xref: /petsc/src/tao/unconstrained/tutorials/minsurf2.c (revision 4e8208cbcbc709572b8abe32f33c78b69c819375)

/* Program usage: mpiexec -n <proc> minsurf2 [-help] [all TAO options] */

/*
  Include "petsctao.h" so we can use TAO solvers.
  petscdmda.h for distributed array
*/
#include <petsctao.h>
#include <petscdmda.h>

static char help[] = "This example demonstrates use of the TAO package to \n\
solve an unconstrained minimization problem. This example is based on a \n\
problem from the MINPACK-2 test suite. Given a rectangular 2-D domain and \n\
boundary values along the edges of the domain, the objective is to find the\n\
surface with the minimal area that satisfies the boundary conditions.\n\
The command line options are:\n\
  -da_grid_x <xg>, where <xg> = number of grid points in the 1st coordinate direction\n\
  -da_grid_y <yg>, where <yg> = number of grid points in the 2nd coordinate direction\n\
  -start <st>, where <st> =0 for zero vector, <st> >0 for random start, and <st> <0 \n\
               for an average of the boundary conditions\n\n";

/*
   User-defined application context - contains data needed by the
   application-provided call-back routines, FormFunction(),
   FormFunctionGradient(), and FormHessian().
*/
typedef struct {
  PetscInt   mx, my;                      /* discretization in x, y directions */
  PetscReal *bottom, *top, *left, *right; /* boundary values */
  DM         dm;                          /* distributed array data structure */
  Mat        H;                           /* Hessian */
} AppCtx;

/* -------- User-defined Routines --------- */

static PetscErrorCode MSA_BoundaryConditions(AppCtx *);
static PetscErrorCode MSA_InitialPoint(AppCtx *, Vec);
static PetscErrorCode QuadraticH(AppCtx *, Vec, Mat);
static PetscErrorCode FormFunction(Tao, Vec, PetscReal *, void *);
static PetscErrorCode FormFunctionGradient(Tao, Vec, PetscReal *, Vec, void *);
static PetscErrorCode FormGradient(Tao, Vec, Vec, void *);
static PetscErrorCode FormHessian(Tao, Vec, Mat, Mat, void *);
static PetscErrorCode My_Monitor(Tao, void *);

int main(int argc, char **argv)
{
  Vec           x;                     /* solution, gradient vectors */
  PetscBool     viewmat;               /* flags */
  PetscBool     fddefault, fdcoloring; /* flags */
  Tao           tao;                   /* TAO solver context */
  AppCtx        user;                  /* user-defined work context */
  ISColoring    iscoloring;
  MatFDColoring matfdcoloring;

  /* Initialize TAO */
  PetscFunctionBeginUser;
  PetscCall(PetscInitialize(&argc, &argv, NULL, help));

  /* Create distributed array (DM) to manage parallel grid and vectors  */
  PetscCall(DMDACreate2d(PETSC_COMM_WORLD, DM_BOUNDARY_NONE, DM_BOUNDARY_NONE, DMDA_STENCIL_BOX, 10, 10, PETSC_DECIDE, PETSC_DECIDE, 1, 1, NULL, NULL, &user.dm));
  PetscCall(DMSetFromOptions(user.dm));
  PetscCall(DMSetUp(user.dm));
  PetscCall(DMDASetUniformCoordinates(user.dm, -0.5, 0.5, -0.5, 0.5, PETSC_DECIDE, PETSC_DECIDE));
  PetscCall(DMDAGetInfo(user.dm, PETSC_IGNORE, &user.mx, &user.my, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE));

  PetscCall(PetscPrintf(MPI_COMM_WORLD, "\n---- Minimum Surface Area Problem -----\n"));
  PetscCall(PetscPrintf(MPI_COMM_WORLD, "mx: %" PetscInt_FMT "     my: %" PetscInt_FMT "   \n\n", user.mx, user.my));

  /* Create TAO solver and set desired solution method.*/
  PetscCall(TaoCreate(PETSC_COMM_WORLD, &tao));
  PetscCall(TaoSetType(tao, TAOCG));

  /*
     Extract global vector from DA for the vector of variables --  PETSc routine
     Compute the initial solution                              --  application specific, see below
     Set this vector for use by TAO                            --  TAO routine
  */
  PetscCall(DMCreateGlobalVector(user.dm, &x));
  PetscCall(MSA_BoundaryConditions(&user));
  PetscCall(MSA_InitialPoint(&user, x));
  PetscCall(TaoSetSolution(tao, x));

  /*
     Initialize the Application context for use in function evaluations  --  application specific, see below.
     Set routines for function and gradient evaluation
  */
  PetscCall(TaoSetObjective(tao, FormFunction, (void *)&user));
  PetscCall(TaoSetObjectiveAndGradient(tao, NULL, FormFunctionGradient, (void *)&user));

  /*
     Given the command line arguments, calculate the hessian with either the user-
     provided function FormHessian, or the default finite-difference driven Hessian
     functions
  */
  PetscCall(PetscOptionsHasName(NULL, NULL, "-tao_fddefault", &fddefault));
  PetscCall(PetscOptionsHasName(NULL, NULL, "-tao_fdcoloring", &fdcoloring));

  /*
     Create a matrix data structure to store the Hessian and set
     the Hessian evaluation routine.
     Set the matrix nonzero structure to be used for Hessian evaluations
  */
  PetscCall(DMCreateMatrix(user.dm, &user.H));
  PetscCall(MatSetOption(user.H, MAT_SYMMETRIC, PETSC_TRUE));

  if (fdcoloring) {
    PetscCall(DMCreateColoring(user.dm, IS_COLORING_GLOBAL, &iscoloring));
    PetscCall(MatFDColoringCreate(user.H, iscoloring, &matfdcoloring));
    PetscCall(ISColoringDestroy(&iscoloring));
    PetscCall(MatFDColoringSetFunction(matfdcoloring, (MatFDColoringFn *)FormGradient, (void *)&user));
    PetscCall(MatFDColoringSetFromOptions(matfdcoloring));
    PetscCall(TaoSetHessian(tao, user.H, user.H, TaoDefaultComputeHessianColor, (void *)matfdcoloring));
  } else if (fddefault) {
    PetscCall(TaoSetHessian(tao, user.H, user.H, TaoDefaultComputeHessian, (void *)NULL));
  } else {
    PetscCall(TaoSetHessian(tao, user.H, user.H, FormHessian, (void *)&user));
  }

  /*
     If my_monitor option is in command line, then use the user-provided
     monitoring function
  */
  PetscCall(PetscOptionsHasName(NULL, NULL, "-my_monitor", &viewmat));
  if (viewmat) PetscCall(TaoMonitorSet(tao, My_Monitor, NULL, NULL));

  /* Check for any tao command line options */
  PetscCall(TaoSetFromOptions(tao));

  /* SOLVE THE APPLICATION */
  PetscCall(TaoSolve(tao));

  PetscCall(TaoView(tao, PETSC_VIEWER_STDOUT_WORLD));

  /* Free TAO data structures */
  PetscCall(TaoDestroy(&tao));

  /* Free PETSc data structures */
  PetscCall(VecDestroy(&x));
  PetscCall(MatDestroy(&user.H));
  if (fdcoloring) PetscCall(MatFDColoringDestroy(&matfdcoloring));
  PetscCall(PetscFree(user.bottom));
  PetscCall(PetscFree(user.top));
  PetscCall(PetscFree(user.left));
  PetscCall(PetscFree(user.right));
  PetscCall(DMDestroy(&user.dm));
  PetscCall(PetscFinalize());
  return 0;
}

/* -------------------------------------------------------------------- */
/*
    FormFunction - Evaluates the objective function.

    Input Parameters:
.   tao     - the Tao context
.   X       - input vector
.   userCtx - optional user-defined context, as set by TaoSetObjective()

    Output Parameters:
.   fcn     - the newly evaluated function
*/
PetscErrorCode FormFunction(Tao tao, Vec X, PetscReal *fcn, void *userCtx)
{
  AppCtx     *user = (AppCtx *)userCtx;
  PetscInt    i, j;
  PetscInt    mx = user->mx, my = user->my;
  PetscInt    xs, xm, gxs, gxm, ys, ym, gys, gym;
  PetscReal   ft = 0.0;
  PetscReal   hx = 1.0 / (mx + 1), hy = 1.0 / (my + 1), area = 0.5 * hx * hy;
  PetscReal   rhx = mx + 1, rhy = my + 1;
  PetscReal   f2, f4, d1, d2, d3, d4, xc, xl, xr, xt, xb;
  PetscReal **x;
  Vec         localX;

  PetscFunctionBegin;
  /* Get local mesh boundaries */
  PetscCall(DMGetLocalVector(user->dm, &localX));
  PetscCall(DMDAGetCorners(user->dm, &xs, &ys, NULL, &xm, &ym, NULL));
  PetscCall(DMDAGetGhostCorners(user->dm, &gxs, &gys, NULL, &gxm, &gym, NULL));

  /* Scatter ghost points to local vector */
  PetscCall(DMGlobalToLocalBegin(user->dm, X, INSERT_VALUES, localX));
  PetscCall(DMGlobalToLocalEnd(user->dm, X, INSERT_VALUES, localX));

  /* Get pointers to vector data */
  PetscCall(DMDAVecGetArray(user->dm, localX, (void **)&x));

  /* Compute function and gradient over the locally owned part of the mesh */
  for (j = ys; j < ys + ym; j++) {
    for (i = xs; i < xs + xm; i++) {
      xc = x[j][i];

      if (i == 0) { /* left side */
        xl = user->left[j - ys + 1];
      } else {
        xl = x[j][i - 1];
      }

      if (j == 0) { /* bottom side */
        xb = user->bottom[i - xs + 1];
      } else {
        xb = x[j - 1][i];
      }

      if (i + 1 == gxs + gxm) { /* right side */
        xr = user->right[j - ys + 1];
      } else {
        xr = x[j][i + 1];
      }

      if (j + 1 == gys + gym) { /* top side */
        xt = user->top[i - xs + 1];
      } else {
        xt = x[j + 1][i];
      }

      d1 = (xc - xl);
      d2 = (xc - xr);
      d3 = (xc - xt);
      d4 = (xc - xb);

      d1 *= rhx;
      d2 *= rhx;
      d3 *= rhy;
      d4 *= rhy;

      f2 = PetscSqrtReal(1.0 + d1 * d1 + d4 * d4);
      f4 = PetscSqrtReal(1.0 + d3 * d3 + d2 * d2);

      ft = ft + (f2 + f4);
    }
  }

  /* Compute triangular areas along the border of the domain. */
  if (xs == 0) { /* left side */
    for (j = ys; j < ys + ym; j++) {
      d3 = (user->left[j - ys + 1] - user->left[j - ys + 2]) * rhy;
      d2 = (user->left[j - ys + 1] - x[j][0]) * rhx;
      ft = ft + PetscSqrtReal(1.0 + d3 * d3 + d2 * d2);
    }
  }
  if (ys == 0) { /* bottom side */
    for (i = xs; i < xs + xm; i++) {
      d2 = (user->bottom[i + 1 - xs] - user->bottom[i - xs + 2]) * rhx;
      d3 = (user->bottom[i - xs + 1] - x[0][i]) * rhy;
      ft = ft + PetscSqrtReal(1.0 + d3 * d3 + d2 * d2);
    }
  }
  if (xs + xm == mx) { /* right side */
    for (j = ys; j < ys + ym; j++) {
      d1 = (x[j][mx - 1] - user->right[j - ys + 1]) * rhx;
      d4 = (user->right[j - ys] - user->right[j - ys + 1]) * rhy;
      ft = ft + PetscSqrtReal(1.0 + d1 * d1 + d4 * d4);
    }
  }
  if (ys + ym == my) { /* top side */
    for (i = xs; i < xs + xm; i++) {
      d1 = (x[my - 1][i] - user->top[i - xs + 1]) * rhy;
      d4 = (user->top[i - xs + 1] - user->top[i - xs]) * rhx;
      ft = ft + PetscSqrtReal(1.0 + d1 * d1 + d4 * d4);
    }
  }
  if (ys == 0 && xs == 0) {
    d1 = (user->left[0] - user->left[1]) / hy;
    d2 = (user->bottom[0] - user->bottom[1]) * rhx;
    ft += PetscSqrtReal(1.0 + d1 * d1 + d2 * d2);
  }
  if (ys + ym == my && xs + xm == mx) {
    d1 = (user->right[ym + 1] - user->right[ym]) * rhy;
    d2 = (user->top[xm + 1] - user->top[xm]) * rhx;
    ft += PetscSqrtReal(1.0 + d1 * d1 + d2 * d2);
  }

  ft = ft * area;
  PetscCallMPI(MPIU_Allreduce(&ft, fcn, 1, MPIU_REAL, MPIU_SUM, MPI_COMM_WORLD));

  /* Restore vectors */
  PetscCall(DMDAVecRestoreArray(user->dm, localX, (void **)&x));
  PetscCall(DMRestoreLocalVector(user->dm, &localX));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/* -------------------------------------------------------------------- */
/*
    FormFunctionGradient - Evaluates the function and corresponding gradient.

    Input Parameters:
.   tao     - the Tao context
.   X      - input vector
.   userCtx - optional user-defined context, as set by TaoSetObjectiveAndGradient()

    Output Parameters:
.   fcn     - the newly evaluated function
.   G       - vector containing the newly evaluated gradient
*/
PetscErrorCode FormFunctionGradient(Tao tao, Vec X, PetscReal *fcn, Vec G, void *userCtx)
{
  AppCtx     *user = (AppCtx *)userCtx;
  PetscInt    i, j;
  PetscInt    mx = user->mx, my = user->my;
  PetscInt    xs, xm, gxs, gxm, ys, ym, gys, gym;
  PetscReal   ft = 0.0;
  PetscReal   hx = 1.0 / (mx + 1), hy = 1.0 / (my + 1), hydhx = hy / hx, hxdhy = hx / hy, area = 0.5 * hx * hy;
  PetscReal   rhx = mx + 1, rhy = my + 1;
  PetscReal   f1, f2, f3, f4, f5, f6, d1, d2, d3, d4, d5, d6, d7, d8, xc, xl, xr, xt, xb, xlt, xrb;
  PetscReal   df1dxc, df2dxc, df3dxc, df4dxc, df5dxc, df6dxc;
  PetscReal **g, **x;
  Vec         localX;

  PetscFunctionBegin;
  /* Get local mesh boundaries */
  PetscCall(DMGetLocalVector(user->dm, &localX));
  PetscCall(DMDAGetCorners(user->dm, &xs, &ys, NULL, &xm, &ym, NULL));
  PetscCall(DMDAGetGhostCorners(user->dm, &gxs, &gys, NULL, &gxm, &gym, NULL));

  /* Scatter ghost points to local vector */
  PetscCall(DMGlobalToLocalBegin(user->dm, X, INSERT_VALUES, localX));
  PetscCall(DMGlobalToLocalEnd(user->dm, X, INSERT_VALUES, localX));

  /* Get pointers to vector data */
  PetscCall(DMDAVecGetArray(user->dm, localX, (void **)&x));
  PetscCall(DMDAVecGetArray(user->dm, G, (void **)&g));

  /* Compute function and gradient over the locally owned part of the mesh */
  for (j = ys; j < ys + ym; j++) {
    for (i = xs; i < xs + xm; i++) {
      xc  = x[j][i];
      xlt = xrb = xl = xr = xb = xt = xc;

      if (i == 0) { /* left side */
        xl  = user->left[j - ys + 1];
        xlt = user->left[j - ys + 2];
      } else {
        xl = x[j][i - 1];
      }

      if (j == 0) { /* bottom side */
        xb  = user->bottom[i - xs + 1];
        xrb = user->bottom[i - xs + 2];
      } else {
        xb = x[j - 1][i];
      }

      if (i + 1 == gxs + gxm) { /* right side */
        xr  = user->right[j - ys + 1];
        xrb = user->right[j - ys];
      } else {
        xr = x[j][i + 1];
      }

      if (j + 1 == gys + gym) { /* top side */
        xt  = user->top[i - xs + 1];
        xlt = user->top[i - xs];
      } else {
        xt = x[j + 1][i];
      }

      if (i > gxs && j + 1 < gys + gym) xlt = x[j + 1][i - 1];
      if (j > gys && i + 1 < gxs + gxm) xrb = x[j - 1][i + 1];

      d1 = (xc - xl);
      d2 = (xc - xr);
      d3 = (xc - xt);
      d4 = (xc - xb);
      d5 = (xr - xrb);
      d6 = (xrb - xb);
      d7 = (xlt - xl);
      d8 = (xt - xlt);

      df1dxc = d1 * hydhx;
      df2dxc = (d1 * hydhx + d4 * hxdhy);
      df3dxc = d3 * hxdhy;
      df4dxc = (d2 * hydhx + d3 * hxdhy);
      df5dxc = d2 * hydhx;
      df6dxc = d4 * hxdhy;

      d1 *= rhx;
      d2 *= rhx;
      d3 *= rhy;
      d4 *= rhy;
      d5 *= rhy;
      d6 *= rhx;
      d7 *= rhy;
      d8 *= rhx;

      f1 = PetscSqrtReal(1.0 + d1 * d1 + d7 * d7);
      f2 = PetscSqrtReal(1.0 + d1 * d1 + d4 * d4);
      f3 = PetscSqrtReal(1.0 + d3 * d3 + d8 * d8);
      f4 = PetscSqrtReal(1.0 + d3 * d3 + d2 * d2);
      f5 = PetscSqrtReal(1.0 + d2 * d2 + d5 * d5);
      f6 = PetscSqrtReal(1.0 + d4 * d4 + d6 * d6);

      ft = ft + (f2 + f4);

      df1dxc /= f1;
      df2dxc /= f2;
      df3dxc /= f3;
      df4dxc /= f4;
      df5dxc /= f5;
      df6dxc /= f6;

      g[j][i] = (df1dxc + df2dxc + df3dxc + df4dxc + df5dxc + df6dxc) * 0.5;
    }
  }

  /* Compute triangular areas along the border of the domain. */
  if (xs == 0) { /* left side */
    for (j = ys; j < ys + ym; j++) {
      d3 = (user->left[j - ys + 1] - user->left[j - ys + 2]) * rhy;
      d2 = (user->left[j - ys + 1] - x[j][0]) * rhx;
      ft = ft + PetscSqrtReal(1.0 + d3 * d3 + d2 * d2);
    }
  }
  if (ys == 0) { /* bottom side */
    for (i = xs; i < xs + xm; i++) {
      d2 = (user->bottom[i + 1 - xs] - user->bottom[i - xs + 2]) * rhx;
      d3 = (user->bottom[i - xs + 1] - x[0][i]) * rhy;
      ft = ft + PetscSqrtReal(1.0 + d3 * d3 + d2 * d2);
    }
  }

  if (xs + xm == mx) { /* right side */
    for (j = ys; j < ys + ym; j++) {
      d1 = (x[j][mx - 1] - user->right[j - ys + 1]) * rhx;
      d4 = (user->right[j - ys] - user->right[j - ys + 1]) * rhy;
      ft = ft + PetscSqrtReal(1.0 + d1 * d1 + d4 * d4);
    }
  }
  if (ys + ym == my) { /* top side */
    for (i = xs; i < xs + xm; i++) {
      d1 = (x[my - 1][i] - user->top[i - xs + 1]) * rhy;
      d4 = (user->top[i - xs + 1] - user->top[i - xs]) * rhx;
      ft = ft + PetscSqrtReal(1.0 + d1 * d1 + d4 * d4);
    }
  }

  if (ys == 0 && xs == 0) {
    d1 = (user->left[0] - user->left[1]) / hy;
    d2 = (user->bottom[0] - user->bottom[1]) * rhx;
    ft += PetscSqrtReal(1.0 + d1 * d1 + d2 * d2);
  }
  if (ys + ym == my && xs + xm == mx) {
    d1 = (user->right[ym + 1] - user->right[ym]) * rhy;
    d2 = (user->top[xm + 1] - user->top[xm]) * rhx;
    ft += PetscSqrtReal(1.0 + d1 * d1 + d2 * d2);
  }

  ft = ft * area;
  PetscCallMPI(MPIU_Allreduce(&ft, fcn, 1, MPIU_REAL, MPIU_SUM, MPI_COMM_WORLD));

  /* Restore vectors */
  PetscCall(DMDAVecRestoreArray(user->dm, localX, (void **)&x));
  PetscCall(DMDAVecRestoreArray(user->dm, G, (void **)&g));
  PetscCall(DMRestoreLocalVector(user->dm, &localX));
  PetscCall(PetscLogFlops(67.0 * xm * ym));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode FormGradient(Tao tao, Vec X, Vec G, void *userCtx)
{
  PetscReal fcn;

  PetscFunctionBegin;
  PetscCall(FormFunctionGradient(tao, X, &fcn, G, userCtx));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/* ------------------------------------------------------------------- */
/*
   FormHessian - Evaluates Hessian matrix.

   Input Parameters:
.  tao  - the Tao context
.  x    - input vector
.  ptr  - optional user-defined context, as set by TaoSetHessian()

   Output Parameters:
.  H    - Hessian matrix
.  Hpre - optionally different matrix used to compute the preconditioner

*/
PetscErrorCode FormHessian(Tao tao, Vec X, Mat H, Mat Hpre, void *ptr)
{
  AppCtx *user = (AppCtx *)ptr;

  PetscFunctionBegin;
  /* Evaluate the Hessian entries*/
  PetscCall(QuadraticH(user, X, H));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/* ------------------------------------------------------------------- */
/*
   QuadraticH - Evaluates Hessian matrix.

   Input Parameters:
.  user - user-defined context, as set by TaoSetHessian()
.  X    - input vector

   Output Parameter:
.  H    - Hessian matrix
*/
PetscErrorCode QuadraticH(AppCtx *user, Vec X, Mat Hessian)
{
  PetscInt    i, j, k;
  PetscInt    mx = user->mx, my = user->my;
  PetscInt    xs, xm, gxs, gxm, ys, ym, gys, gym;
  PetscReal   hx = 1.0 / (mx + 1), hy = 1.0 / (my + 1), hydhx = hy / hx, hxdhy = hx / hy;
  PetscReal   f1, f2, f3, f4, f5, f6, d1, d2, d3, d4, d5, d6, d7, d8, xc, xl, xr, xt, xb, xlt, xrb;
  PetscReal   hl, hr, ht, hb, hc, htl, hbr;
  PetscReal **x, v[7];
  MatStencil  col[7], row;
  Vec         localX;
  PetscBool   assembled;

  PetscFunctionBegin;
  /* Get local mesh boundaries */
  PetscCall(DMGetLocalVector(user->dm, &localX));

  PetscCall(DMDAGetCorners(user->dm, &xs, &ys, NULL, &xm, &ym, NULL));
  PetscCall(DMDAGetGhostCorners(user->dm, &gxs, &gys, NULL, &gxm, &gym, NULL));

  /* Scatter ghost points to local vector */
  PetscCall(DMGlobalToLocalBegin(user->dm, X, INSERT_VALUES, localX));
  PetscCall(DMGlobalToLocalEnd(user->dm, X, INSERT_VALUES, localX));

  /* Get pointers to vector data */
  PetscCall(DMDAVecGetArray(user->dm, localX, (void **)&x));

  /* Initialize matrix entries to zero */
  PetscCall(MatAssembled(Hessian, &assembled));
  if (assembled) PetscCall(MatZeroEntries(Hessian));

  /* Set various matrix options */
  PetscCall(MatSetOption(Hessian, MAT_IGNORE_OFF_PROC_ENTRIES, PETSC_TRUE));

  /* Compute Hessian over the locally owned part of the mesh */
  for (j = ys; j < ys + ym; j++) {
    for (i = xs; i < xs + xm; i++) {
      xc  = x[j][i];
      xlt = xrb = xl = xr = xb = xt = xc;

      /* Left side */
      if (i == 0) {
        xl  = user->left[j - ys + 1];
        xlt = user->left[j - ys + 2];
      } else {
        xl = x[j][i - 1];
      }

      if (j == 0) {
        xb  = user->bottom[i - xs + 1];
        xrb = user->bottom[i - xs + 2];
      } else {
        xb = x[j - 1][i];
      }

      if (i + 1 == mx) {
        xr  = user->right[j - ys + 1];
        xrb = user->right[j - ys];
      } else {
        xr = x[j][i + 1];
      }

      if (j + 1 == my) {
        xt  = user->top[i - xs + 1];
        xlt = user->top[i - xs];
      } else {
        xt = x[j + 1][i];
      }

      if (i > 0 && j + 1 < my) xlt = x[j + 1][i - 1];
      if (j > 0 && i + 1 < mx) xrb = x[j - 1][i + 1];

      d1 = (xc - xl) / hx;
      d2 = (xc - xr) / hx;
      d3 = (xc - xt) / hy;
      d4 = (xc - xb) / hy;
      d5 = (xrb - xr) / hy;
      d6 = (xrb - xb) / hx;
      d7 = (xlt - xl) / hy;
      d8 = (xlt - xt) / hx;

      f1 = PetscSqrtReal(1.0 + d1 * d1 + d7 * d7);
      f2 = PetscSqrtReal(1.0 + d1 * d1 + d4 * d4);
      f3 = PetscSqrtReal(1.0 + d3 * d3 + d8 * d8);
      f4 = PetscSqrtReal(1.0 + d3 * d3 + d2 * d2);
      f5 = PetscSqrtReal(1.0 + d2 * d2 + d5 * d5);
      f6 = PetscSqrtReal(1.0 + d4 * d4 + d6 * d6);

      hl = (-hydhx * (1.0 + d7 * d7) + d1 * d7) / (f1 * f1 * f1) + (-hydhx * (1.0 + d4 * d4) + d1 * d4) / (f2 * f2 * f2);
      hr = (-hydhx * (1.0 + d5 * d5) + d2 * d5) / (f5 * f5 * f5) + (-hydhx * (1.0 + d3 * d3) + d2 * d3) / (f4 * f4 * f4);
      ht = (-hxdhy * (1.0 + d8 * d8) + d3 * d8) / (f3 * f3 * f3) + (-hxdhy * (1.0 + d2 * d2) + d2 * d3) / (f4 * f4 * f4);
      hb = (-hxdhy * (1.0 + d6 * d6) + d4 * d6) / (f6 * f6 * f6) + (-hxdhy * (1.0 + d1 * d1) + d1 * d4) / (f2 * f2 * f2);

      hbr = -d2 * d5 / (f5 * f5 * f5) - d4 * d6 / (f6 * f6 * f6);
      htl = -d1 * d7 / (f1 * f1 * f1) - d3 * d8 / (f3 * f3 * f3);

      hc = hydhx * (1.0 + d7 * d7) / (f1 * f1 * f1) + hxdhy * (1.0 + d8 * d8) / (f3 * f3 * f3) + hydhx * (1.0 + d5 * d5) / (f5 * f5 * f5) + hxdhy * (1.0 + d6 * d6) / (f6 * f6 * f6) + (hxdhy * (1.0 + d1 * d1) + hydhx * (1.0 + d4 * d4) - 2 * d1 * d4) / (f2 * f2 * f2) + (hxdhy * (1.0 + d2 * d2) + hydhx * (1.0 + d3 * d3) - 2 * d2 * d3) / (f4 * f4 * f4);

      hl /= 2.0;
      hr /= 2.0;
      ht /= 2.0;
      hb /= 2.0;
      hbr /= 2.0;
      htl /= 2.0;
      hc /= 2.0;

      row.j = j;
      row.i = i;
      k     = 0;
      if (j > 0) {
        v[k]     = hb;
        col[k].j = j - 1;
        col[k].i = i;
        k++;
      }

      if (j > 0 && i < mx - 1) {
        v[k]     = hbr;
        col[k].j = j - 1;
        col[k].i = i + 1;
        k++;
      }

      if (i > 0) {
        v[k]     = hl;
        col[k].j = j;
        col[k].i = i - 1;
        k++;
      }

      v[k]     = hc;
      col[k].j = j;
      col[k].i = i;
      k++;

      if (i < mx - 1) {
        v[k]     = hr;
        col[k].j = j;
        col[k].i = i + 1;
        k++;
      }

      if (i > 0 && j < my - 1) {
        v[k]     = htl;
        col[k].j = j + 1;
        col[k].i = i - 1;
        k++;
      }

      if (j < my - 1) {
        v[k]     = ht;
        col[k].j = j + 1;
        col[k].i = i;
        k++;
      }

      /*
         Set matrix values using local numbering, which was defined
         earlier, in the main routine.
      */
      PetscCall(MatSetValuesStencil(Hessian, 1, &row, k, col, v, INSERT_VALUES));
    }
  }

  PetscCall(DMDAVecRestoreArray(user->dm, localX, (void **)&x));
  PetscCall(DMRestoreLocalVector(user->dm, &localX));

  PetscCall(MatAssemblyBegin(Hessian, MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(Hessian, MAT_FINAL_ASSEMBLY));

  PetscCall(PetscLogFlops(199.0 * xm * ym));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/* ------------------------------------------------------------------- */
/*
   MSA_BoundaryConditions -  Calculates the boundary conditions for
   the region.

   Input Parameter:
.  user - user-defined application context

   Output Parameter:
.  user - user-defined application context
*/
static PetscErrorCode MSA_BoundaryConditions(AppCtx *user)
{
  PetscInt   i, j, k, limit = 0, maxits = 5;
  PetscInt   xs, ys, xm, ym, gxs, gys, gxm, gym;
  PetscInt   mx = user->mx, my = user->my;
  PetscInt   bsize = 0, lsize = 0, tsize = 0, rsize = 0;
  PetscReal  one = 1.0, two = 2.0, three = 3.0, tol = 1e-10;
  PetscReal  fnorm, det, hx, hy, xt = 0, yt = 0;
  PetscReal  u1, u2, nf1, nf2, njac11, njac12, njac21, njac22;
  PetscReal  b = -0.5, t = 0.5, l = -0.5, r = 0.5;
  PetscReal *boundary;
  PetscBool  flg;

  PetscFunctionBegin;
  /* Get local mesh boundaries */
  PetscCall(DMDAGetCorners(user->dm, &xs, &ys, NULL, &xm, &ym, NULL));
  PetscCall(DMDAGetGhostCorners(user->dm, &gxs, &gys, NULL, &gxm, &gym, NULL));

  bsize = xm + 2;
  lsize = ym + 2;
  rsize = ym + 2;
  tsize = xm + 2;

  PetscCall(PetscMalloc1(bsize, &user->bottom));
  PetscCall(PetscMalloc1(tsize, &user->top));
  PetscCall(PetscMalloc1(lsize, &user->left));
  PetscCall(PetscMalloc1(rsize, &user->right));

  hx = (r - l) / (mx + 1);
  hy = (t - b) / (my + 1);

  for (j = 0; j < 4; j++) {
    if (j == 0) {
      yt       = b;
      xt       = l + hx * xs;
      limit    = bsize;
      boundary = user->bottom;
    } else if (j == 1) {
      yt       = t;
      xt       = l + hx * xs;
      limit    = tsize;
      boundary = user->top;
    } else if (j == 2) {
      yt       = b + hy * ys;
      xt       = l;
      limit    = lsize;
      boundary = user->left;
    } else { /* if (j==3) */
      yt       = b + hy * ys;
      xt       = r;
      limit    = rsize;
      boundary = user->right;
    }

    for (i = 0; i < limit; i++) {
      u1 = xt;
      u2 = -yt;
      for (k = 0; k < maxits; k++) {
        nf1   = u1 + u1 * u2 * u2 - u1 * u1 * u1 / three - xt;
        nf2   = -u2 - u1 * u1 * u2 + u2 * u2 * u2 / three - yt;
        fnorm = PetscSqrtReal(nf1 * nf1 + nf2 * nf2);
        if (fnorm <= tol) break;
        njac11 = one + u2 * u2 - u1 * u1;
        njac12 = two * u1 * u2;
        njac21 = -two * u1 * u2;
        njac22 = -one - u1 * u1 + u2 * u2;
        det    = njac11 * njac22 - njac21 * njac12;
        u1     = u1 - (njac22 * nf1 - njac12 * nf2) / det;
        u2     = u2 - (njac11 * nf2 - njac21 * nf1) / det;
      }

      boundary[i] = u1 * u1 - u2 * u2;
      if (j == 0 || j == 1) {
        xt = xt + hx;
      } else { /*  if (j==2 || j==3) */
        yt = yt + hy;
      }
    }
  }

  /* Scale the boundary if desired */
  if (1 == 1) {
    PetscReal scl = 1.0;

    PetscCall(PetscOptionsGetReal(NULL, NULL, "-bottom", &scl, &flg));
    if (flg) {
      for (i = 0; i < bsize; i++) user->bottom[i] *= scl;
    }

    PetscCall(PetscOptionsGetReal(NULL, NULL, "-top", &scl, &flg));
    if (flg) {
      for (i = 0; i < tsize; i++) user->top[i] *= scl;
    }

    PetscCall(PetscOptionsGetReal(NULL, NULL, "-right", &scl, &flg));
    if (flg) {
      for (i = 0; i < rsize; i++) user->right[i] *= scl;
    }

    PetscCall(PetscOptionsGetReal(NULL, NULL, "-left", &scl, &flg));
    if (flg) {
      for (i = 0; i < lsize; i++) user->left[i] *= scl;
    }
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/* ------------------------------------------------------------------- */
/*
   MSA_InitialPoint - Calculates the initial guess in one of three ways.

   Input Parameters:
.  user - user-defined application context
.  X - vector for initial guess

   Output Parameters:
.  X - newly computed initial guess
*/
static PetscErrorCode MSA_InitialPoint(AppCtx *user, Vec X)
{
  PetscInt  start2 = -1, i, j;
  PetscReal start1 = 0;
  PetscBool flg1, flg2;

  PetscFunctionBegin;
  PetscCall(PetscOptionsGetReal(NULL, NULL, "-start", &start1, &flg1));
  PetscCall(PetscOptionsGetInt(NULL, NULL, "-random", &start2, &flg2));

  if (flg1) { /* The zero vector is reasonable */
    PetscCall(VecSet(X, start1));
  } else if (flg2 && start2 > 0) { /* Try a random start between -0.5 and 0.5 */
    PetscRandom rctx;
    PetscReal   np5 = -0.5;

    PetscCall(PetscRandomCreate(PETSC_COMM_WORLD, &rctx));
    PetscCall(VecSetRandom(X, rctx));
    PetscCall(PetscRandomDestroy(&rctx));
    PetscCall(VecShift(X, np5));
  } else { /* Take an average of the boundary conditions */
    PetscInt    xs, xm, ys, ym;
    PetscInt    mx = user->mx, my = user->my;
    PetscReal **x;

    /* Get local mesh boundaries */
    PetscCall(DMDAGetCorners(user->dm, &xs, &ys, NULL, &xm, &ym, NULL));

    /* Get pointers to vector data */
    PetscCall(DMDAVecGetArray(user->dm, X, (void **)&x));

    /* Perform local computations */
    for (j = ys; j < ys + ym; j++) {
      for (i = xs; i < xs + xm; i++) x[j][i] = (((j + 1) * user->bottom[i - xs + 1] + (my - j + 1) * user->top[i - xs + 1]) / (my + 2) + ((i + 1) * user->left[j - ys + 1] + (mx - i + 1) * user->right[j - ys + 1]) / (mx + 2)) / 2.0;
    }
    PetscCall(DMDAVecRestoreArray(user->dm, X, (void **)&x));
    PetscCall(PetscLogFlops(9.0 * xm * ym));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*-----------------------------------------------------------------------*/
PetscErrorCode My_Monitor(Tao tao, PetscCtx ctx)
{
  Vec X;

  PetscFunctionBegin;
  PetscCall(TaoGetSolution(tao, &X));
  PetscCall(VecView(X, PETSC_VIEWER_STDOUT_WORLD));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*TEST

   build:
      requires: !complex

   test:
      args: -tao_monitor_short -tao_type lmvm -da_grid_x 10 -da_grid_y 8 -tao_gatol 1.e-3
      requires: !single

   test:
      suffix: 2
      nsize: 2
      args: -tao_monitor_short -tao_type nls -tao_nls_ksp_max_it 15 -tao_gatol 1.e-4
      filter: grep -v "nls ksp"
      requires: !single

   test:
      suffix: 2_snes
      nsize: 2
      args: -tao_monitor_short -tao_type snes -ksp_converged_maxits -ksp_max_it 15 -snes_atol 1.e-4
      filter: grep -v "nls ksp"
      requires: !single

   test:
      suffix: 3
      nsize: 3
      args: -tao_monitor_short -tao_type cg -tao_cg_type fr -da_grid_x 10 -da_grid_y 10 -tao_gatol 1.e-3
      requires: !single

   test:
      suffix: 3_snes
      nsize: 3
      args: -tao_monitor_short -tao_type snes -snes_type ncg -snes_ncg_type fr -da_grid_x 10 -da_grid_y 10 -snes_atol 1.e-4
      requires: !single

   test:
      suffix: 4_snes_ngmres
      args: -tao_type snes -snes_type ngmres -npc_snes_type ncg -da_grid_x 10 -da_grid_y 10 -snes_atol 1.e-4 -npc_snes_ncg_type fr -snes_converged_reason -snes_monitor ::ascii_info_detail -snes_ngmres_monitor -snes_ngmres_select_type {{linesearch difference}separate output}
      requires: !single

   test:
      suffix: 5
      nsize: 2
      args: -tao_monitor_short -tao_type bmrm -da_grid_x 10 -da_grid_y 8 -tao_gatol 1.e-3
      requires: !single

TEST*/