基于ksp向导的PETSc解线性方程组

Question

我开始用PETSc库并行求解线性方程组。我已经安装了所有的软件包，成功地构建并运行了petsc/src/ksp/ksp/ examples /tutorials/文件夹中的示例，例如ex.c

但是我不能理解如何通过从文件中读取矩阵A，X和B来填充它们。

这里我提供了ex2.c文件中的代码：

/* Program usage:  mpiexec -n <procs> ex2 [-help] [all PETSc options] */ 

static char help[] = "Solves a linear system in parallel with KSP.\n\
Input parameters include:\n\
  -random_exact_sol : use a random exact solution vector\n\
  -view_exact_sol   : write exact solution vector to stdout\n\
  -m <mesh_x>       : number of mesh points in x-direction\n\
  -n <mesh_n>       : number of mesh points in y-direction\n\n";

/*T
   Concepts: KSP^basic parallel example;
   Concepts: KSP^Laplacian, 2d
   Concepts: Laplacian, 2d
   Processors: n
T*/

/* 
  Include "petscksp.h" so that we can use KSP solvers.  Note that this file
  automatically includes:
     petscsys.h       - base PETSc routines   petscvec.h - vectors
     petscmat.h - matrices
     petscis.h     - index sets            petscksp.h - Krylov subspace methods
     petscviewer.h - viewers               petscpc.h  - preconditioners
*/
#include <C:\PETSC\include\petscksp.h>

#undef __FUNCT__
#define __FUNCT__ "main"
int main(int argc,char **args)
{
  Vec            x,b,u;  /* approx solution, RHS, exact solution */
  Mat            A;        /* linear system matrix */
  KSP            ksp;     /* linear solver context */
  PetscRandom    rctx;     /* random number generator context */
  PetscReal      norm;     /* norm of solution error */
  PetscInt       i,j,Ii,J,Istart,Iend,m = 8,n = 7,its;
  PetscErrorCode ierr;
  PetscBool      flg = PETSC_FALSE;
  PetscScalar    v;
#if defined(PETSC_USE_LOG)
  PetscLogStage  stage;
#endif

  PetscInitialize(&argc,&args,(char *)0,help);
  ierr = PetscOptionsGetInt(PETSC_NULL,"-m",&m,PETSC_NULL);CHKERRQ(ierr);
  ierr = PetscOptionsGetInt(PETSC_NULL,"-n",&n,PETSC_NULL);CHKERRQ(ierr);
  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
         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.
  */
  ierr = MatCreate(PETSC_COMM_WORLD,&A);CHKERRQ(ierr);
  ierr = MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,m*n,m*n);CHKERRQ(ierr);
  ierr = MatSetFromOptions(A);CHKERRQ(ierr);
  ierr = MatMPIAIJSetPreallocation(A,5,PETSC_NULL,5,PETSC_NULL);CHKERRQ(ierr);
  ierr = MatSeqAIJSetPreallocation(A,5,PETSC_NULL);CHKERRQ(ierr);

  /* 
     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. 
  */
  ierr = MatGetOwnershipRange(A,&Istart,&Iend);CHKERRQ(ierr);

  /* 
     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.

   */
  ierr = PetscLogStageRegister("Assembly", &stage);CHKERRQ(ierr);
  ierr = PetscLogStagePush(stage);CHKERRQ(ierr);
  for (Ii=Istart; Ii<Iend; Ii++) { 
    v = -1.0; i = Ii/n; j = Ii - i*n;  
    if (i>0)   {J = Ii - n; ierr = MatSetValues(A,1,&Ii,1,&J,&v,INSERT_VALUES);CHKERRQ(ierr);}
    if (i<m-1) {J = Ii + n; ierr = MatSetValues(A,1,&Ii,1,&J,&v,INSERT_VALUES);CHKERRQ(ierr);}
    if (j>0)   {J = Ii - 1; ierr = MatSetValues(A,1,&Ii,1,&J,&v,INSERT_VALUES);CHKERRQ(ierr);}
    if (j<n-1) {J = Ii + 1; ierr = MatSetValues(A,1,&Ii,1,&J,&v,INSERT_VALUES);CHKERRQ(ierr);}
    v = 4.0; ierr = MatSetValues(A,1,&Ii,1,&Ii,&v,INSERT_VALUES);CHKERRQ(ierr);
  }

  /* 
     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.
  */
  ierr = MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = PetscLogStagePop();CHKERRQ(ierr);

  /* A is symmetric. Set symmetric flag to enable ICC/Cholesky preconditioner */
  ierr = MatSetOption(A,MAT_SYMMETRIC,PETSC_TRUE);CHKERRQ(ierr);

  /* 
     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).
  */
  ierr = VecCreate(PETSC_COMM_WORLD,&u);CHKERRQ(ierr);
  ierr = VecSetSizes(u,PETSC_DECIDE,m*n);CHKERRQ(ierr);
  ierr = VecSetFromOptions(u);CHKERRQ(ierr);
  ierr = VecDuplicate(u,&b);CHKERRQ(ierr); 
  ierr = VecDuplicate(b,&x);CHKERRQ(ierr);

  /* 
     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;  Alternatively, using the runtime option
     -random_sol forms a solution vector with random components.
  */
  ierr = PetscOptionsGetBool(PETSC_NULL,"-random_exact_sol",&flg,PETSC_NULL);CHKERRQ(ierr);
  if (flg) {
    ierr = PetscRandomCreate(PETSC_COMM_WORLD,&rctx);CHKERRQ(ierr);
    ierr = PetscRandomSetFromOptions(rctx);CHKERRQ(ierr);
    ierr = VecSetRandom(u,rctx);CHKERRQ(ierr);
    ierr = PetscRandomDestroy(&rctx);CHKERRQ(ierr);
  } else {
    ierr = VecSet(u,1.0);CHKERRQ(ierr);
  }
  ierr = MatMult(A,u,b);CHKERRQ(ierr);

  /*
     View the exact solution vector if desired
  */
  flg  = PETSC_FALSE;
  ierr = PetscOptionsGetBool(PETSC_NULL,"-view_exact_sol",&flg,PETSC_NULL);CHKERRQ(ierr);
  if (flg) {ierr = VecView(u,PETSC_VIEWER_STDOUT_WORLD);CHKERRQ(ierr);}

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
                Create the linear solver and set various options
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

  /* 
     Create linear solver context
  */
  ierr = KSPCreate(PETSC_COMM_WORLD,&ksp);CHKERRQ(ierr);

  /* 
     Set operators. Here the matrix that defines the linear system
     also serves as the preconditioning matrix.
  */
  ierr = KSPSetOperators(ksp,A,A,DIFFERENT_NONZERO_PATTERN);CHKERRQ(ierr);

  /* 
     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.
  */
  ierr = KSPSetTolerances(ksp,1.e-2/((m+1)*(n+1)),1.e-50,PETSC_DEFAULT,
                          PETSC_DEFAULT);CHKERRQ(ierr);

  /* 
    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.
  */
  ierr = KSPSetFromOptions(ksp);CHKERRQ(ierr);

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
                      Solve the linear system
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

  ierr = KSPSolve(ksp,b,x);CHKERRQ(ierr);

  /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
                      Check solution and clean up
     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

  /* 
     Check the error
  */
  ierr = VecAXPY(x,-1.0,u);CHKERRQ(ierr);
  ierr = VecNorm(x,NORM_2,&norm);CHKERRQ(ierr);
  ierr = KSPGetIterationNumber(ksp,&its);CHKERRQ(ierr);
  /* Scale the norm */
  /*  norm *= sqrt(1.0/((m+1)*(n+1))); */

  /*
     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.
  */
  ierr = PetscPrintf(PETSC_COMM_WORLD,"Norm of error %A iterations %D\n",
                     norm,its);CHKERRQ(ierr);

  /*
     Free work space.  All PETSc objects should be destroyed when they
     are no longer needed.
  */
  ierr = KSPDestroy(&ksp);CHKERRQ(ierr);
  ierr = VecDestroy(&u);CHKERRQ(ierr);  ierr = VecDestroy(&x);CHKERRQ(ierr);
  ierr = VecDestroy(&b);CHKERRQ(ierr);  ierr = MatDestroy(&A);CHKERRQ(ierr);

  /*
     Always call PetscFinalize() before exiting a program.  This routine
       - finalizes the PETSc libraries as well as MPI
       - provides summary and diagnostic information if certain runtime
         options are chosen (e.g., -log_summary). 
  */
  ierr = PetscFinalize();
  return 0;
}

有人知道如何在示例中填充自己的矩阵吗？

Answer 1

是的，当你刚开始的时候，这可能有点令人望而生畏。在2006年的this ACTS教程中有一个很好的演练过程；PetSC网页上的tutorials listed通常都很好。

其中的关键部分是：

  ierr = MatCreate(PETSC_COMM_WORLD,&A);CHKERRQ(ierr);

实际创建PetSC矩阵对象Mat A；

  ierr = MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,m*n,m*n);CHKERRQ(ierr);

设置大小；这里的矩阵是m*n x m*n，因为它是在m x n 2d网格上操作的模板

  ierr = MatSetFromOptions(A);CHKERRQ(ierr);

如果你想控制A是如何设置的，这只需要你在运行时提供的任何PetSC命令行选项，并将它们应用于矩阵；否则，你可以使用MatCreateMPIAIJ()将其创建为AIJ格式的矩阵(默认)，如果它将是密集矩阵，则使用MatCreateMPIDense()。

  ierr = MatMPIAIJSetPreallocation(A,5,PETSC_NULL,5,PETSC_NULL);CHKERRQ(ierr);
  ierr = MatSeqAIJSetPreallocation(A,5,PETSC_NULL);CHKERRQ(ierr);

现在我们已经得到了一个AIJ矩阵，这些调用只是预先分配了稀疏矩阵，假设每行有5个非零。这是为了提高性能。请注意，必须同时调用MPI和Seq函数，以确保它同时适用于单处理器和多处理器；这看起来总是很奇怪，但就是这样。

好了，现在矩阵都设置好了，现在我们开始进入实际问题的核心。

首先，我们找出这个特定进程拥有哪些行。分布是按行的，对于典型的稀疏矩阵来说，这是一个很好的分布。

  ierr = MatGetOwnershipRange(A,&Istart,&Iend);CHKERRQ(ierr);

因此，在此调用之后，每个处理器都有自己的Istart和Iend版本，它的this processor作业更新从Istart开始的行，在Iend之前结束，如下面的for循环所示：

  for (Ii=Istart; Ii<Iend; Ii++) { 
    v = -1.0; i = Ii/n; j = Ii - i*n;  

好的，如果我们在行Ii上操作，这对应于网格位置(i,j)，其中i = Ii/n和j = Ii % n。例如，网格位置(i,j)对应于行Ii = i*n + j。合乎道理?

我将在这里去掉if语句，因为它们很重要，但它们只是处理边界值，它们使事情变得更加复杂。

在此行中，对角线上将有一个+4，对应于(i-1,j)、(i+1,j)、(i,j-1)和(i,j+1)的列上将有一个-1。假设我们还没有脱离网格的末端(例如，1 < i < m-1和1 < j < n-1)，这意味着

    J = Ii - n; ierr = MatSetValues(A,1,&Ii,1,&J,&v,INSERT_VALUES);CHKERRQ(ierr);
    J = Ii + n; ierr = MatSetValues(A,1,&Ii,1,&J,&v,INSERT_VALUES);CHKERRQ(ierr);
    J = Ii - 1; ierr = MatSetValues(A,1,&Ii,1,&J,&v,INSERT_VALUES);CHKERRQ(ierr);
    J = Ii + 1; ierr = MatSetValues(A,1,&Ii,1,&J,&v,INSERT_VALUES);CHKERRQ(ierr);

    v = 4.0; ierr = MatSetValues(A,1,&Ii,1,&Ii,&v,INSERT_VALUES);CHKERRQ(ierr);
  }

我去掉的if语句只是避免设置那些不存在的值，如果ierr != 0，CHKERRQ宏会打印出一个有用的错误，例如set values调用失败(因为我们试图设置一个无效值)。

现在，我们已经设置了本地值；MatAssembly调用开始通信，以确保在处理器之间交换任何必要的值。如果你有任何不相关的工作要做，它可以卡在开始和结束之间，试图重叠通信和计算：

  ierr = MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);

现在你完成了，可以调用你的求解器。

因此，典型的工作流程是：

• Create your matrix (MatCreate)
• Set its (MatSetSizes)
• Set各种矩阵选项(MatSetFromOptions是一个很好的选择，而不是硬编码的东西)
• 对于稀疏矩阵，将预分配设置为每行非零数的合理猜测；您可以使用单个值(如下所示)或表示每行非零数的数组(此处用MatSeqAIJSetPreallocation)
• Find填充了PETSC_NULL)来完成此操作：( elements)
• Then，MatAssemblyEnd).

输出哪些行由您负责：(MatGetOwnershipRange)

• Set这些值(对于每个值调用MatSetValues一次，或者传入一组值；INSERT_VALUES设置新元素，ADD_VALUES递增任何现有的elements)

• Then执行程序集(MatSetValues，ADD_VALUES

其他更复杂的用例也是可能的。