Date: 2022-03-31T04:48:59+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: /*T
     23:    Concepts: TAO^Solving an unconstrained minimization problem
     24:    Routines: TaoCreate(); TaoSetType();
     25:    Routines: TaoSetSolution();
     26:    Routines: TaoSetObjectiveAndGradient();
     27:    Routines: TaoSetHessian(); TaoSetFromOptions();
     28:    Routines: TaoSetMonitor();
     29:    Routines: TaoSolve(); TaoView();
     30:    Routines: TaoDestroy();
     31:    Processors: n
     32: T*/

     34: /*
     35:    User-defined application context - contains data needed by the
     36:    application-provided call-back routines, FormFunctionGradient()
     37:    and FormHessian().
     38: */
     39: typedef struct {
     40:   PetscInt    mx, my;                 /* discretization in x, y directions */
     41:   PetscReal   *bottom, *top, *left, *right;             /* boundary values */
     42:   DM          dm;                      /* distributed array data structure */
     43:   Mat         H;                       /* Hessian */
     44: } AppCtx;

     46: /* -------- User-defined Routines --------- */

     48: static PetscErrorCode MSA_BoundaryConditions(AppCtx*);
     49: static PetscErrorCode MSA_InitialPoint(AppCtx*,Vec);
     50: PetscErrorCode QuadraticH(AppCtx*,Vec,Mat);
     51: PetscErrorCode FormFunctionGradient(Tao,Vec,PetscReal *,Vec,void*);
     52: PetscErrorCode FormGradient(Tao,Vec,Vec,void*);
     53: PetscErrorCode FormHessian(Tao,Vec,Mat,Mat,void*);
     54: PetscErrorCode My_Monitor(Tao, void *);

     56: int main(int argc, char **argv)
     57: {
     58:   PetscInt           Nx, Ny;              /* number of processors in x- and y- directions */
     59:   Vec                x;                   /* solution, gradient vectors */
     60:   PetscBool          flg, viewmat;        /* flags */
     61:   PetscBool          fddefault, fdcoloring;   /* flags */
     62:   Tao                tao;                 /* TAO solver context */
     63:   AppCtx             user;                /* user-defined work context */
     64:   ISColoring         iscoloring;
     65:   MatFDColoring      matfdcoloring;

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

     70:   /* Specify dimension of the problem */
     71:   user.mx = 10; user.my = 10;

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

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

     80:   /* Let PETSc determine the vector distribution */
     81:   Nx = PETSC_DECIDE; Ny = PETSC_DECIDE;

     83:   /* Create distributed array (DM) to manage parallel grid and vectors  */
     84:   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);
     85:   DMSetFromOptions(user.dm);
     86:   DMSetUp(user.dm);

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

     92:   /*
     93:      Extract global vector from DA for the vector of variables --  PETSC routine
     94:      Compute the initial solution                              --  application specific, see below
     95:      Set this vector for use by TAO                            --  TAO routine
     96:   */
     97:   DMCreateGlobalVector(user.dm,&x);
     98:   MSA_BoundaryConditions(&user);
     99:   MSA_InitialPoint(&user,x);
    100:   TaoSetSolution(tao,x);

    102:   /*
    103:      Initialize the Application context for use in function evaluations  --  application specific, see below.
    104:      Set routines for function and gradient evaluation
    105:   */
    106:   TaoSetObjectiveAndGradient(tao,NULL,FormFunctionGradient,(void *)&user);

    108:   /*
    109:      Given the command line arguments, calculate the hessian with either the user-
    110:      provided function FormHessian, or the default finite-difference driven Hessian
    111:      functions
    112:   */
    113:   PetscOptionsHasName(NULL,NULL,"-tao_fddefault",&fddefault);
    114:   PetscOptionsHasName(NULL,NULL,"-tao_fdcoloring",&fdcoloring);

    116:   /*
    117:      Create a matrix data structure to store the Hessian and set
    118:      the Hessian evalution routine.
    119:      Set the matrix structure to be used for Hessian evalutions
    120:   */
    121:   DMCreateMatrix(user.dm,&user.H);
    122:   MatSetOption(user.H,MAT_SYMMETRIC,PETSC_TRUE);

    124:   if (fdcoloring) {
    125:     DMCreateColoring(user.dm,IS_COLORING_GLOBAL,&iscoloring);
    126:     MatFDColoringCreate(user.H,iscoloring,&matfdcoloring);
    127:     ISColoringDestroy(&iscoloring);
    128:     MatFDColoringSetFunction(matfdcoloring,(PetscErrorCode (*)(void))FormGradient,(void*)&user);
    129:     MatFDColoringSetFromOptions(matfdcoloring);
    130:     TaoSetHessian(tao,user.H,user.H,TaoDefaultComputeHessianColor,(void *)matfdcoloring);
    131:   } else if (fddefault) {
    132:     TaoSetHessian(tao,user.H,user.H,TaoDefaultComputeHessian,(void *)NULL);
    133:   } else {
    134:     TaoSetHessian(tao,user.H,user.H,FormHessian,(void *)&user);
    135:   }

    137:   /*
    138:      If my_monitor option is in command line, then use the user-provided
    139:      monitoring function
    140:   */
    141:   PetscOptionsHasName(NULL,NULL,"-my_monitor",&viewmat);
    142:   if (viewmat) {
    143:     TaoSetMonitor(tao,My_Monitor,NULL,NULL);
    144:   }

    146:   /* Check for any tao command line options */
    147:   TaoSetFromOptions(tao);

    149:   /* SOLVE THE APPLICATION */
    150:   TaoSolve(tao);

    152:   TaoView(tao,PETSC_VIEWER_STDOUT_WORLD);

    154:   /* Free TAO data structures */
    155:   TaoDestroy(&tao);

    157:   /* Free PETSc data structures */
    158:   VecDestroy(&x);
    159:   MatDestroy(&user.H);
    160:   if (fdcoloring) {
    161:     MatFDColoringDestroy(&matfdcoloring);
    162:   }
    163:   PetscFree(user.bottom);
    164:   PetscFree(user.top);
    165:   PetscFree(user.left);
    166:   PetscFree(user.right);
    167:   DMDestroy(&user.dm);
    168:   PetscFinalize();
    169:   return 0;
    170: }

    172: PetscErrorCode FormGradient(Tao tao, Vec X, Vec G,void *userCtx)
    173: {
    174:   PetscReal      fcn;

    176:   FormFunctionGradient(tao,X,&fcn,G,userCtx);
    177:   return 0;
    178: }

    180: /* -------------------------------------------------------------------- */
    181: /*  FormFunctionGradient - Evaluates the function and corresponding gradient.

    183:     Input Parameters:
    184: .   tao     - the Tao context
    185: .   XX      - input vector
    186: .   userCtx - optional user-defined context, as set by TaoSetObjectiveAndGradient()

    188:     Output Parameters:
    189: .   fcn     - the newly evaluated function
    190: .   GG       - vector containing the newly evaluated gradient
    191: */
    192: PetscErrorCode FormFunctionGradient(Tao tao, Vec X, PetscReal *fcn,Vec G,void *userCtx)
    193: {
    194:   AppCtx         *user = (AppCtx *) userCtx;
    195:   PetscInt       i,j;
    196:   PetscInt       mx=user->mx, my=user->my;
    197:   PetscInt       xs,xm,gxs,gxm,ys,ym,gys,gym;
    198:   PetscReal      ft=0.0;
    199:   PetscReal      hx=1.0/(mx+1),hy=1.0/(my+1), hydhx=hy/hx, hxdhy=hx/hy, area=0.5*hx*hy;
    200:   PetscReal      rhx=mx+1, rhy=my+1;
    201:   PetscReal      f1,f2,f3,f4,f5,f6,d1,d2,d3,d4,d5,d6,d7,d8,xc,xl,xr,xt,xb,xlt,xrb;
    202:   PetscReal      df1dxc,df2dxc,df3dxc,df4dxc,df5dxc,df6dxc;
    203:   PetscReal      **g, **x;
    204:   Vec            localX;

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

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

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

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

    223:       xc = x[j][i];
    224:       xlt=xrb=xl=xr=xb=xt=xc;

    226:       if (i==0) { /* left side */
    227:         xl= user->left[j-ys+1];
    228:         xlt = user->left[j-ys+2];
    229:       } else {
    230:         xl = x[j][i-1];
    231:       }

    233:       if (j==0) { /* bottom side */
    234:         xb=user->bottom[i-xs+1];
    235:         xrb =user->bottom[i-xs+2];
    236:       } else {
    237:         xb = x[j-1][i];
    238:       }

    240:       if (i+1 == gxs+gxm) { /* right side */
    241:         xr=user->right[j-ys+1];
    242:         xrb = user->right[j-ys];
    243:       } else {
    244:         xr = x[j][i+1];
    245:       }

    247:       if (j+1==gys+gym) { /* top side */
    248:         xt=user->top[i-xs+1];
    249:         xlt = user->top[i-xs];
    250:       }else {
    251:         xt = x[j+1][i];
    252:       }

    254:       if (i>gxs && j+1<gys+gym) {
    255:         xlt = x[j+1][i-1];
    256:       }
    257:       if (j>gys && i+1<gxs+gxm) {
    258:         xrb = x[j-1][i+1];
    259:       }

    261:       d1 = (xc-xl);
    262:       d2 = (xc-xr);
    263:       d3 = (xc-xt);
    264:       d4 = (xc-xb);
    265:       d5 = (xr-xrb);
    266:       d6 = (xrb-xb);
    267:       d7 = (xlt-xl);
    268:       d8 = (xt-xlt);

    270:       df1dxc = d1*hydhx;
    271:       df2dxc = (d1*hydhx + d4*hxdhy);
    272:       df3dxc = d3*hxdhy;
    273:       df4dxc = (d2*hydhx + d3*hxdhy);
    274:       df5dxc = d2*hydhx;
    275:       df6dxc = d4*hxdhy;

    277:       d1 *= rhx;
    278:       d2 *= rhx;
    279:       d3 *= rhy;
    280:       d4 *= rhy;
    281:       d5 *= rhy;
    282:       d6 *= rhx;
    283:       d7 *= rhy;
    284:       d8 *= rhx;

    286:       f1 = PetscSqrtReal(1.0 + d1*d1 + d7*d7);
    287:       f2 = PetscSqrtReal(1.0 + d1*d1 + d4*d4);
    288:       f3 = PetscSqrtReal(1.0 + d3*d3 + d8*d8);
    289:       f4 = PetscSqrtReal(1.0 + d3*d3 + d2*d2);
    290:       f5 = PetscSqrtReal(1.0 + d2*d2 + d5*d5);
    291:       f6 = PetscSqrtReal(1.0 + d4*d4 + d6*d6);

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

    295:       df1dxc /= f1;
    296:       df2dxc /= f2;
    297:       df3dxc /= f3;
    298:       df4dxc /= f4;
    299:       df5dxc /= f5;
    300:       df6dxc /= f6;

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

    304:     }
    305:   }

    307:   /* Compute triangular areas along the border of the domain. */
    308:   if (xs==0) { /* left side */
    309:     for (j=ys; j<ys+ym; j++) {
    310:       d3=(user->left[j-ys+1] - user->left[j-ys+2])*rhy;
    311:       d2=(user->left[j-ys+1] - x[j][0]) *rhx;
    312:       ft = ft+PetscSqrtReal(1.0 + d3*d3 + d2*d2);
    313:     }
    314:   }
    315:   if (ys==0) { /* bottom side */
    316:     for (i=xs; i<xs+xm; i++) {
    317:       d2=(user->bottom[i+1-xs]-user->bottom[i-xs+2])*rhx;
    318:       d3=(user->bottom[i-xs+1]-x[0][i])*rhy;
    319:       ft = ft+PetscSqrtReal(1.0 + d3*d3 + d2*d2);
    320:     }
    321:   }

    323:   if (xs+xm==mx) { /* right side */
    324:     for (j=ys; j< ys+ym; j++) {
    325:       d1=(x[j][mx-1] - user->right[j-ys+1])*rhx;
    326:       d4=(user->right[j-ys]-user->right[j-ys+1])*rhy;
    327:       ft = ft+PetscSqrtReal(1.0 + d1*d1 + d4*d4);
    328:     }
    329:   }
    330:   if (ys+ym==my) { /* top side */
    331:     for (i=xs; i<xs+xm; i++) {
    332:       d1=(x[my-1][i] - user->top[i-xs+1])*rhy;
    333:       d4=(user->top[i-xs+1] - user->top[i-xs])*rhx;
    334:       ft = ft+PetscSqrtReal(1.0 + d1*d1 + d4*d4);
    335:     }
    336:   }

    338:   if (ys==0 && xs==0) {
    339:     d1=(user->left[0]-user->left[1])/hy;
    340:     d2=(user->bottom[0]-user->bottom[1])*rhx;
    341:     ft +=PetscSqrtReal(1.0 + d1*d1 + d2*d2);
    342:   }
    343:   if (ys+ym == my && xs+xm == mx) {
    344:     d1=(user->right[ym+1] - user->right[ym])*rhy;
    345:     d2=(user->top[xm+1] - user->top[xm])*rhx;
    346:     ft +=PetscSqrtReal(1.0 + d1*d1 + d2*d2);
    347:   }

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

    352:   /* Restore vectors */
    353:   DMDAVecRestoreArray(user->dm,localX,(void**)&x);
    354:   DMDAVecRestoreArray(user->dm,G,(void**)&g);

    356:   /* Scatter values to global vector */
    357:   DMRestoreLocalVector(user->dm,&localX);
    358:   PetscLogFlops(67.0*xm*ym);
    359:   return 0;
    360: }

    362: /* ------------------------------------------------------------------- */
    363: /*
    364:    FormHessian - Evaluates Hessian matrix.

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

    371:    Output Parameters:
    372: .  H    - Hessian matrix
    373: .  Hpre - optionally different preconditioning matrix
    374: .  flg  - flag indicating matrix structure

    376: */
    377: PetscErrorCode FormHessian(Tao tao,Vec X,Mat H, Mat Hpre, void *ptr)
    378: {
    379:   AppCtx         *user = (AppCtx *) ptr;

    381:   /* Evaluate the Hessian entries*/
    382:   QuadraticH(user,X,H);
    383:   return 0;
    384: }

    386: /* ------------------------------------------------------------------- */
    387: /*
    388:    QuadraticH - Evaluates Hessian matrix.

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

    394:    Output Parameter:
    395: .  H    - Hessian matrix
    396: */
    397: PetscErrorCode QuadraticH(AppCtx *user, Vec X, Mat Hessian)
    398: {
    399:   PetscInt i,j,k;
    400:   PetscInt mx=user->mx, my=user->my;
    401:   PetscInt xs,xm,gxs,gxm,ys,ym,gys,gym;
    402:   PetscReal hx=1.0/(mx+1), hy=1.0/(my+1), hydhx=hy/hx, hxdhy=hx/hy;
    403:   PetscReal f1,f2,f3,f4,f5,f6,d1,d2,d3,d4,d5,d6,d7,d8,xc,xl,xr,xt,xb,xlt,xrb;
    404:   PetscReal hl,hr,ht,hb,hc,htl,hbr;
    405:   PetscReal **x, v[7];
    406:   MatStencil col[7],row;
    407:   Vec    localX;
    408:   PetscBool assembled;

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

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

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

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

    423:   /* Initialize matrix entries to zero */
    424:   MatAssembled(Hessian,&assembled);
    425:   if (assembled) MatZeroEntries(Hessian);

    427:   /* Set various matrix options */
    428:   MatSetOption(Hessian,MAT_IGNORE_OFF_PROC_ENTRIES,PETSC_TRUE);

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

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

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

    436:       xc = x[j][i];
    437:       xlt=xrb=xl=xr=xb=xt=xc;

    439:       /* Left side */
    440:       if (i==0) {
    441:         xl  = user->left[j-ys+1];
    442:         xlt = user->left[j-ys+2];
    443:       } else {
    444:         xl  = x[j][i-1];
    445:       }

    447:       if (j==0) {
    448:         xb  = user->bottom[i-xs+1];
    449:         xrb = user->bottom[i-xs+2];
    450:       } else {
    451:         xb  = x[j-1][i];
    452:       }

    454:       if (i+1 == mx) {
    455:         xr  = user->right[j-ys+1];
    456:         xrb = user->right[j-ys];
    457:       } else {
    458:         xr  = x[j][i+1];
    459:       }

    461:       if (j+1==my) {
    462:         xt  = user->top[i-xs+1];
    463:         xlt = user->top[i-xs];
    464:       }else {
    465:         xt  = x[j+1][i];
    466:       }

    468:       if (i>0 && j+1<my) {
    469:         xlt = x[j+1][i-1];
    470:       }
    471:       if (j>0 && i+1<mx) {
    472:         xrb = x[j-1][i+1];
    473:       }

    475:       d1 = (xc-xl)/hx;
    476:       d2 = (xc-xr)/hx;
    477:       d3 = (xc-xt)/hy;
    478:       d4 = (xc-xb)/hy;
    479:       d5 = (xrb-xr)/hy;
    480:       d6 = (xrb-xb)/hx;
    481:       d7 = (xlt-xl)/hy;
    482:       d8 = (xlt-xt)/hx;

    484:       f1 = PetscSqrtReal(1.0 + d1*d1 + d7*d7);
    485:       f2 = PetscSqrtReal(1.0 + d1*d1 + d4*d4);
    486:       f3 = PetscSqrtReal(1.0 + d3*d3 + d8*d8);
    487:       f4 = PetscSqrtReal(1.0 + d3*d3 + d2*d2);
    488:       f5 = PetscSqrtReal(1.0 + d2*d2 + d5*d5);
    489:       f6 = PetscSqrtReal(1.0 + d4*d4 + d6*d6);

    491:       hl = (-hydhx*(1.0+d7*d7)+d1*d7)/(f1*f1*f1)+
    492:         (-hydhx*(1.0+d4*d4)+d1*d4)/(f2*f2*f2);
    493:       hr = (-hydhx*(1.0+d5*d5)+d2*d5)/(f5*f5*f5)+
    494:         (-hydhx*(1.0+d3*d3)+d2*d3)/(f4*f4*f4);
    495:       ht = (-hxdhy*(1.0+d8*d8)+d3*d8)/(f3*f3*f3)+
    496:         (-hxdhy*(1.0+d2*d2)+d2*d3)/(f4*f4*f4);
    497:       hb = (-hxdhy*(1.0+d6*d6)+d4*d6)/(f6*f6*f6)+
    498:         (-hxdhy*(1.0+d1*d1)+d1*d4)/(f2*f2*f2);

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

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

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

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

    518:       if (j>0 && i < mx -1) {
    519:         v[k]=hbr;
    520:         col[k].j = j - 1; col[k].i = i+1;
    521:         k++;
    522:       }

    524:       if (i>0) {
    525:         v[k]= hl;
    526:         col[k].j = j; col[k].i = i-1;
    527:         k++;
    528:       }

    530:       v[k]= hc;
    531:       col[k].j = j; col[k].i = i;
    532:       k++;

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

    540:       if (i>0 && j < my-1) {
    541:         v[k]= htl;
    542:         col[k].j = j+1; col[k].i = i-1;
    543:         k++;
    544:       }

    546:       if (j < my-1) {
    547:         v[k]= ht;
    548:         col[k].j = j+1; col[k].i = i;
    549:         k++;
    550:       }

    552:       /*
    553:          Set matrix values using local numbering, which was defined
    554:          earlier, in the main routine.
    555:       */
    556:       MatSetValuesStencil(Hessian,1,&row,k,col,v,INSERT_VALUES);
    557:     }
    558:   }

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

    563:   MatAssemblyBegin(Hessian,MAT_FINAL_ASSEMBLY);
    564:   MatAssemblyEnd(Hessian,MAT_FINAL_ASSEMBLY);

    566:   PetscLogFlops(199.0*xm*ym);
    567:   return 0;
    568: }

    570: /* ------------------------------------------------------------------- */
    571: /*
    572:    MSA_BoundaryConditions -  Calculates the boundary conditions for
    573:    the region.

    575:    Input Parameter:
    576: .  user - user-defined application context

    578:    Output Parameter:
    579: .  user - user-defined application context
    580: */
    581: static PetscErrorCode MSA_BoundaryConditions(AppCtx * user)
    582: {
    583:   PetscInt       i,j,k,limit=0,maxits=5;
    584:   PetscInt       xs,ys,xm,ym,gxs,gys,gxm,gym;
    585:   PetscInt       mx=user->mx,my=user->my;
    586:   PetscInt       bsize=0, lsize=0, tsize=0, rsize=0;
    587:   PetscReal      one=1.0, two=2.0, three=3.0, tol=1e-10;
    588:   PetscReal      fnorm,det,hx,hy,xt=0,yt=0;
    589:   PetscReal      u1,u2,nf1,nf2,njac11,njac12,njac21,njac22;
    590:   PetscReal      b=-0.5, t=0.5, l=-0.5, r=0.5;
    591:   PetscReal      *boundary;
    592:   PetscBool      flg;

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

    598:   bsize=xm+2;
    599:   lsize=ym+2;
    600:   rsize=ym+2;
    601:   tsize=xm+2;

    603:   PetscMalloc1(bsize,&user->bottom);
    604:   PetscMalloc1(tsize,&user->top);
    605:   PetscMalloc1(lsize,&user->left);
    606:   PetscMalloc1(rsize,&user->right);

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

    610:   for (j=0; j<4; j++) {
    611:     if (j==0) {
    612:       yt=b;
    613:       xt=l+hx*xs;
    614:       limit=bsize;
    615:       boundary=user->bottom;
    616:     } else if (j==1) {
    617:       yt=t;
    618:       xt=l+hx*xs;
    619:       limit=tsize;
    620:       boundary=user->top;
    621:     } else if (j==2) {
    622:       yt=b+hy*ys;
    623:       xt=l;
    624:       limit=lsize;
    625:       boundary=user->left;
    626:     } else { /* if (j==3) */
    627:       yt=b+hy*ys;
    628:       xt=r;
    629:       limit=rsize;
    630:       boundary=user->right;
    631:     }

    633:     for (i=0; i<limit; i++) {
    634:       u1=xt;
    635:       u2=-yt;
    636:       for (k=0; k<maxits; k++) {
    637:         nf1=u1 + u1*u2*u2 - u1*u1*u1/three-xt;
    638:         nf2=-u2 - u1*u1*u2 + u2*u2*u2/three-yt;
    639:         fnorm=PetscSqrtReal(nf1*nf1+nf2*nf2);
    640:         if (fnorm <= tol) break;
    641:         njac11=one+u2*u2-u1*u1;
    642:         njac12=two*u1*u2;
    643:         njac21=-two*u1*u2;
    644:         njac22=-one - u1*u1 + u2*u2;
    645:         det = njac11*njac22-njac21*njac12;
    646:         u1 = u1-(njac22*nf1-njac12*nf2)/det;
    647:         u2 = u2-(njac11*nf2-njac21*nf1)/det;
    648:       }

    650:       boundary[i]=u1*u1-u2*u2;
    651:       if (j==0 || j==1) {
    652:         xt=xt+hx;
    653:       } else { /*  if (j==2 || j==3) */
    654:         yt=yt+hy;
    655:       }

    657:     }

    659:   }

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

    665:     PetscOptionsGetReal(NULL,NULL,"-bottom",&scl,&flg);
    666:     if (flg) {
    667:       for (i=0;i<bsize;i++) user->bottom[i]*=scl;
    668:     }

    670:     PetscOptionsGetReal(NULL,NULL,"-top",&scl,&flg);
    671:     if (flg) {
    672:       for (i=0;i<tsize;i++) user->top[i]*=scl;
    673:     }

    675:     PetscOptionsGetReal(NULL,NULL,"-right",&scl,&flg);
    676:     if (flg) {
    677:       for (i=0;i<rsize;i++) user->right[i]*=scl;
    678:     }

    680:     PetscOptionsGetReal(NULL,NULL,"-left",&scl,&flg);
    681:     if (flg) {
    682:       for (i=0;i<lsize;i++) user->left[i]*=scl;
    683:     }
    684:   }
    685:   return 0;
    686: }

    688: /* ------------------------------------------------------------------- */
    689: /*
    690:    MSA_InitialPoint - Calculates the initial guess in one of three ways.

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

    696:    Output Parameters:
    697: .  X - newly computed initial guess
    698: */
    699: static PetscErrorCode MSA_InitialPoint(AppCtx * user, Vec X)
    700: {
    701:   PetscInt       start2=-1,i,j;
    702:   PetscReal      start1=0;
    703:   PetscBool      flg1,flg2;

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

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

    710:     VecSet(X, start1);

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

    715:     PetscRandomCreate(PETSC_COMM_WORLD,&rctx);
    716:     VecSetRandom(X, rctx);
    717:     PetscRandomDestroy(&rctx);
    718:     VecShift(X, np5);

    720:   } else { /* Take an average of the boundary conditions */
    721:     PetscInt  xs,xm,ys,ym;
    722:     PetscInt  mx=user->mx,my=user->my;
    723:     PetscReal **x;

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

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

    731:     /* Perform local computations */
    732:     for (j=ys; j<ys+ym; j++) {
    733:       for (i=xs; i< xs+xm; i++) {
    734:         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;
    735:       }
    736:     }
    737:     DMDAVecRestoreArray(user->dm,X,(void**)&x);
    738:     PetscLogFlops(9.0*xm*ym);
    739:   }
    740:   return 0;
    741: }

    743: /*-----------------------------------------------------------------------*/
    744: PetscErrorCode My_Monitor(Tao tao, void *ctx)
    745: {
    746:   Vec            X;

    748:   TaoGetSolution(tao,&X);
    749:   VecView(X,PETSC_VIEWER_STDOUT_WORLD);
    750:   return 0;
    751: }

    753: /*TEST

    755:    build:
    756:       requires: !complex

    758:    test:
    759:       args: -tao_smonitor -tao_type lmvm -mx 10 -my 8 -tao_gatol 1.e-3
    760:       requires: !single

    762:    test:
    763:       suffix: 2
    764:       nsize: 2
    765:       args: -tao_smonitor -tao_type nls -tao_nls_ksp_max_it 15 -tao_gatol 1.e-4
    766:       filter: grep -v "nls ksp"
    767:       requires: !single

    769:    test:
    770:       suffix: 3
    771:       nsize: 3
    772:       args: -tao_smonitor -tao_type cg -tao_cg_type fr -mx 10 -my 10 -tao_gatol 1.e-3
    773:       requires: !single

    775:    test:
    776:       suffix: 5
    777:       nsize: 2
    778:       args: -tao_smonitor -tao_type bmrm -mx 10 -my 8 -tao_gatol 1.e-3
    779:       requires: !single

    781: TEST*/