Date: 2022-08-02T03:15:12+00:00

Actual source code: minsurf2.c

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

      3: /*
      4:   Include "petsctao.h" so we can use TAO solvers.
      5:   petscdmda.h for distributed array
      6: */
      7: #include <petsctao.h>
      8: #include <petscdmda.h>

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

     22: /*
     23:    User-defined application context - contains data needed by the
     24:    application-provided call-back routines, FormFunctionGradient()
     25:    and FormHessian().
     26: */
     27: typedef struct {
     28:   PetscInt    mx, my;                 /* discretization in x, y directions */
     29:   PetscReal   *bottom, *top, *left, *right;             /* boundary values */
     30:   DM          dm;                      /* distributed array data structure */
     31:   Mat         H;                       /* Hessian */
     32: } AppCtx;

     34: /* -------- User-defined Routines --------- */

     36: static PetscErrorCode MSA_BoundaryConditions(AppCtx*);
     37: static PetscErrorCode MSA_InitialPoint(AppCtx*,Vec);
     38: PetscErrorCode QuadraticH(AppCtx*,Vec,Mat);
     39: PetscErrorCode FormFunctionGradient(Tao,Vec,PetscReal *,Vec,void*);
     40: PetscErrorCode FormGradient(Tao,Vec,Vec,void*);
     41: PetscErrorCode FormHessian(Tao,Vec,Mat,Mat,void*);
     42: PetscErrorCode My_Monitor(Tao, void *);

     44: int main(int argc, char **argv)
     45: {
     46:   PetscInt           Nx, Ny;              /* number of processors in x- and y- directions */
     47:   Vec                x;                   /* solution, gradient vectors */
     48:   PetscBool          flg, viewmat;        /* flags */
     49:   PetscBool          fddefault, fdcoloring;   /* flags */
     50:   Tao                tao;                 /* TAO solver context */
     51:   AppCtx             user;                /* user-defined work context */
     52:   ISColoring         iscoloring;
     53:   MatFDColoring      matfdcoloring;

     55:   /* Initialize TAO */
     56:   PetscInitialize(&argc, &argv,(char *)0,help);

     58:   /* Specify dimension of the problem */
     59:   user.mx = 10; user.my = 10;

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

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

     68:   /* Let PETSc determine the vector distribution */
     69:   Nx = PETSC_DECIDE; Ny = PETSC_DECIDE;

     71:   /* Create distributed array (DM) to manage parallel grid and vectors  */
     72:   DMDACreate2d(PETSC_COMM_WORLD,DM_BOUNDARY_NONE,DM_BOUNDARY_NONE,DMDA_STENCIL_BOX,user.mx, user.my,Nx,Ny,1,1,NULL,NULL,&user.dm);
     73:   DMSetFromOptions(user.dm);
     74:   DMSetUp(user.dm);

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

     80:   /*
     81:      Extract global vector from DA for the vector of variables --  PETSC routine
     82:      Compute the initial solution                              --  application specific, see below
     83:      Set this vector for use by TAO                            --  TAO routine
     84:   */
     85:   DMCreateGlobalVector(user.dm,&x);
     86:   MSA_BoundaryConditions(&user);
     87:   MSA_InitialPoint(&user,x);
     88:   TaoSetSolution(tao,x);

     90:   /*
     91:      Initialize the Application context for use in function evaluations  --  application specific, see below.
     92:      Set routines for function and gradient evaluation
     93:   */
     94:   TaoSetObjectiveAndGradient(tao,NULL,FormFunctionGradient,(void *)&user);

     96:   /*
     97:      Given the command line arguments, calculate the hessian with either the user-
     98:      provided function FormHessian, or the default finite-difference driven Hessian
     99:      functions
    100:   */
    101:   PetscOptionsHasName(NULL,NULL,"-tao_fddefault",&fddefault);
    102:   PetscOptionsHasName(NULL,NULL,"-tao_fdcoloring",&fdcoloring);

    104:   /*
    105:      Create a matrix data structure to store the Hessian and set
    106:      the Hessian evalution routine.
    107:      Set the matrix structure to be used for Hessian evalutions
    108:   */
    109:   DMCreateMatrix(user.dm,&user.H);
    110:   MatSetOption(user.H,MAT_SYMMETRIC,PETSC_TRUE);

    112:   if (fdcoloring) {
    113:     DMCreateColoring(user.dm,IS_COLORING_GLOBAL,&iscoloring);
    114:     MatFDColoringCreate(user.H,iscoloring,&matfdcoloring);
    115:     ISColoringDestroy(&iscoloring);
    116:     MatFDColoringSetFunction(matfdcoloring,(PetscErrorCode (*)(void))FormGradient,(void*)&user);
    117:     MatFDColoringSetFromOptions(matfdcoloring);
    118:     TaoSetHessian(tao,user.H,user.H,TaoDefaultComputeHessianColor,(void *)matfdcoloring);
    119:   } else if (fddefault) {
    120:     TaoSetHessian(tao,user.H,user.H,TaoDefaultComputeHessian,(void *)NULL);
    121:   } else {
    122:     TaoSetHessian(tao,user.H,user.H,FormHessian,(void *)&user);
    123:   }

    125:   /*
    126:      If my_monitor option is in command line, then use the user-provided
    127:      monitoring function
    128:   */
    129:   PetscOptionsHasName(NULL,NULL,"-my_monitor",&viewmat);
    130:   if (viewmat) {
    131:     TaoSetMonitor(tao,My_Monitor,NULL,NULL);
    132:   }

    134:   /* Check for any tao command line options */
    135:   TaoSetFromOptions(tao);

    137:   /* SOLVE THE APPLICATION */
    138:   TaoSolve(tao);

    140:   TaoView(tao,PETSC_VIEWER_STDOUT_WORLD);

    142:   /* Free TAO data structures */
    143:   TaoDestroy(&tao);

    145:   /* Free PETSc data structures */
    146:   VecDestroy(&x);
    147:   MatDestroy(&user.H);
    148:   if (fdcoloring) {
    149:     MatFDColoringDestroy(&matfdcoloring);
    150:   }
    151:   PetscFree(user.bottom);
    152:   PetscFree(user.top);
    153:   PetscFree(user.left);
    154:   PetscFree(user.right);
    155:   DMDestroy(&user.dm);
    156:   PetscFinalize();
    157:   return 0;
    158: }

    160: PetscErrorCode FormGradient(Tao tao, Vec X, Vec G,void *userCtx)
    161: {
    162:   PetscReal      fcn;

    164:   FormFunctionGradient(tao,X,&fcn,G,userCtx);
    165:   return 0;
    166: }

    168: /* -------------------------------------------------------------------- */
    169: /*  FormFunctionGradient - Evaluates the function and corresponding gradient.

    171:     Input Parameters:
    172: .   tao     - the Tao context
    173: .   XX      - input vector
    174: .   userCtx - optional user-defined context, as set by TaoSetObjectiveAndGradient()

    176:     Output Parameters:
    177: .   fcn     - the newly evaluated function
    178: .   GG       - vector containing the newly evaluated gradient
    179: */
    180: PetscErrorCode FormFunctionGradient(Tao tao, Vec X, PetscReal *fcn,Vec G,void *userCtx)
    181: {
    182:   AppCtx         *user = (AppCtx *) userCtx;
    183:   PetscInt       i,j;
    184:   PetscInt       mx=user->mx, my=user->my;
    185:   PetscInt       xs,xm,gxs,gxm,ys,ym,gys,gym;
    186:   PetscReal      ft=0.0;
    187:   PetscReal      hx=1.0/(mx+1),hy=1.0/(my+1), hydhx=hy/hx, hxdhy=hx/hy, area=0.5*hx*hy;
    188:   PetscReal      rhx=mx+1, rhy=my+1;
    189:   PetscReal      f1,f2,f3,f4,f5,f6,d1,d2,d3,d4,d5,d6,d7,d8,xc,xl,xr,xt,xb,xlt,xrb;
    190:   PetscReal      df1dxc,df2dxc,df3dxc,df4dxc,df5dxc,df6dxc;
    191:   PetscReal      **g, **x;
    192:   Vec            localX;

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

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

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

    207:   /* Compute function and gradient over the locally owned part of the mesh */
    208:   for (j=ys; j<ys+ym; j++) {
    209:     for (i=xs; i< xs+xm; i++) {

    211:       xc = x[j][i];
    212:       xlt=xrb=xl=xr=xb=xt=xc;

    214:       if (i==0) { /* left side */
    215:         xl= user->left[j-ys+1];
    216:         xlt = user->left[j-ys+2];
    217:       } else {
    218:         xl = x[j][i-1];
    219:       }

    221:       if (j==0) { /* bottom side */
    222:         xb=user->bottom[i-xs+1];
    223:         xrb =user->bottom[i-xs+2];
    224:       } else {
    225:         xb = x[j-1][i];
    226:       }

    228:       if (i+1 == gxs+gxm) { /* right side */
    229:         xr=user->right[j-ys+1];
    230:         xrb = user->right[j-ys];
    231:       } else {
    232:         xr = x[j][i+1];
    233:       }

    235:       if (j+1==gys+gym) { /* top side */
    236:         xt=user->top[i-xs+1];
    237:         xlt = user->top[i-xs];
    238:       }else {
    239:         xt = x[j+1][i];
    240:       }

    242:       if (i>gxs && j+1<gys+gym) {
    243:         xlt = x[j+1][i-1];
    244:       }
    245:       if (j>gys && i+1<gxs+gxm) {
    246:         xrb = x[j-1][i+1];
    247:       }

    249:       d1 = (xc-xl);
    250:       d2 = (xc-xr);
    251:       d3 = (xc-xt);
    252:       d4 = (xc-xb);
    253:       d5 = (xr-xrb);
    254:       d6 = (xrb-xb);
    255:       d7 = (xlt-xl);
    256:       d8 = (xt-xlt);

    258:       df1dxc = d1*hydhx;
    259:       df2dxc = (d1*hydhx + d4*hxdhy);
    260:       df3dxc = d3*hxdhy;
    261:       df4dxc = (d2*hydhx + d3*hxdhy);
    262:       df5dxc = d2*hydhx;
    263:       df6dxc = d4*hxdhy;

    265:       d1 *= rhx;
    266:       d2 *= rhx;
    267:       d3 *= rhy;
    268:       d4 *= rhy;
    269:       d5 *= rhy;
    270:       d6 *= rhx;
    271:       d7 *= rhy;
    272:       d8 *= rhx;

    274:       f1 = PetscSqrtReal(1.0 + d1*d1 + d7*d7);
    275:       f2 = PetscSqrtReal(1.0 + d1*d1 + d4*d4);
    276:       f3 = PetscSqrtReal(1.0 + d3*d3 + d8*d8);
    277:       f4 = PetscSqrtReal(1.0 + d3*d3 + d2*d2);
    278:       f5 = PetscSqrtReal(1.0 + d2*d2 + d5*d5);
    279:       f6 = PetscSqrtReal(1.0 + d4*d4 + d6*d6);

    281:       ft = ft + (f2 + f4);

    283:       df1dxc /= f1;
    284:       df2dxc /= f2;
    285:       df3dxc /= f3;
    286:       df4dxc /= f4;
    287:       df5dxc /= f5;
    288:       df6dxc /= f6;

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

    292:     }
    293:   }

    295:   /* Compute triangular areas along the border of the domain. */
    296:   if (xs==0) { /* left side */
    297:     for (j=ys; j<ys+ym; j++) {
    298:       d3=(user->left[j-ys+1] - user->left[j-ys+2])*rhy;
    299:       d2=(user->left[j-ys+1] - x[j][0]) *rhx;
    300:       ft = ft+PetscSqrtReal(1.0 + d3*d3 + d2*d2);
    301:     }
    302:   }
    303:   if (ys==0) { /* bottom side */
    304:     for (i=xs; i<xs+xm; i++) {
    305:       d2=(user->bottom[i+1-xs]-user->bottom[i-xs+2])*rhx;
    306:       d3=(user->bottom[i-xs+1]-x[0][i])*rhy;
    307:       ft = ft+PetscSqrtReal(1.0 + d3*d3 + d2*d2);
    308:     }
    309:   }

    311:   if (xs+xm==mx) { /* right side */
    312:     for (j=ys; j< ys+ym; j++) {
    313:       d1=(x[j][mx-1] - user->right[j-ys+1])*rhx;
    314:       d4=(user->right[j-ys]-user->right[j-ys+1])*rhy;
    315:       ft = ft+PetscSqrtReal(1.0 + d1*d1 + d4*d4);
    316:     }
    317:   }
    318:   if (ys+ym==my) { /* top side */
    319:     for (i=xs; i<xs+xm; i++) {
    320:       d1=(x[my-1][i] - user->top[i-xs+1])*rhy;
    321:       d4=(user->top[i-xs+1] - user->top[i-xs])*rhx;
    322:       ft = ft+PetscSqrtReal(1.0 + d1*d1 + d4*d4);
    323:     }
    324:   }

    326:   if (ys==0 && xs==0) {
    327:     d1=(user->left[0]-user->left[1])/hy;
    328:     d2=(user->bottom[0]-user->bottom[1])*rhx;
    329:     ft +=PetscSqrtReal(1.0 + d1*d1 + d2*d2);
    330:   }
    331:   if (ys+ym == my && xs+xm == mx) {
    332:     d1=(user->right[ym+1] - user->right[ym])*rhy;
    333:     d2=(user->top[xm+1] - user->top[xm])*rhx;
    334:     ft +=PetscSqrtReal(1.0 + d1*d1 + d2*d2);
    335:   }

    337:   ft=ft*area;
    338:   MPI_Allreduce(&ft,fcn,1,MPIU_REAL,MPIU_SUM,MPI_COMM_WORLD);

    340:   /* Restore vectors */
    341:   DMDAVecRestoreArray(user->dm,localX,(void**)&x);
    342:   DMDAVecRestoreArray(user->dm,G,(void**)&g);

    344:   /* Scatter values to global vector */
    345:   DMRestoreLocalVector(user->dm,&localX);
    346:   PetscLogFlops(67.0*xm*ym);
    347:   return 0;
    348: }

    350: /* ------------------------------------------------------------------- */
    351: /*
    352:    FormHessian - Evaluates Hessian matrix.

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

    359:    Output Parameters:
    360: .  H    - Hessian matrix
    361: .  Hpre - optionally different preconditioning matrix
    362: .  flg  - flag indicating matrix structure

    364: */
    365: PetscErrorCode FormHessian(Tao tao,Vec X,Mat H, Mat Hpre, void *ptr)
    366: {
    367:   AppCtx         *user = (AppCtx *) ptr;

    369:   /* Evaluate the Hessian entries*/
    370:   QuadraticH(user,X,H);
    371:   return 0;
    372: }

    374: /* ------------------------------------------------------------------- */
    375: /*
    376:    QuadraticH - Evaluates Hessian matrix.

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

    382:    Output Parameter:
    383: .  H    - Hessian matrix
    384: */
    385: PetscErrorCode QuadraticH(AppCtx *user, Vec X, Mat Hessian)
    386: {
    387:   PetscInt i,j,k;
    388:   PetscInt mx=user->mx, my=user->my;
    389:   PetscInt xs,xm,gxs,gxm,ys,ym,gys,gym;
    390:   PetscReal hx=1.0/(mx+1), hy=1.0/(my+1), hydhx=hy/hx, hxdhy=hx/hy;
    391:   PetscReal f1,f2,f3,f4,f5,f6,d1,d2,d3,d4,d5,d6,d7,d8,xc,xl,xr,xt,xb,xlt,xrb;
    392:   PetscReal hl,hr,ht,hb,hc,htl,hbr;
    393:   PetscReal **x, v[7];
    394:   MatStencil col[7],row;
    395:   Vec    localX;
    396:   PetscBool assembled;

    398:   /* Get local mesh boundaries */
    399:   DMGetLocalVector(user->dm,&localX);

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

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

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

    411:   /* Initialize matrix entries to zero */
    412:   MatAssembled(Hessian,&assembled);
    413:   if (assembled) MatZeroEntries(Hessian);

    415:   /* Set various matrix options */
    416:   MatSetOption(Hessian,MAT_IGNORE_OFF_PROC_ENTRIES,PETSC_TRUE);

    418:   /* Compute Hessian over the locally owned part of the mesh */

    420:   for (j=ys; j<ys+ym; j++) {

    422:     for (i=xs; i< xs+xm; i++) {

    424:       xc = x[j][i];
    425:       xlt=xrb=xl=xr=xb=xt=xc;

    427:       /* Left side */
    428:       if (i==0) {
    429:         xl  = user->left[j-ys+1];
    430:         xlt = user->left[j-ys+2];
    431:       } else {
    432:         xl  = x[j][i-1];
    433:       }

    435:       if (j==0) {
    436:         xb  = user->bottom[i-xs+1];
    437:         xrb = user->bottom[i-xs+2];
    438:       } else {
    439:         xb  = x[j-1][i];
    440:       }

    442:       if (i+1 == mx) {
    443:         xr  = user->right[j-ys+1];
    444:         xrb = user->right[j-ys];
    445:       } else {
    446:         xr  = x[j][i+1];
    447:       }

    449:       if (j+1==my) {
    450:         xt  = user->top[i-xs+1];
    451:         xlt = user->top[i-xs];
    452:       }else {
    453:         xt  = x[j+1][i];
    454:       }

    456:       if (i>0 && j+1<my) {
    457:         xlt = x[j+1][i-1];
    458:       }
    459:       if (j>0 && i+1<mx) {
    460:         xrb = x[j-1][i+1];
    461:       }

    463:       d1 = (xc-xl)/hx;
    464:       d2 = (xc-xr)/hx;
    465:       d3 = (xc-xt)/hy;
    466:       d4 = (xc-xb)/hy;
    467:       d5 = (xrb-xr)/hy;
    468:       d6 = (xrb-xb)/hx;
    469:       d7 = (xlt-xl)/hy;
    470:       d8 = (xlt-xt)/hx;

    472:       f1 = PetscSqrtReal(1.0 + d1*d1 + d7*d7);
    473:       f2 = PetscSqrtReal(1.0 + d1*d1 + d4*d4);
    474:       f3 = PetscSqrtReal(1.0 + d3*d3 + d8*d8);
    475:       f4 = PetscSqrtReal(1.0 + d3*d3 + d2*d2);
    476:       f5 = PetscSqrtReal(1.0 + d2*d2 + d5*d5);
    477:       f6 = PetscSqrtReal(1.0 + d4*d4 + d6*d6);

    479:       hl = (-hydhx*(1.0+d7*d7)+d1*d7)/(f1*f1*f1)+
    480:         (-hydhx*(1.0+d4*d4)+d1*d4)/(f2*f2*f2);
    481:       hr = (-hydhx*(1.0+d5*d5)+d2*d5)/(f5*f5*f5)+
    482:         (-hydhx*(1.0+d3*d3)+d2*d3)/(f4*f4*f4);
    483:       ht = (-hxdhy*(1.0+d8*d8)+d3*d8)/(f3*f3*f3)+
    484:         (-hxdhy*(1.0+d2*d2)+d2*d3)/(f4*f4*f4);
    485:       hb = (-hxdhy*(1.0+d6*d6)+d4*d6)/(f6*f6*f6)+
    486:         (-hxdhy*(1.0+d1*d1)+d1*d4)/(f2*f2*f2);

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

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

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

    498:       row.j = j; row.i = i;
    499:       k=0;
    500:       if (j>0) {
    501:         v[k]=hb;
    502:         col[k].j = j - 1; col[k].i = i;
    503:         k++;
    504:       }

    506:       if (j>0 && i < mx -1) {
    507:         v[k]=hbr;
    508:         col[k].j = j - 1; col[k].i = i+1;
    509:         k++;
    510:       }

    512:       if (i>0) {
    513:         v[k]= hl;
    514:         col[k].j = j; col[k].i = i-1;
    515:         k++;
    516:       }

    518:       v[k]= hc;
    519:       col[k].j = j; col[k].i = i;
    520:       k++;

    522:       if (i < mx-1) {
    523:         v[k]= hr;
    524:         col[k].j = j; col[k].i = i+1;
    525:         k++;
    526:       }

    528:       if (i>0 && j < my-1) {
    529:         v[k]= htl;
    530:         col[k].j = j+1; col[k].i = i-1;
    531:         k++;
    532:       }

    534:       if (j < my-1) {
    535:         v[k]= ht;
    536:         col[k].j = j+1; col[k].i = i;
    537:         k++;
    538:       }

    540:       /*
    541:          Set matrix values using local numbering, which was defined
    542:          earlier, in the main routine.
    543:       */
    544:       MatSetValuesStencil(Hessian,1,&row,k,col,v,INSERT_VALUES);
    545:     }
    546:   }

    548:   DMDAVecRestoreArray(user->dm,localX,(void**)&x);
    549:   DMRestoreLocalVector(user->dm,&localX);

    551:   MatAssemblyBegin(Hessian,MAT_FINAL_ASSEMBLY);
    552:   MatAssemblyEnd(Hessian,MAT_FINAL_ASSEMBLY);

    554:   PetscLogFlops(199.0*xm*ym);
    555:   return 0;
    556: }

    558: /* ------------------------------------------------------------------- */
    559: /*
    560:    MSA_BoundaryConditions -  Calculates the boundary conditions for
    561:    the region.

    563:    Input Parameter:
    564: .  user - user-defined application context

    566:    Output Parameter:
    567: .  user - user-defined application context
    568: */
    569: static PetscErrorCode MSA_BoundaryConditions(AppCtx * user)
    570: {
    571:   PetscInt       i,j,k,limit=0,maxits=5;
    572:   PetscInt       xs,ys,xm,ym,gxs,gys,gxm,gym;
    573:   PetscInt       mx=user->mx,my=user->my;
    574:   PetscInt       bsize=0, lsize=0, tsize=0, rsize=0;
    575:   PetscReal      one=1.0, two=2.0, three=3.0, tol=1e-10;
    576:   PetscReal      fnorm,det,hx,hy,xt=0,yt=0;
    577:   PetscReal      u1,u2,nf1,nf2,njac11,njac12,njac21,njac22;
    578:   PetscReal      b=-0.5, t=0.5, l=-0.5, r=0.5;
    579:   PetscReal      *boundary;
    580:   PetscBool      flg;

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

    586:   bsize=xm+2;
    587:   lsize=ym+2;
    588:   rsize=ym+2;
    589:   tsize=xm+2;

    591:   PetscMalloc1(bsize,&user->bottom);
    592:   PetscMalloc1(tsize,&user->top);
    593:   PetscMalloc1(lsize,&user->left);
    594:   PetscMalloc1(rsize,&user->right);

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

    598:   for (j=0; j<4; j++) {
    599:     if (j==0) {
    600:       yt=b;
    601:       xt=l+hx*xs;
    602:       limit=bsize;
    603:       boundary=user->bottom;
    604:     } else if (j==1) {
    605:       yt=t;
    606:       xt=l+hx*xs;
    607:       limit=tsize;
    608:       boundary=user->top;
    609:     } else if (j==2) {
    610:       yt=b+hy*ys;
    611:       xt=l;
    612:       limit=lsize;
    613:       boundary=user->left;
    614:     } else { /* if (j==3) */
    615:       yt=b+hy*ys;
    616:       xt=r;
    617:       limit=rsize;
    618:       boundary=user->right;
    619:     }

    621:     for (i=0; i<limit; i++) {
    622:       u1=xt;
    623:       u2=-yt;
    624:       for (k=0; k<maxits; k++) {
    625:         nf1=u1 + u1*u2*u2 - u1*u1*u1/three-xt;
    626:         nf2=-u2 - u1*u1*u2 + u2*u2*u2/three-yt;
    627:         fnorm=PetscSqrtReal(nf1*nf1+nf2*nf2);
    628:         if (fnorm <= tol) break;
    629:         njac11=one+u2*u2-u1*u1;
    630:         njac12=two*u1*u2;
    631:         njac21=-two*u1*u2;
    632:         njac22=-one - u1*u1 + u2*u2;
    633:         det = njac11*njac22-njac21*njac12;
    634:         u1 = u1-(njac22*nf1-njac12*nf2)/det;
    635:         u2 = u2-(njac11*nf2-njac21*nf1)/det;
    636:       }

    638:       boundary[i]=u1*u1-u2*u2;
    639:       if (j==0 || j==1) {
    640:         xt=xt+hx;
    641:       } else { /*  if (j==2 || j==3) */
    642:         yt=yt+hy;
    643:       }

    645:     }

    647:   }

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

    653:     PetscOptionsGetReal(NULL,NULL,"-bottom",&scl,&flg);
    654:     if (flg) {
    655:       for (i=0;i<bsize;i++) user->bottom[i]*=scl;
    656:     }

    658:     PetscOptionsGetReal(NULL,NULL,"-top",&scl,&flg);
    659:     if (flg) {
    660:       for (i=0;i<tsize;i++) user->top[i]*=scl;
    661:     }

    663:     PetscOptionsGetReal(NULL,NULL,"-right",&scl,&flg);
    664:     if (flg) {
    665:       for (i=0;i<rsize;i++) user->right[i]*=scl;
    666:     }

    668:     PetscOptionsGetReal(NULL,NULL,"-left",&scl,&flg);
    669:     if (flg) {
    670:       for (i=0;i<lsize;i++) user->left[i]*=scl;
    671:     }
    672:   }
    673:   return 0;
    674: }

    676: /* ------------------------------------------------------------------- */
    677: /*
    678:    MSA_InitialPoint - Calculates the initial guess in one of three ways.

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

    684:    Output Parameters:
    685: .  X - newly computed initial guess
    686: */
    687: static PetscErrorCode MSA_InitialPoint(AppCtx * user, Vec X)
    688: {
    689:   PetscInt       start2=-1,i,j;
    690:   PetscReal      start1=0;
    691:   PetscBool      flg1,flg2;

    693:   PetscOptionsGetReal(NULL,NULL,"-start",&start1,&flg1);
    694:   PetscOptionsGetInt(NULL,NULL,"-random",&start2,&flg2);

    696:   if (flg1) { /* The zero vector is reasonable */

    698:     VecSet(X, start1);

    700:   } else if (flg2 && start2>0) { /* Try a random start between -0.5 and 0.5 */
    701:     PetscRandom rctx;  PetscReal np5=-0.5;

    703:     PetscRandomCreate(PETSC_COMM_WORLD,&rctx);
    704:     VecSetRandom(X, rctx);
    705:     PetscRandomDestroy(&rctx);
    706:     VecShift(X, np5);

    708:   } else { /* Take an average of the boundary conditions */
    709:     PetscInt  xs,xm,ys,ym;
    710:     PetscInt  mx=user->mx,my=user->my;
    711:     PetscReal **x;

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

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

    719:     /* Perform local computations */
    720:     for (j=ys; j<ys+ym; j++) {
    721:       for (i=xs; i< xs+xm; i++) {
    722:         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;
    723:       }
    724:     }
    725:     DMDAVecRestoreArray(user->dm,X,(void**)&x);
    726:     PetscLogFlops(9.0*xm*ym);
    727:   }
    728:   return 0;
    729: }

    731: /*-----------------------------------------------------------------------*/
    732: PetscErrorCode My_Monitor(Tao tao, void *ctx)
    733: {
    734:   Vec            X;

    736:   TaoGetSolution(tao,&X);
    737:   VecView(X,PETSC_VIEWER_STDOUT_WORLD);
    738:   return 0;
    739: }

    741: /*TEST

    743:    build:
    744:       requires: !complex

    746:    test:
    747:       args: -tao_smonitor -tao_type lmvm -mx 10 -my 8 -tao_gatol 1.e-3
    748:       requires: !single

    750:    test:
    751:       suffix: 2
    752:       nsize: 2
    753:       args: -tao_smonitor -tao_type nls -tao_nls_ksp_max_it 15 -tao_gatol 1.e-4
    754:       filter: grep -v "nls ksp"
    755:       requires: !single

    757:    test:
    758:       suffix: 3
    759:       nsize: 3
    760:       args: -tao_smonitor -tao_type cg -tao_cg_type fr -mx 10 -my 10 -tao_gatol 1.e-3
    761:       requires: !single

    763:    test:
    764:       suffix: 5
    765:       nsize: 2
    766:       args: -tao_smonitor -tao_type bmrm -mx 10 -my 8 -tao_gatol 1.e-3
    767:       requires: !single

    769: TEST*/