xref: /petsc/src/tao/complementarity/tutorials/minsurf1.c (revision 732aec7a18f2199fb53bb9a2f3aef439a834ce31)

#include <petsctao.h>

static char help[] = "This example demonstrates use of the TAO package to\n\
solve an unconstrained system of equations.  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\
This application solves this problem using complimentarity -- We are actually\n\
solving the system  (grad f)_i >= 0, if x_i == l_i \n\
                    (grad f)_i = 0, if l_i < x_i < u_i \n\
                    (grad f)_i <= 0, if x_i == u_i  \n\
where f is the function to be minimized. \n\
\n\
The command line options are:\n\
  -mx <xg>, where <xg> = number of grid points in the 1st coordinate direction\n\
  -my <yg>, where <yg> = number of grid points in the 2nd coordinate direction\n\
  -start <st>, where <st> =0 for zero vector, and an average of the boundary conditions otherwise \n\n";

/*
   User-defined application context - contains data needed by the
   application-provided call-back routines, FormFunctionGradient(),
   FormHessian().
*/
typedef struct {
  PetscInt   mx, my;
  PetscReal *bottom, *top, *left, *right;
} AppCtx;

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

static PetscErrorCode MSA_BoundaryConditions(AppCtx *);
static PetscErrorCode MSA_InitialPoint(AppCtx *, Vec);
PetscErrorCode        FormConstraints(Tao, Vec, Vec, void *);
PetscErrorCode        FormJacobian(Tao, Vec, Mat, Mat, void *);

int main(int argc, char **argv)
{
  Vec         x;                    /* solution vector */
  Vec         c;                    /* Constraints function vector */
  Vec         xl, xu;               /* Bounds on the variables */
  PetscBool   flg;                  /* A return variable when checking for user options */
  Tao         tao;                  /* TAO solver context */
  Mat         J;                    /* Jacobian matrix */
  PetscInt    N;                    /* Number of elements in vector */
  PetscScalar lb = PETSC_NINFINITY; /* lower bound constant */
  PetscScalar ub = PETSC_INFINITY;  /* upper bound constant */
  AppCtx      user;                 /* user-defined work context */

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

  /* Specify default dimension of the problem */
  user.mx = 4;
  user.my = 4;

  /* Check for any command line arguments that override defaults */
  PetscCall(PetscOptionsGetInt(NULL, NULL, "-mx", &user.mx, &flg));
  PetscCall(PetscOptionsGetInt(NULL, NULL, "-my", &user.my, &flg));

  /* Calculate any derived values from parameters */
  N = user.mx * user.my;

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

  /* Create appropriate vectors and matrices */
  PetscCall(VecCreateSeq(MPI_COMM_SELF, N, &x));
  PetscCall(VecDuplicate(x, &c));
  PetscCall(MatCreateSeqAIJ(MPI_COMM_SELF, N, N, 7, NULL, &J));

  /* The TAO code begins here */

  /* Create TAO solver and set desired solution method */
  PetscCall(TaoCreate(PETSC_COMM_SELF, &tao));
  PetscCall(TaoSetType(tao, TAOSSILS));

  /* Set data structure */
  PetscCall(TaoSetSolution(tao, x));

  /*  Set routines for constraints function and Jacobian evaluation */
  PetscCall(TaoSetConstraintsRoutine(tao, c, FormConstraints, (void *)&user));
  PetscCall(TaoSetJacobianRoutine(tao, J, J, FormJacobian, (void *)&user));

  /* Set the variable bounds */
  PetscCall(MSA_BoundaryConditions(&user));

  /* Set initial solution guess */
  PetscCall(MSA_InitialPoint(&user, x));

  /* Set Bounds on variables */
  PetscCall(VecDuplicate(x, &xl));
  PetscCall(VecDuplicate(x, &xu));
  PetscCall(VecSet(xl, lb));
  PetscCall(VecSet(xu, ub));
  PetscCall(TaoSetVariableBounds(tao, xl, xu));

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

  /* Solve the application */
  PetscCall(TaoSolve(tao));

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

  /* Free PETSc data structures */
  PetscCall(VecDestroy(&x));
  PetscCall(VecDestroy(&xl));
  PetscCall(VecDestroy(&xu));
  PetscCall(VecDestroy(&c));
  PetscCall(MatDestroy(&J));

  /* Free user-created data structures */
  PetscCall(PetscFree(user.bottom));
  PetscCall(PetscFree(user.top));
  PetscCall(PetscFree(user.left));
  PetscCall(PetscFree(user.right));

  PetscCall(PetscFinalize());
  return 0;
}

/* -------------------------------------------------------------------- */

/*  FormConstraints - Evaluates gradient of f.

    Input Parameters:
.   tao  - the TAO_APPLICATION context
.   X    - input vector
.   ptr  - optional user-defined context, as set by TaoSetConstraintsRoutine()

    Output Parameters:
.   G - vector containing the newly evaluated gradient
*/
PetscErrorCode FormConstraints(Tao tao, Vec X, Vec G, void *ptr)
{
  AppCtx      *user = (AppCtx *)ptr;
  PetscInt     i, j, row;
  PetscInt     mx = user->mx, my = user->my;
  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    df1dxc, df2dxc, df3dxc, df4dxc, df5dxc, df6dxc;
  PetscScalar  zero = 0.0;
  PetscScalar *g, *x;

  PetscFunctionBegin;
  /* Initialize vector to zero */
  PetscCall(VecSet(G, zero));

  /* Get pointers to vector data */
  PetscCall(VecGetArray(X, &x));
  PetscCall(VecGetArray(G, &g));

  /* Compute function over the locally owned part of the mesh */
  for (j = 0; j < my; j++) {
    for (i = 0; i < mx; i++) {
      row = j * mx + i;

      xc  = x[row];
      xlt = xrb = xl = xr = xb = xt = xc;

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

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

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

      if (j + 1 == 0 + my) { /* top side */
        xt  = user->top[i + 1];
        xlt = user->top[i];
      } else {
        xt = x[row + mx];
      }

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

      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 /= hx;
      d2 /= hx;
      d3 /= hy;
      d4 /= hy;
      d5 /= hy;
      d6 /= hx;
      d7 /= hy;
      d8 /= hx;

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

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

      g[row] = (df1dxc + df2dxc + df3dxc + df4dxc + df5dxc + df6dxc) / 2.0;
    }
  }

  /* Restore vectors */
  PetscCall(VecRestoreArray(X, &x));
  PetscCall(VecRestoreArray(G, &g));
  PetscCall(PetscLogFlops(67 * mx * my));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/* ------------------------------------------------------------------- */
/*
   FormJacobian - Evaluates Jacobian matrix.

   Input Parameters:
.  tao  - the TAO_APPLICATION context
.  X    - input vector
.  ptr  - optional user-defined context, as set by TaoSetJacobian()

   Output Parameters:
.  tH    - Jacobian matrix

*/
PetscErrorCode FormJacobian(Tao tao, Vec X, Mat H, Mat tHPre, void *ptr)
{
  AppCtx            *user = (AppCtx *)ptr;
  PetscInt           i, j, k, row;
  PetscInt           mx = user->mx, my = user->my;
  PetscInt           col[7];
  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;
  const PetscScalar *x;
  PetscScalar        v[7];
  PetscBool          assembled;

  /* Set various matrix options */
  PetscFunctionBegin;
  PetscCall(MatSetOption(H, MAT_IGNORE_OFF_PROC_ENTRIES, PETSC_TRUE));
  PetscCall(MatAssembled(H, &assembled));
  if (assembled) PetscCall(MatZeroEntries(H));

  /* Get pointers to vector data */
  PetscCall(VecGetArrayRead(X, &x));

  /* Compute Jacobian over the locally owned part of the mesh */
  for (i = 0; i < mx; i++) {
    for (j = 0; j < my; j++) {
      row = j * mx + i;

      xc  = x[row];
      xlt = xrb = xl = xr = xb = xt = xc;

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

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

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

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

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

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

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

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

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

      v[k]   = hc;
      col[k] = row;
      k++;

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

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

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

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

  /* Restore vectors */
  PetscCall(VecRestoreArrayRead(X, &x));

  /* Assemble the matrix */
  PetscCall(MatAssemblyBegin(H, MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(H, MAT_FINAL_ASSEMBLY));
  PetscCall(PetscLogFlops(199 * mx * my));
  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   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;

  PetscFunctionBegin;
  bsize = mx + 2;
  lsize = my + 2;
  rsize = my + 2;
  tsize = mx + 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;
      limit    = bsize;
      boundary = user->bottom;
    } else if (j == 1) {
      yt       = t;
      xt       = l;
      limit    = tsize;
      boundary = user->top;
    } else if (j == 2) {
      yt       = b;
      xt       = l;
      limit    = lsize;
      boundary = user->left;
    } else { /* if  (j==3) */
      yt       = b;
      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 = PetscSqrtScalar(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;
      }
    }
  }
  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    start = -1, i, j;
  PetscScalar zero  = 0.0;
  PetscBool   flg;

  PetscFunctionBegin;
  PetscCall(PetscOptionsGetInt(NULL, NULL, "-start", &start, &flg));

  if (flg && start == 0) { /* The zero vector is reasonable */
    PetscCall(VecSet(X, zero));
  } else { /* Take an average of the boundary conditions */
    PetscInt     row;
    PetscInt     mx = user->mx, my = user->my;
    PetscScalar *x;

    /* Get pointers to vector data */
    PetscCall(VecGetArray(X, &x));

    /* Perform local computations */
    for (j = 0; j < my; j++) {
      for (i = 0; i < mx; i++) {
        row    = (j)*mx + (i);
        x[row] = (((j + 1) * user->bottom[i + 1] + (my - j + 1) * user->top[i + 1]) / (my + 2) + ((i + 1) * user->left[j + 1] + (mx - i + 1) * user->right[j + 1]) / (mx + 2)) / 2.0;
      }
    }

    /* Restore vectors */
    PetscCall(VecRestoreArray(X, &x));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*TEST

   build:
      requires: !complex

   test:
      args: -tao_monitor -tao_view -tao_type ssils -tao_gttol 1.e-5
      requires: !single

   test:
      suffix: 2
      args: -tao_monitor -tao_view -tao_type ssfls -tao_gttol 1.e-5

TEST*/