) deltah = 1.d0/(idim-1) deltah2 = deltah*deltah ! initialize array descriptor and sparse matrix storage. provide an ! estimate of the number of non zeroes ipoints=idim-2 m = ipoints*ipoints*ipoints n = m nnz = ((n*9)/(np)) if(iam == psb_root_) write(psb_out_unit,'("Generating Matrix (size=",i0,")...")')n ! ! Using a simple BLOCK distribution. ! nt = (m+np-1)/np nr = max(0,min(nt,m-(iam*nt))) nt = nr call psb_sum(ictxt,nt) if (nt /= m) write(psb_err_unit,*) iam, 'Initialization error ',nr,nt,m call psb_barrier(ictxt) t0 = psb_wtime() call psb_cdall(ictxt,desc_a,info,nl=nr) if (info == psb_success_) call psb_spall(a,desc_a,info,nnz=nnz) ! define rhs from boundary conditions; also build initial guess if (info == psb_success_) call psb_geall(b,desc_a,info) if (info == psb_success_) call psb_geall(xv,desc_a,info) nlr = psb_cd_get_local_rows(desc_a) call psb_barrier(ictxt) talc = psb_wtime()-t0 if (info /= psb_success_) then info=psb_err_from_subroutine_ ch_err='allocation rout.' call psb_errpush(info,name,a_err=ch_err) goto 9999 end if ! we build an auxiliary matrix consisting of one row at a ! time; just a small matrix. might be extended to generate ! a bunch of rows per call. ! allocate(val(20*nb),irow(20*nb),& &icol(20*nb),myidx(nlr),stat=info) if (info /= psb_success_ ) then info=psb_err_alloc_dealloc_ call psb_errpush(info,name) goto 9999 endif do i=1,nlr myidx(i) = i end do call psb_loc_to_glob(myidx,desc_a,info) ! loop over rows belonging to current process in a block ! distribution. call psb_barrier(ictxt) t1 = psb_wtime() do ii=1, nlr,nb ib = min(nb,nlr-ii+1) element = 1 do k=1,ib i=ii+k-1 ! local matrix pointer glob_row=myidx(i) ! compute gridpoint coordinates if (mod(glob_row,ipoints*ipoints) == 0) then ix = glob_row/(ipoints*ipoints) else ix = glob_row/(ipoints*ipoints)+1 endif if (mod((glob_row-(ix-1)*ipoints*ipoints),ipoints) == 0) then iy = (glob_row-(ix-1)*ipoints*ipoints)/ipoints else iy = (glob_row-(ix-1)*ipoints*ipoints)/ipoints+1 endif iz = glob_row-(ix-1)*ipoints*ipoints-(iy-1)*ipoints ! x, y, x coordinates x=ix*deltah y=iy*deltah z=iz*deltah ! check on boundary points zt(k) = 0.d0 ! internal point: build discretization ! ! term depending on (x-1,y,z) ! if (ix == 1) then val(element) = -b1(x,y,z)/deltah2-a1(x,y,z)/deltah zt(k) = exp(-x**2-y**2-z**2)*(-val(element)) else val(element) = -b1(x,y,z)/deltah2-a1(x,y,z)/deltah icol(element) = (ix-2)*ipoints*ipoints+(iy-1)*ipoints+(iz) irow(element) = glob_row element = element+1 endif ! term depending on (x,y-1,z) if (iy == 1) then val(element) = -b2(x,y,z)/deltah2-a2(x,y,z)/deltah zt(k) = exp(-x**2-y**2-z**2)*exp(-x)*(-val(element)) else val(element) = -b2(x,y,z)/deltah2-a2(x,y,z)/deltah icol(element) = (ix-1)*ipoints*ipoints+(iy-2)*ipoints+(iz) irow(element) = glob_row element = element+1 endif ! term depending on (x,y,z-1) if (iz == 1) then val(element)=-b3(x,y,z)/deltah2-a3(x,y,z)/deltah zt(k) = exp(-x**2-y**2-z**2)*exp(-x)*(-val(element)) else val(element)=-b3(x,y,z)/deltah2-a3(x,y,z)/deltah icol(element) = (ix-1)*ipoints*ipoints+(iy-1)*ipoints+(iz-1) irow(element) = glob_row element = element+1 endif ! term depending on (x,y,z) val(element)=(2*b1(x,y,z) + 2*b2(x,y,z) + 2*b3(x,y,z))/deltah2& & + (a1(x,y,z) + a2(x,y,z) + a3(x,y,z)+ a4(x,y,z))/deltah icol(element) = (ix-1)*ipoints*ipoints+(iy-1)*ipoints+(iz) irow(element) = glob_row element = element+1 ! term depending on (x,y,z+1) if (iz == ipoints) then val(element)=-b1(x,y,z)/deltah2 zt(k) = exp(-x**2-y**2-z**2)*exp(-x)*(-val(element)) else val(element)=-b1(x,y,z)/deltah2 icol(element) = (ix-1)*ipoints*ipoints+(iy-1)*ipoints+(iz+1) irow(element) = glob_row element = element+1 endif ! term depending on (x,y+1,z) if (iy == ipoints) then val(element)=-b2(x,y,z)/deltah2 zt(k) = exp(-x**2-y**2-z**2)*exp(-x)*(-val(element)) else val(element)=-b2(x,y,z)/deltah2 icol(element) = (ix-1)*ipoints*ipoints+(iy)*ipoints+(iz) irow(element) = glob_row element = element+1 endif ! term depending on (x+1,y,z) if (ix==ipoints) then val(element)=-b3(x,y,z)/deltah2 zt(k) = exp(-y**2-z**2)*exp(-x)*(-val(element)) else val(element)=-b3(x,y,z)/deltah2 icol(element) = (ix)*ipoints*ipoints+(iy-1)*ipoints+(iz) irow(element) = glob_row element = element+1 endif end do call psb_spins(element-1,irow,icol,val,a,desc_a,info) if(info /= psb_success_) exit call psb_geins(ib,myidx(ii:ii+ib-1),zt(1:ib),b,desc_a,info) if(info /= psb_success_) exit zt(:)=0.d0 call psb_geins(ib,myidx(ii:ii+ib-1),zt(1:ib),xv,desc_a,info) if(info /= psb_success_) exit end do tgen = psb_wtime()-t1 if(info /= psb_success_) then info=psb_err_from_subroutine_ call psb_errpush(info,name) goto 9999 end if deallocate(val,irow,icol) call psb_barrier(ictxt) t1 = psb_wtime() call psb_cdasb(desc_a,info) if (info == psb_success_) & & call psb_spasb(a,desc_a,info,dupl=psb_dupl_err_) call psb_barrier(ictxt) if(info /= psb_success_) then info=psb_err_from_subroutine_ call psb_errpush(info,name) goto 9999 end if call psb_geasb(b,desc_a,info) call psb_geasb(xv,desc_a,info) if(info /= psb_success_) then info=psb_err_from_subroutine_ call psb_errpush(info,name) goto 9999 end if tasb = psb_wtime()-t1 call psb_barrier(ictxt) ttot = psb_wtime() - t0 call psb_amx(ictxt,talc) call psb_amx(ictxt,tgen) call psb_amx(ictxt,tasb) call psb_amx(ictxt,ttot) if(iam == psb_root_) then write(*,'("The matrix has been generated and assembled in ",a3," format.")')& & a%get_fmt() write(*,'("-allocation time : ",es12.5)') talc write(*,'("-coeff. gen. time : ",es12.5)') tgen write(*,'("-assembly time : ",es12.5)') tasb write(*,'("-total time : ",es12.5)') ttot end if call psb_erractionrestore(err_act) return 9999 continue call psb_erractionrestore(err_act) if (err_act == psb_act_abort_) then call psb_error(ictxt) return end if return end subroutine create_matrix end program mld_dexample_ml ! ! functions parametrizing the differential equation ! function a1(x,y,z) use psb_base_mod, only : psb_dpk_ real(psb_dpk_) :: a1 real(psb_dpk_) :: x,y,z ! a1=1.d0 a1=0.d0 end function a1 function a2(x,y,z) use psb_base_mod, only : psb_dpk_ real(psb_dpk_) :: a2 real(psb_dpk_) :: x,y,z ! a2=2.d1*y a2=0.d0 end function a2 function a3(x,y,z) use psb_base_mod, only : psb_dpk_ real(psb_dpk_) :: a3 real(psb_dpk_) :: x,y,z ! a3=1.d0 a3=0.d0 end function a3 function a4(x,y,z) use psb_base_mod, only : psb_dpk_ real(psb_dpk_) :: a4 real(psb_dpk_) :: x,y,z ! a4=1.d0 a4=0.d0 end function a4 function b1(x,y,z) use psb_base_mod, only : psb_dpk_ real(psb_dpk_) :: b1 real(psb_dpk_) :: x,y,z b1=1.d0 end function b1 function b2(x,y,z) use psb_base_mod, only : psb_dpk_ real(psb_dpk_) :: b2 real(psb_dpk_) :: x,y,z b2=1.d0 end function b2 function b3(x,y,z) use psb_base_mod, only : psb_dpk_ real(psb_dpk_) :: b3 real(psb_dpk_) :: x,y,z b3=1.d0 end function b3