aij/mpi/fdmpiaij.c

#include <../src/mat/impls/sell/mpi/mpisell.h>
#include <../src/mat/impls/aij/mpi/mpiaij.h>
#include <../src/mat/impls/baij/mpi/mpibaij.h>
#include <petsc/private/isimpl.h>

PetscErrorCode MatFDColoringApply_BAIJ(Mat J, MatFDColoring coloring, Vec x1, void *sctx) {
  PetscErrorCode (*f)(void *, Vec, Vec, void *) = (PetscErrorCode(*)(void *, Vec, Vec, void *))coloring->f;
  PetscInt           k, cstart, cend, l, row, col, nz, spidx, i, j;
  PetscScalar        dx = 0.0, *w3_array, *dy_i, *dy = coloring->dy;
  PetscScalar       *vscale_array;
  const PetscScalar *xx;
  PetscReal          epsilon = coloring->error_rel, umin = coloring->umin, unorm;
  Vec                w1 = coloring->w1, w2 = coloring->w2, w3, vscale = coloring->vscale;
  void              *fctx  = coloring->fctx;
  PetscInt           ctype = coloring->ctype, nxloc, nrows_k;
  PetscScalar       *valaddr;
  MatEntry          *Jentry  = coloring->matentry;
  MatEntry2         *Jentry2 = coloring->matentry2;
  const PetscInt     ncolors = coloring->ncolors, *ncolumns = coloring->ncolumns, *nrows = coloring->nrows;
  PetscInt           bs = J->rmap->bs;

  PetscFunctionBegin;
  PetscCall(VecBindToCPU(x1, PETSC_TRUE));
  /* (1) Set w1 = F(x1) */
  if (!coloring->fset) {
    PetscCall(PetscLogEventBegin(MAT_FDColoringFunction, coloring, 0, 0, 0));
    PetscCall((*f)(sctx, x1, w1, fctx));
    PetscCall(PetscLogEventEnd(MAT_FDColoringFunction, coloring, 0, 0, 0));
  } else {
    coloring->fset = PETSC_FALSE;
  }

  /* (2) Compute vscale = 1./dx - the local scale factors, including ghost points */
  PetscCall(VecGetLocalSize(x1, &nxloc));
  if (coloring->htype[0] == 'w') {
    /* vscale = dx is a constant scalar */
    PetscCall(VecNorm(x1, NORM_2, &unorm));
    dx = 1.0 / (PetscSqrtReal(1.0 + unorm) * epsilon);
  } else {
    PetscCall(VecGetArrayRead(x1, &xx));
    PetscCall(VecGetArray(vscale, &vscale_array));
    for (col = 0; col < nxloc; col++) {
      dx = xx[col];
      if (PetscAbsScalar(dx) < umin) {
        if (PetscRealPart(dx) >= 0.0) dx = umin;
        else if (PetscRealPart(dx) < 0.0) dx = -umin;
      }
      dx *= epsilon;
      vscale_array[col] = 1.0 / dx;
    }
    PetscCall(VecRestoreArrayRead(x1, &xx));
    PetscCall(VecRestoreArray(vscale, &vscale_array));
  }
  if (ctype == IS_COLORING_GLOBAL && coloring->htype[0] == 'd') {
    PetscCall(VecGhostUpdateBegin(vscale, INSERT_VALUES, SCATTER_FORWARD));
    PetscCall(VecGhostUpdateEnd(vscale, INSERT_VALUES, SCATTER_FORWARD));
  }

  /* (3) Loop over each color */
  if (!coloring->w3) {
    PetscCall(VecDuplicate(x1, &coloring->w3));
    /* Vec is used intensively in particular piece of scalar CPU code; won't benefit from bouncing back and forth to the GPU */
    PetscCall(VecBindToCPU(coloring->w3, PETSC_TRUE));
    PetscCall(PetscLogObjectParent((PetscObject)coloring, (PetscObject)coloring->w3));
  }
  w3 = coloring->w3;

  PetscCall(VecGetOwnershipRange(x1, &cstart, &cend)); /* used by ghosted vscale */
  if (vscale) PetscCall(VecGetArray(vscale, &vscale_array));
  nz = 0;
  for (k = 0; k < ncolors; k++) {
    coloring->currentcolor = k;

    /*
      (3-1) Loop over each column associated with color
      adding the perturbation to the vector w3 = x1 + dx.
    */
    PetscCall(VecCopy(x1, w3));
    dy_i = dy;
    for (i = 0; i < bs; i++) { /* Loop over a block of columns */
      PetscCall(VecGetArray(w3, &w3_array));
      if (ctype == IS_COLORING_GLOBAL) w3_array -= cstart; /* shift pointer so global index can be used */
      if (coloring->htype[0] == 'w') {
        for (l = 0; l < ncolumns[k]; l++) {
          col = i + bs * coloring->columns[k][l]; /* local column (in global index!) of the matrix we are probing for */
          w3_array[col] += 1.0 / dx;
          if (i) w3_array[col - 1] -= 1.0 / dx; /* resume original w3[col-1] */
        }
      } else {                  /* htype == 'ds' */
        vscale_array -= cstart; /* shift pointer so global index can be used */
        for (l = 0; l < ncolumns[k]; l++) {
          col = i + bs * coloring->columns[k][l]; /* local column (in global index!) of the matrix we are probing for */
          w3_array[col] += 1.0 / vscale_array[col];
          if (i) w3_array[col - 1] -= 1.0 / vscale_array[col - 1]; /* resume original w3[col-1] */
        }
        vscale_array += cstart;
      }
      if (ctype == IS_COLORING_GLOBAL) w3_array += cstart;
      PetscCall(VecRestoreArray(w3, &w3_array));

      /*
       (3-2) Evaluate function at w3 = x1 + dx (here dx is a vector of perturbations)
                           w2 = F(x1 + dx) - F(x1)
       */
      PetscCall(PetscLogEventBegin(MAT_FDColoringFunction, 0, 0, 0, 0));
      PetscCall(VecPlaceArray(w2, dy_i)); /* place w2 to the array dy_i */
      PetscCall((*f)(sctx, w3, w2, fctx));
      PetscCall(PetscLogEventEnd(MAT_FDColoringFunction, 0, 0, 0, 0));
      PetscCall(VecAXPY(w2, -1.0, w1));
      PetscCall(VecResetArray(w2));
      dy_i += nxloc; /* points to dy+i*nxloc */
    }

    /*
     (3-3) Loop over rows of vector, putting results into Jacobian matrix
    */
    nrows_k = nrows[k];
    if (coloring->htype[0] == 'w') {
      for (l = 0; l < nrows_k; l++) {
        row     = bs * Jentry2[nz].row; /* local row index */
        valaddr = Jentry2[nz++].valaddr;
        spidx   = 0;
        dy_i    = dy;
        for (i = 0; i < bs; i++) {   /* column of the block */
          for (j = 0; j < bs; j++) { /* row of the block */
            valaddr[spidx++] = dy_i[row + j] * dx;
          }
          dy_i += nxloc; /* points to dy+i*nxloc */
        }
      }
    } else { /* htype == 'ds' */
      for (l = 0; l < nrows_k; l++) {
        row     = bs * Jentry[nz].row; /* local row index */
        col     = bs * Jentry[nz].col; /* local column index */
        valaddr = Jentry[nz++].valaddr;
        spidx   = 0;
        dy_i    = dy;
        for (i = 0; i < bs; i++) {   /* column of the block */
          for (j = 0; j < bs; j++) { /* row of the block */
            valaddr[spidx++] = dy_i[row + j] * vscale_array[col + i];
          }
          dy_i += nxloc; /* points to dy+i*nxloc */
        }
      }
    }
  }
  PetscCall(MatAssemblyBegin(J, MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(J, MAT_FINAL_ASSEMBLY));
  if (vscale) PetscCall(VecRestoreArray(vscale, &vscale_array));

  coloring->currentcolor = -1;
  PetscCall(VecBindToCPU(x1, PETSC_FALSE));
  PetscFunctionReturn(0);
}

/* this is declared PETSC_EXTERN because it is used by MatFDColoringUseDM() which is in the DM library */
PetscErrorCode MatFDColoringApply_AIJ(Mat J, MatFDColoring coloring, Vec x1, void *sctx) {
  PetscErrorCode (*f)(void *, Vec, Vec, void *) = (PetscErrorCode(*)(void *, Vec, Vec, void *))coloring->f;
  PetscInt           k, cstart, cend, l, row, col, nz;
  PetscScalar        dx = 0.0, *y, *w3_array;
  const PetscScalar *xx;
  PetscScalar       *vscale_array;
  PetscReal          epsilon = coloring->error_rel, umin = coloring->umin, unorm;
  Vec                w1 = coloring->w1, w2 = coloring->w2, w3, vscale = coloring->vscale;
  void              *fctx  = coloring->fctx;
  ISColoringType     ctype = coloring->ctype;
  PetscInt           nxloc, nrows_k;
  MatEntry          *Jentry  = coloring->matentry;
  MatEntry2         *Jentry2 = coloring->matentry2;
  const PetscInt     ncolors = coloring->ncolors, *ncolumns = coloring->ncolumns, *nrows = coloring->nrows;
  PetscBool          alreadyboundtocpu;

  PetscFunctionBegin;
  PetscCall(VecBoundToCPU(x1, &alreadyboundtocpu));
  PetscCall(VecBindToCPU(x1, PETSC_TRUE));
  PetscCheck(!(ctype == IS_COLORING_LOCAL) || !(J->ops->fdcoloringapply == MatFDColoringApply_AIJ), PetscObjectComm((PetscObject)J), PETSC_ERR_SUP, "Must call MatColoringUseDM() with IS_COLORING_LOCAL");
  /* (1) Set w1 = F(x1) */
  if (!coloring->fset) {
    PetscCall(PetscLogEventBegin(MAT_FDColoringFunction, 0, 0, 0, 0));
    PetscCall((*f)(sctx, x1, w1, fctx));
    PetscCall(PetscLogEventEnd(MAT_FDColoringFunction, 0, 0, 0, 0));
  } else {
    coloring->fset = PETSC_FALSE;
  }

  /* (2) Compute vscale = 1./dx - the local scale factors, including ghost points */
  if (coloring->htype[0] == 'w') {
    /* vscale = 1./dx is a constant scalar */
    PetscCall(VecNorm(x1, NORM_2, &unorm));
    dx = 1.0 / (PetscSqrtReal(1.0 + unorm) * epsilon);
  } else {
    PetscCall(VecGetLocalSize(x1, &nxloc));
    PetscCall(VecGetArrayRead(x1, &xx));
    PetscCall(VecGetArray(vscale, &vscale_array));
    for (col = 0; col < nxloc; col++) {
      dx = xx[col];
      if (PetscAbsScalar(dx) < umin) {
        if (PetscRealPart(dx) >= 0.0) dx = umin;
        else if (PetscRealPart(dx) < 0.0) dx = -umin;
      }
      dx *= epsilon;
      vscale_array[col] = 1.0 / dx;
    }
    PetscCall(VecRestoreArrayRead(x1, &xx));
    PetscCall(VecRestoreArray(vscale, &vscale_array));
  }
  if (ctype == IS_COLORING_GLOBAL && coloring->htype[0] == 'd') {
    PetscCall(VecGhostUpdateBegin(vscale, INSERT_VALUES, SCATTER_FORWARD));
    PetscCall(VecGhostUpdateEnd(vscale, INSERT_VALUES, SCATTER_FORWARD));
  }

  /* (3) Loop over each color */
  if (!coloring->w3) {
    PetscCall(VecDuplicate(x1, &coloring->w3));
    PetscCall(PetscLogObjectParent((PetscObject)coloring, (PetscObject)coloring->w3));
  }
  w3 = coloring->w3;

  PetscCall(VecGetOwnershipRange(x1, &cstart, &cend)); /* used by ghosted vscale */
  if (vscale) PetscCall(VecGetArray(vscale, &vscale_array));
  nz = 0;

  if (coloring->bcols > 1) { /* use blocked insertion of Jentry */
    PetscInt     i, m = J->rmap->n, nbcols, bcols = coloring->bcols;
    PetscScalar *dy = coloring->dy, *dy_k;

    nbcols = 0;
    for (k = 0; k < ncolors; k += bcols) {
      /*
       (3-1) Loop over each column associated with color
       adding the perturbation to the vector w3 = x1 + dx.
       */

      dy_k = dy;
      if (k + bcols > ncolors) bcols = ncolors - k;
      for (i = 0; i < bcols; i++) {
        coloring->currentcolor = k + i;

        PetscCall(VecCopy(x1, w3));
        PetscCall(VecGetArray(w3, &w3_array));
        if (ctype == IS_COLORING_GLOBAL) w3_array -= cstart; /* shift pointer so global index can be used */
        if (coloring->htype[0] == 'w') {
          for (l = 0; l < ncolumns[k + i]; l++) {
            col = coloring->columns[k + i][l]; /* local column (in global index!) of the matrix we are probing for */
            w3_array[col] += 1.0 / dx;
          }
        } else {                  /* htype == 'ds' */
          vscale_array -= cstart; /* shift pointer so global index can be used */
          for (l = 0; l < ncolumns[k + i]; l++) {
            col = coloring->columns[k + i][l]; /* local column (in global index!) of the matrix we are probing for */
            w3_array[col] += 1.0 / vscale_array[col];
          }
          vscale_array += cstart;
        }
        if (ctype == IS_COLORING_GLOBAL) w3_array += cstart;
        PetscCall(VecRestoreArray(w3, &w3_array));

        /*
         (3-2) Evaluate function at w3 = x1 + dx (here dx is a vector of perturbations)
                           w2 = F(x1 + dx) - F(x1)
         */
        PetscCall(PetscLogEventBegin(MAT_FDColoringFunction, 0, 0, 0, 0));
        PetscCall(VecPlaceArray(w2, dy_k)); /* place w2 to the array dy_i */
        PetscCall((*f)(sctx, w3, w2, fctx));
        PetscCall(PetscLogEventEnd(MAT_FDColoringFunction, 0, 0, 0, 0));
        PetscCall(VecAXPY(w2, -1.0, w1));
        PetscCall(VecResetArray(w2));
        dy_k += m; /* points to dy+i*nxloc */
      }

      /*
       (3-3) Loop over block rows of vector, putting results into Jacobian matrix
       */
      nrows_k = nrows[nbcols++];

      if (coloring->htype[0] == 'w') {
        for (l = 0; l < nrows_k; l++) {
          row = Jentry2[nz].row; /* local row index */
                                 /* The 'useless' ifdef is due to a bug in NVIDIA nvc 21.11, which triggers a segfault on this line. We write it in
             another way, and it seems work. See https://lists.mcs.anl.gov/pipermail/petsc-users/2021-December/045158.html
           */
#if defined(PETSC_USE_COMPLEX)
          PetscScalar *tmp = Jentry2[nz].valaddr;
          *tmp             = dy[row] * dx;
#else
          *(Jentry2[nz].valaddr) = dy[row] * dx;
#endif
          nz++;
        }
      } else { /* htype == 'ds' */
        for (l = 0; l < nrows_k; l++) {
          row = Jentry[nz].row; /* local row index */
#if defined(PETSC_USE_COMPLEX)  /* See https://lists.mcs.anl.gov/pipermail/petsc-users/2021-December/045158.html */
          PetscScalar *tmp = Jentry[nz].valaddr;
          *tmp             = dy[row] * vscale_array[Jentry[nz].col];
#else
          *(Jentry[nz].valaddr)  = dy[row] * vscale_array[Jentry[nz].col];
#endif
          nz++;
        }
      }
    }
  } else { /* bcols == 1 */
    for (k = 0; k < ncolors; k++) {
      coloring->currentcolor = k;

      /*
       (3-1) Loop over each column associated with color
       adding the perturbation to the vector w3 = x1 + dx.
       */
      PetscCall(VecCopy(x1, w3));
      PetscCall(VecGetArray(w3, &w3_array));
      if (ctype == IS_COLORING_GLOBAL) w3_array -= cstart; /* shift pointer so global index can be used */
      if (coloring->htype[0] == 'w') {
        for (l = 0; l < ncolumns[k]; l++) {
          col = coloring->columns[k][l]; /* local column (in global index!) of the matrix we are probing for */
          w3_array[col] += 1.0 / dx;
        }
      } else {                  /* htype == 'ds' */
        vscale_array -= cstart; /* shift pointer so global index can be used */
        for (l = 0; l < ncolumns[k]; l++) {
          col = coloring->columns[k][l]; /* local column (in global index!) of the matrix we are probing for */
          w3_array[col] += 1.0 / vscale_array[col];
        }
        vscale_array += cstart;
      }
      if (ctype == IS_COLORING_GLOBAL) w3_array += cstart;
      PetscCall(VecRestoreArray(w3, &w3_array));

      /*
       (3-2) Evaluate function at w3 = x1 + dx (here dx is a vector of perturbations)
                           w2 = F(x1 + dx) - F(x1)
       */
      PetscCall(PetscLogEventBegin(MAT_FDColoringFunction, 0, 0, 0, 0));
      PetscCall((*f)(sctx, w3, w2, fctx));
      PetscCall(PetscLogEventEnd(MAT_FDColoringFunction, 0, 0, 0, 0));
      PetscCall(VecAXPY(w2, -1.0, w1));

      /*
       (3-3) Loop over rows of vector, putting results into Jacobian matrix
       */
      nrows_k = nrows[k];
      PetscCall(VecGetArray(w2, &y));
      if (coloring->htype[0] == 'w') {
        for (l = 0; l < nrows_k; l++) {
          row = Jentry2[nz].row; /* local row index */
#if defined(PETSC_USE_COMPLEX)   /* See https://lists.mcs.anl.gov/pipermail/petsc-users/2021-December/045158.html */
          PetscScalar *tmp = Jentry2[nz].valaddr;
          *tmp             = y[row] * dx;
#else
          *(Jentry2[nz].valaddr) = y[row] * dx;
#endif
          nz++;
        }
      } else { /* htype == 'ds' */
        for (l = 0; l < nrows_k; l++) {
          row = Jentry[nz].row; /* local row index */
#if defined(PETSC_USE_COMPLEX)  /* See https://lists.mcs.anl.gov/pipermail/petsc-users/2021-December/045158.html */
          PetscScalar *tmp = Jentry[nz].valaddr;
          *tmp             = y[row] * vscale_array[Jentry[nz].col];
#else
          *(Jentry[nz].valaddr)  = y[row] * vscale_array[Jentry[nz].col];
#endif
          nz++;
        }
      }
      PetscCall(VecRestoreArray(w2, &y));
    }
  }

#if defined(PETSC_HAVE_VIENNACL) || defined(PETSC_HAVE_CUDA)
  if (J->offloadmask != PETSC_OFFLOAD_UNALLOCATED) J->offloadmask = PETSC_OFFLOAD_CPU;
#endif
  PetscCall(MatAssemblyBegin(J, MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(J, MAT_FINAL_ASSEMBLY));
  if (vscale) PetscCall(VecRestoreArray(vscale, &vscale_array));
  coloring->currentcolor = -1;
  if (!alreadyboundtocpu) PetscCall(VecBindToCPU(x1, PETSC_FALSE));
  PetscFunctionReturn(0);
}

PetscErrorCode MatFDColoringSetUp_MPIXAIJ(Mat mat, ISColoring iscoloring, MatFDColoring c) {
  PetscMPIInt            size, *ncolsonproc, *disp, nn;
  PetscInt               i, n, nrows, nrows_i, j, k, m, ncols, col, *rowhit, cstart, cend, colb;
  const PetscInt        *is, *A_ci, *A_cj, *B_ci, *B_cj, *row = NULL, *ltog = NULL;
  PetscInt               nis = iscoloring->n, nctot, *cols, tmp = 0;
  ISLocalToGlobalMapping map   = mat->cmap->mapping;
  PetscInt               ctype = c->ctype, *spidxA, *spidxB, nz, bs, bs2, spidx;
  Mat                    A, B;
  PetscScalar           *A_val, *B_val, **valaddrhit;
  MatEntry              *Jentry;
  MatEntry2             *Jentry2;
  PetscBool              isBAIJ, isSELL;
  PetscInt               bcols = c->bcols;
#if defined(PETSC_USE_CTABLE)
  PetscTable colmap = NULL;
#else
  PetscInt *colmap = NULL;      /* local col number of off-diag col */
#endif

  PetscFunctionBegin;
  if (ctype == IS_COLORING_LOCAL) {
    PetscCheck(map, PetscObjectComm((PetscObject)mat), PETSC_ERR_ARG_INCOMP, "When using ghosted differencing matrix must have local to global mapping provided with MatSetLocalToGlobalMapping");
    PetscCall(ISLocalToGlobalMappingGetIndices(map, &ltog));
  }

  PetscCall(MatGetBlockSize(mat, &bs));
  PetscCall(PetscObjectBaseTypeCompare((PetscObject)mat, MATMPIBAIJ, &isBAIJ));
  PetscCall(PetscObjectTypeCompare((PetscObject)mat, MATMPISELL, &isSELL));
  if (isBAIJ) {
    Mat_MPIBAIJ *baij = (Mat_MPIBAIJ *)mat->data;
    Mat_SeqBAIJ *spA, *spB;
    A     = baij->A;
    spA   = (Mat_SeqBAIJ *)A->data;
    A_val = spA->a;
    B     = baij->B;
    spB   = (Mat_SeqBAIJ *)B->data;
    B_val = spB->a;
    nz    = spA->nz + spB->nz; /* total nonzero entries of mat */
    if (!baij->colmap) PetscCall(MatCreateColmap_MPIBAIJ_Private(mat));
    colmap = baij->colmap;
    PetscCall(MatGetColumnIJ_SeqBAIJ_Color(A, 0, PETSC_FALSE, PETSC_FALSE, &ncols, &A_ci, &A_cj, &spidxA, NULL));
    PetscCall(MatGetColumnIJ_SeqBAIJ_Color(B, 0, PETSC_FALSE, PETSC_FALSE, &ncols, &B_ci, &B_cj, &spidxB, NULL));

    if (ctype == IS_COLORING_GLOBAL && c->htype[0] == 'd') { /* create vscale for storing dx */
      PetscInt *garray;
      PetscCall(PetscMalloc1(B->cmap->n, &garray));
      for (i = 0; i < baij->B->cmap->n / bs; i++) {
        for (j = 0; j < bs; j++) garray[i * bs + j] = bs * baij->garray[i] + j;
      }
      PetscCall(VecCreateGhost(PetscObjectComm((PetscObject)mat), mat->cmap->n, PETSC_DETERMINE, B->cmap->n, garray, &c->vscale));
      PetscCall(VecBindToCPU(c->vscale, PETSC_TRUE));
      PetscCall(PetscFree(garray));
    }
  } else if (isSELL) {
    Mat_MPISELL *sell = (Mat_MPISELL *)mat->data;
    Mat_SeqSELL *spA, *spB;
    A     = sell->A;
    spA   = (Mat_SeqSELL *)A->data;
    A_val = spA->val;
    B     = sell->B;
    spB   = (Mat_SeqSELL *)B->data;
    B_val = spB->val;
    nz    = spA->nz + spB->nz; /* total nonzero entries of mat */
    if (!sell->colmap) {
      /* Allow access to data structures of local part of matrix
       - creates aij->colmap which maps global column number to local number in part B */
      PetscCall(MatCreateColmap_MPISELL_Private(mat));
    }
    colmap = sell->colmap;
    PetscCall(MatGetColumnIJ_SeqSELL_Color(A, 0, PETSC_FALSE, PETSC_FALSE, &ncols, &A_ci, &A_cj, &spidxA, NULL));
    PetscCall(MatGetColumnIJ_SeqSELL_Color(B, 0, PETSC_FALSE, PETSC_FALSE, &ncols, &B_ci, &B_cj, &spidxB, NULL));

    bs = 1; /* only bs=1 is supported for non MPIBAIJ matrix */

    if (ctype == IS_COLORING_GLOBAL && c->htype[0] == 'd') { /* create vscale for storing dx */
      PetscCall(VecCreateGhost(PetscObjectComm((PetscObject)mat), mat->cmap->n, PETSC_DETERMINE, B->cmap->n, sell->garray, &c->vscale));
      PetscCall(VecBindToCPU(c->vscale, PETSC_TRUE));
    }
  } else {
    Mat_MPIAIJ *aij = (Mat_MPIAIJ *)mat->data;
    Mat_SeqAIJ *spA, *spB;
    A     = aij->A;
    spA   = (Mat_SeqAIJ *)A->data;
    A_val = spA->a;
    B     = aij->B;
    spB   = (Mat_SeqAIJ *)B->data;
    B_val = spB->a;
    nz    = spA->nz + spB->nz; /* total nonzero entries of mat */
    if (!aij->colmap) {
      /* Allow access to data structures of local part of matrix
       - creates aij->colmap which maps global column number to local number in part B */
      PetscCall(MatCreateColmap_MPIAIJ_Private(mat));
    }
    colmap = aij->colmap;
    PetscCall(MatGetColumnIJ_SeqAIJ_Color(A, 0, PETSC_FALSE, PETSC_FALSE, &ncols, &A_ci, &A_cj, &spidxA, NULL));
    PetscCall(MatGetColumnIJ_SeqAIJ_Color(B, 0, PETSC_FALSE, PETSC_FALSE, &ncols, &B_ci, &B_cj, &spidxB, NULL));

    bs = 1; /* only bs=1 is supported for non MPIBAIJ matrix */

    if (ctype == IS_COLORING_GLOBAL && c->htype[0] == 'd') { /* create vscale for storing dx */
      PetscCall(VecCreateGhost(PetscObjectComm((PetscObject)mat), mat->cmap->n, PETSC_DETERMINE, B->cmap->n, aij->garray, &c->vscale));
      PetscCall(VecBindToCPU(c->vscale, PETSC_TRUE));
    }
  }

  m      = mat->rmap->n / bs;
  cstart = mat->cmap->rstart / bs;
  cend   = mat->cmap->rend / bs;

  PetscCall(PetscMalloc2(nis, &c->ncolumns, nis, &c->columns));
  PetscCall(PetscMalloc1(nis, &c->nrows));
  PetscCall(PetscLogObjectMemory((PetscObject)c, 3 * nis * sizeof(PetscInt)));

  if (c->htype[0] == 'd') {
    PetscCall(PetscMalloc1(nz, &Jentry));
    PetscCall(PetscLogObjectMemory((PetscObject)c, nz * sizeof(MatEntry)));
    c->matentry = Jentry;
  } else if (c->htype[0] == 'w') {
    PetscCall(PetscMalloc1(nz, &Jentry2));
    PetscCall(PetscLogObjectMemory((PetscObject)c, nz * sizeof(MatEntry2)));
    c->matentry2 = Jentry2;
  } else SETERRQ(PetscObjectComm((PetscObject)mat), PETSC_ERR_SUP, "htype is not supported");

  PetscCall(PetscMalloc2(m + 1, &rowhit, m + 1, &valaddrhit));
  nz = 0;
  PetscCall(ISColoringGetIS(iscoloring, PETSC_OWN_POINTER, PETSC_IGNORE, &c->isa));

  if (ctype == IS_COLORING_GLOBAL) {
    PetscCallMPI(MPI_Comm_size(PetscObjectComm((PetscObject)mat), &size));
    PetscCall(PetscMalloc2(size, &ncolsonproc, size, &disp));
  }

  for (i = 0; i < nis; i++) { /* for each local color */
    PetscCall(ISGetLocalSize(c->isa[i], &n));
    PetscCall(ISGetIndices(c->isa[i], &is));

    c->ncolumns[i] = n; /* local number of columns of this color on this process */
    c->columns[i]  = (PetscInt *)is;

    if (ctype == IS_COLORING_GLOBAL) {
      /* Determine nctot, the total (parallel) number of columns of this color */
      /* ncolsonproc[j]: local ncolumns on proc[j] of this color */
      PetscCall(PetscMPIIntCast(n, &nn));
      PetscCallMPI(MPI_Allgather(&nn, 1, MPI_INT, ncolsonproc, 1, MPI_INT, PetscObjectComm((PetscObject)mat)));
      nctot = 0;
      for (j = 0; j < size; j++) nctot += ncolsonproc[j];
      if (!nctot) PetscCall(PetscInfo(mat, "Coloring of matrix has some unneeded colors with no corresponding rows\n"));

      disp[0] = 0;
      for (j = 1; j < size; j++) disp[j] = disp[j - 1] + ncolsonproc[j - 1];

      /* Get cols, the complete list of columns for this color on each process */
      PetscCall(PetscMalloc1(nctot + 1, &cols));
      PetscCallMPI(MPI_Allgatherv((void *)is, n, MPIU_INT, cols, ncolsonproc, disp, MPIU_INT, PetscObjectComm((PetscObject)mat)));
    } else if (ctype == IS_COLORING_LOCAL) {
      /* Determine local number of columns of this color on this process, including ghost points */
      nctot = n;
      cols  = (PetscInt *)is;
    } else SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "Not provided for this MatFDColoring type");

    /* Mark all rows affect by these columns */
    PetscCall(PetscArrayzero(rowhit, m));
    bs2     = bs * bs;
    nrows_i = 0;
    for (j = 0; j < nctot; j++) { /* loop over columns*/
      if (ctype == IS_COLORING_LOCAL) {
        col = ltog[cols[j]];
      } else {
        col = cols[j];
      }
      if (col >= cstart && col < cend) { /* column is in A, diagonal block of mat */
        tmp   = A_ci[col - cstart];
        row   = A_cj + tmp;
        nrows = A_ci[col - cstart + 1] - tmp;
        nrows_i += nrows;

        /* loop over columns of A marking them in rowhit */
        for (k = 0; k < nrows; k++) {
          /* set valaddrhit for part A */
          spidx            = bs2 * spidxA[tmp + k];
          valaddrhit[*row] = &A_val[spidx];
          rowhit[*row++]   = col - cstart + 1; /* local column index */
        }
      } else { /* column is in B, off-diagonal block of mat */
#if defined(PETSC_USE_CTABLE)
        PetscCall(PetscTableFind(colmap, col + 1, &colb));
        colb--;
#else
        colb = colmap[col] - 1; /* local column index */
#endif
        if (colb == -1) {
          nrows = 0;
        } else {
          colb  = colb / bs;
          tmp   = B_ci[colb];
          row   = B_cj + tmp;
          nrows = B_ci[colb + 1] - tmp;
        }
        nrows_i += nrows;
        /* loop over columns of B marking them in rowhit */
        for (k = 0; k < nrows; k++) {
          /* set valaddrhit for part B */
          spidx            = bs2 * spidxB[tmp + k];
          valaddrhit[*row] = &B_val[spidx];
          rowhit[*row++]   = colb + 1 + cend - cstart; /* local column index */
        }
      }
    }
    c->nrows[i] = nrows_i;

    if (c->htype[0] == 'd') {
      for (j = 0; j < m; j++) {
        if (rowhit[j]) {
          Jentry[nz].row     = j;             /* local row index */
          Jentry[nz].col     = rowhit[j] - 1; /* local column index */
          Jentry[nz].valaddr = valaddrhit[j]; /* address of mat value for this entry */
          nz++;
        }
      }
    } else { /* c->htype == 'wp' */
      for (j = 0; j < m; j++) {
        if (rowhit[j]) {
          Jentry2[nz].row     = j;             /* local row index */
          Jentry2[nz].valaddr = valaddrhit[j]; /* address of mat value for this entry */
          nz++;
        }
      }
    }
    if (ctype == IS_COLORING_GLOBAL) PetscCall(PetscFree(cols));
  }
  if (ctype == IS_COLORING_GLOBAL) PetscCall(PetscFree2(ncolsonproc, disp));

  if (bcols > 1) { /* reorder Jentry for faster MatFDColoringApply() */
    PetscCall(MatFDColoringSetUpBlocked_AIJ_Private(mat, c, nz));
  }

  if (isBAIJ) {
    PetscCall(MatRestoreColumnIJ_SeqBAIJ_Color(A, 0, PETSC_FALSE, PETSC_FALSE, &ncols, &A_ci, &A_cj, &spidxA, NULL));
    PetscCall(MatRestoreColumnIJ_SeqBAIJ_Color(B, 0, PETSC_FALSE, PETSC_FALSE, &ncols, &B_ci, &B_cj, &spidxB, NULL));
    PetscCall(PetscMalloc1(bs * mat->rmap->n, &c->dy));
  } else if (isSELL) {
    PetscCall(MatRestoreColumnIJ_SeqSELL_Color(A, 0, PETSC_FALSE, PETSC_FALSE, &ncols, &A_ci, &A_cj, &spidxA, NULL));
    PetscCall(MatRestoreColumnIJ_SeqSELL_Color(B, 0, PETSC_FALSE, PETSC_FALSE, &ncols, &B_ci, &B_cj, &spidxB, NULL));
  } else {
    PetscCall(MatRestoreColumnIJ_SeqAIJ_Color(A, 0, PETSC_FALSE, PETSC_FALSE, &ncols, &A_ci, &A_cj, &spidxA, NULL));
    PetscCall(MatRestoreColumnIJ_SeqAIJ_Color(B, 0, PETSC_FALSE, PETSC_FALSE, &ncols, &B_ci, &B_cj, &spidxB, NULL));
  }

  PetscCall(ISColoringRestoreIS(iscoloring, PETSC_OWN_POINTER, &c->isa));
  PetscCall(PetscFree2(rowhit, valaddrhit));

  if (ctype == IS_COLORING_LOCAL) PetscCall(ISLocalToGlobalMappingRestoreIndices(map, &ltog));
  PetscCall(PetscInfo(c, "ncolors %" PetscInt_FMT ", brows %" PetscInt_FMT " and bcols %" PetscInt_FMT " are used.\n", c->ncolors, c->brows, c->bcols));
  PetscFunctionReturn(0);
}

PetscErrorCode MatFDColoringCreate_MPIXAIJ(Mat mat, ISColoring iscoloring, MatFDColoring c) {
  PetscInt  bs, nis = iscoloring->n, m = mat->rmap->n;
  PetscBool isBAIJ, isSELL;

  PetscFunctionBegin;
  /* set default brows and bcols for speedup inserting the dense matrix into sparse Jacobian;
   bcols is chosen s.t. dy-array takes 50% of memory space as mat */
  PetscCall(MatGetBlockSize(mat, &bs));
  PetscCall(PetscObjectBaseTypeCompare((PetscObject)mat, MATMPIBAIJ, &isBAIJ));
  PetscCall(PetscObjectTypeCompare((PetscObject)mat, MATMPISELL, &isSELL));
  if (isBAIJ || m == 0) {
    c->brows = m;
    c->bcols = 1;
  } else if (isSELL) {
    /* bcols is chosen s.t. dy-array takes 50% of local memory space as mat */
    Mat_MPISELL *sell = (Mat_MPISELL *)mat->data;
    Mat_SeqSELL *spA, *spB;
    Mat          A, B;
    PetscInt     nz, brows, bcols;
    PetscReal    mem;

    bs = 1; /* only bs=1 is supported for MPISELL matrix */

    A     = sell->A;
    spA   = (Mat_SeqSELL *)A->data;
    B     = sell->B;
    spB   = (Mat_SeqSELL *)B->data;
    nz    = spA->nz + spB->nz; /* total local nonzero entries of mat */
    mem   = nz * (sizeof(PetscScalar) + sizeof(PetscInt)) + 3 * m * sizeof(PetscInt);
    bcols = (PetscInt)(0.5 * mem / (m * sizeof(PetscScalar)));
    brows = 1000 / bcols;
    if (bcols > nis) bcols = nis;
    if (brows == 0 || brows > m) brows = m;
    c->brows = brows;
    c->bcols = bcols;
  } else { /* mpiaij matrix */
    /* bcols is chosen s.t. dy-array takes 50% of local memory space as mat */
    Mat_MPIAIJ *aij = (Mat_MPIAIJ *)mat->data;
    Mat_SeqAIJ *spA, *spB;
    Mat         A, B;
    PetscInt    nz, brows, bcols;
    PetscReal   mem;

    bs = 1; /* only bs=1 is supported for MPIAIJ matrix */

    A     = aij->A;
    spA   = (Mat_SeqAIJ *)A->data;
    B     = aij->B;
    spB   = (Mat_SeqAIJ *)B->data;
    nz    = spA->nz + spB->nz; /* total local nonzero entries of mat */
    mem   = nz * (sizeof(PetscScalar) + sizeof(PetscInt)) + 3 * m * sizeof(PetscInt);
    bcols = (PetscInt)(0.5 * mem / (m * sizeof(PetscScalar)));
    brows = 1000 / bcols;
    if (bcols > nis) bcols = nis;
    if (brows == 0 || brows > m) brows = m;
    c->brows = brows;
    c->bcols = bcols;
  }

  c->M       = mat->rmap->N / bs; /* set the global rows and columns and local rows */
  c->N       = mat->cmap->N / bs;
  c->m       = mat->rmap->n / bs;
  c->rstart  = mat->rmap->rstart / bs;
  c->ncolors = nis;
  PetscFunctionReturn(0);
}

/*@C

    MatFDColoringSetValues - takes a matrix in compressed color format and enters the matrix into a PETSc `Mat`

   Collective on J

   Input Parameters:
+    J - the sparse matrix
.    coloring - created with `MatFDColoringCreate()` and a local coloring
-    y - column major storage of matrix values with one color of values per column, the number of rows of y should match
         the number of local rows of J and the number of columns is the number of colors.

   Level: intermediate

   Notes: the matrix in compressed color format may come from an automatic differentiation code

   The code will be slightly faster if `MatFDColoringSetBlockSize`(coloring,`PETSC_DEFAULT`,nc); is called immediately after creating the coloring

.seealso: `MatFDColoringCreate()`, `ISColoring`, `ISColoringCreate()`, `ISColoringSetType()`, `IS_COLORING_LOCAL`, `MatFDColoringSetBlockSize()`
@*/
PetscErrorCode MatFDColoringSetValues(Mat J, MatFDColoring coloring, const PetscScalar *y) {
  MatEntry2      *Jentry2;
  PetscInt        row, i, nrows_k, l, ncolors, nz = 0, bcols, nbcols = 0;
  const PetscInt *nrows;
  PetscBool       eq;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(J, MAT_CLASSID, 1);
  PetscValidHeaderSpecific(coloring, MAT_FDCOLORING_CLASSID, 2);
  PetscCall(PetscObjectCompareId((PetscObject)J, coloring->matid, &eq));
  PetscCheck(eq, PetscObjectComm((PetscObject)J), PETSC_ERR_ARG_WRONG, "Matrix used with MatFDColoringSetValues() must be that used with MatFDColoringCreate()");
  Jentry2 = coloring->matentry2;
  nrows   = coloring->nrows;
  ncolors = coloring->ncolors;
  bcols   = coloring->bcols;

  for (i = 0; i < ncolors; i += bcols) {
    nrows_k = nrows[nbcols++];
    for (l = 0; l < nrows_k; l++) {
      row                      = Jentry2[nz].row; /* local row index */
      *(Jentry2[nz++].valaddr) = y[row];
    }
    y += bcols * coloring->m;
  }
  PetscCall(MatAssemblyBegin(J, MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(J, MAT_FINAL_ASSEMBLY));
  PetscFunctionReturn(0);
}