Date: 2022-03-31T04:49:04+00:00

Actual source code: minsurf1.c

      1: #include <petsctao.h>

      3: static char  help[] =
      4: "This example demonstrates use of the TAO package to\n\
      5: solve an unconstrained system of equations.  This example is based on a\n\
      6: problem from the MINPACK-2 test suite.  Given a rectangular 2-D domain and\n\
      7: boundary values along the edges of the domain, the objective is to find the\n\
      8: surface with the minimal area that satisfies the boundary conditions.\n\
      9: This application solves this problem using complimentarity -- We are actually\n\
     10: solving the system  (grad f)_i >= 0, if x_i == l_i \n\
     11:                     (grad f)_i = 0, if l_i < x_i < u_i \n\
     12:                     (grad f)_i <= 0, if x_i == u_i  \n\
     13: where f is the function to be minimized. \n\
     14: \n\
     15: The command line options are:\n\
     16:   -mx <xg>, where <xg> = number of grid points in the 1st coordinate direction\n\
     17:   -my <yg>, where <yg> = number of grid points in the 2nd coordinate direction\n\
     18:   -start <st>, where <st> =0 for zero vector, and an average of the boundary conditions otherwise \n\n";

     20: /*T
     21:    Concepts: TAO^Solving a complementarity problem
     22:    Routines: TaoCreate(); TaoDestroy();

     24:    Processors: 1
     25: T*/

     27: /*
     28:    User-defined application context - contains data needed by the
     29:    application-provided call-back routines, FormFunctionGradient(),
     30:    FormHessian().
     31: */
     32: typedef struct {
     33:   PetscInt  mx, my;
     34:   PetscReal *bottom, *top, *left, *right;
     35: } AppCtx;

     37: /* -------- User-defined Routines --------- */

     39: static PetscErrorCode MSA_BoundaryConditions(AppCtx *);
     40: static PetscErrorCode MSA_InitialPoint(AppCtx *, Vec);
     41: PetscErrorCode FormConstraints(Tao, Vec, Vec, void *);
     42: PetscErrorCode FormJacobian(Tao, Vec, Mat, Mat, void *);

     44: int main(int argc, char **argv)
     45: {
     46:   Vec            x;                 /* solution vector */
     47:   Vec            c;                 /* Constraints function vector */
     48:   Vec            xl,xu;             /* Bounds on the variables */
     49:   PetscBool      flg;               /* A return variable when checking for user options */
     50:   Tao            tao;               /* TAO solver context */
     51:   Mat            J;                 /* Jacobian matrix */
     52:   PetscInt       N;                 /* Number of elements in vector */
     53:   PetscScalar    lb =  PETSC_NINFINITY;      /* lower bound constant */
     54:   PetscScalar    ub =  PETSC_INFINITY;      /* upper bound constant */
     55:   AppCtx         user;                    /* user-defined work context */

     57:   /* Initialize PETSc, TAO */
     58:   PetscInitialize(&argc, &argv, (char *)0, help);

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

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

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

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

     73:   /* Create appropriate vectors and matrices */
     74:   VecCreateSeq(MPI_COMM_SELF, N, &x);
     75:   VecDuplicate(x, &c);
     76:   MatCreateSeqAIJ(MPI_COMM_SELF, N, N, 7, NULL, &J);

     78:   /* The TAO code begins here */

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

     84:   /* Set data structure */
     85:   TaoSetSolution(tao, x);

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

     91:   /* Set the variable bounds */
     92:   MSA_BoundaryConditions(&user);

     94:   /* Set initial solution guess */
     95:   MSA_InitialPoint(&user, x);

     97:   /* Set Bounds on variables */
     98:   VecDuplicate(x, &xl);
     99:   VecDuplicate(x, &xu);
    100:   VecSet(xl, lb);
    101:   VecSet(xu, ub);
    102:   TaoSetVariableBounds(tao,xl,xu);

    104:   /* Check for any tao command line options */
    105:   TaoSetFromOptions(tao);

    107:   /* Solve the application */
    108:   TaoSolve(tao);

    110:   /* Free Tao data structures */
    111:   TaoDestroy(&tao);

    113:   /* Free PETSc data structures */
    114:   VecDestroy(&x);
    115:   VecDestroy(&xl);
    116:   VecDestroy(&xu);
    117:   VecDestroy(&c);
    118:   MatDestroy(&J);

    120:   /* Free user-created data structures */
    121:   PetscFree(user.bottom);
    122:   PetscFree(user.top);
    123:   PetscFree(user.left);
    124:   PetscFree(user.right);

    126:   PetscFinalize();
    127:   return 0;
    128: }

    130: /* -------------------------------------------------------------------- */

    132: /*  FormConstraints - Evaluates gradient of f.

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

    139:     Output Parameters:
    140: .   G - vector containing the newly evaluated gradient
    141: */
    142: PetscErrorCode FormConstraints(Tao tao, Vec X, Vec G, void *ptr)
    143: {
    144:   AppCtx         *user = (AppCtx *) ptr;
    145:   PetscInt       i,j,row;
    146:   PetscInt       mx=user->mx, my=user->my;
    147:   PetscReal      hx=1.0/(mx+1),hy=1.0/(my+1), hydhx=hy/hx, hxdhy=hx/hy;
    148:   PetscReal      f1,f2,f3,f4,f5,f6,d1,d2,d3,d4,d5,d6,d7,d8,xc,xl,xr,xt,xb,xlt,xrb;
    149:   PetscReal      df1dxc,df2dxc,df3dxc,df4dxc,df5dxc,df6dxc;
    150:   PetscScalar    zero=0.0;
    151:   PetscScalar    *g, *x;

    153:   /* Initialize vector to zero */
    154:   VecSet(G, zero);

    156:   /* Get pointers to vector data */
    157:   VecGetArray(X, &x);
    158:   VecGetArray(G, &g);

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

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

    168:       if (i==0) { /* left side */
    169:         xl= user->left[j+1];
    170:         xlt = user->left[j+2];
    171:       } else {
    172:         xl = x[row-1];
    173:       }

    175:       if (j==0) { /* bottom side */
    176:         xb=user->bottom[i+1];
    177:         xrb = user->bottom[i+2];
    178:       } else {
    179:         xb = x[row-mx];
    180:       }

    182:       if (i+1 == mx) { /* right side */
    183:         xr=user->right[j+1];
    184:         xrb = user->right[j];
    185:       } else {
    186:         xr = x[row+1];
    187:       }

    189:       if (j+1==0+my) { /* top side */
    190:         xt=user->top[i+1];
    191:         xlt = user->top[i];
    192:       }else {
    193:         xt = x[row+mx];
    194:       }

    196:       if (i>0 && j+1<my) {
    197:         xlt = x[row-1+mx];
    198:       }
    199:       if (j>0 && i+1<mx) {
    200:         xrb = x[row+1-mx];
    201:       }

    203:       d1 = (xc-xl);
    204:       d2 = (xc-xr);
    205:       d3 = (xc-xt);
    206:       d4 = (xc-xb);
    207:       d5 = (xr-xrb);
    208:       d6 = (xrb-xb);
    209:       d7 = (xlt-xl);
    210:       d8 = (xt-xlt);

    212:       df1dxc = d1*hydhx;
    213:       df2dxc = (d1*hydhx + d4*hxdhy);
    214:       df3dxc = d3*hxdhy;
    215:       df4dxc = (d2*hydhx + d3*hxdhy);
    216:       df5dxc = d2*hydhx;
    217:       df6dxc = d4*hxdhy;

    219:       d1 /= hx;
    220:       d2 /= hx;
    221:       d3 /= hy;
    222:       d4 /= hy;
    223:       d5 /= hy;
    224:       d6 /= hx;
    225:       d7 /= hy;
    226:       d8 /= hx;

    228:       f1 = PetscSqrtScalar(1.0 + d1*d1 + d7*d7);
    229:       f2 = PetscSqrtScalar(1.0 + d1*d1 + d4*d4);
    230:       f3 = PetscSqrtScalar(1.0 + d3*d3 + d8*d8);
    231:       f4 = PetscSqrtScalar(1.0 + d3*d3 + d2*d2);
    232:       f5 = PetscSqrtScalar(1.0 + d2*d2 + d5*d5);
    233:       f6 = PetscSqrtScalar(1.0 + d4*d4 + d6*d6);

    235:       df1dxc /= f1;
    236:       df2dxc /= f2;
    237:       df3dxc /= f3;
    238:       df4dxc /= f4;
    239:       df5dxc /= f5;
    240:       df6dxc /= f6;

    242:       g[row] = (df1dxc+df2dxc+df3dxc+df4dxc+df5dxc+df6dxc)/2.0;
    243:     }
    244:   }

    246:   /* Restore vectors */
    247:   VecRestoreArray(X, &x);
    248:   VecRestoreArray(G, &g);
    249:   PetscLogFlops(67*mx*my);
    250:   return 0;
    251: }

    253: /* ------------------------------------------------------------------- */
    254: /*
    255:    FormJacobian - Evaluates Jacobian matrix.

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

    262:    Output Parameters:
    263: .  tH    - Jacobian matrix

    265: */
    266: PetscErrorCode FormJacobian(Tao tao, Vec X, Mat H, Mat tHPre, void *ptr)
    267: {
    268:   AppCtx            *user = (AppCtx *) ptr;
    269:   PetscInt          i,j,k,row;
    270:   PetscInt          mx=user->mx, my=user->my;
    271:   PetscInt          col[7];
    272:   PetscReal         hx=1.0/(mx+1), hy=1.0/(my+1), hydhx=hy/hx, hxdhy=hx/hy;
    273:   PetscReal         f1,f2,f3,f4,f5,f6,d1,d2,d3,d4,d5,d6,d7,d8,xc,xl,xr,xt,xb,xlt,xrb;
    274:   PetscReal         hl,hr,ht,hb,hc,htl,hbr;
    275:   const PetscScalar *x;
    276:   PetscScalar       v[7];
    277:   PetscBool         assembled;

    279:   /* Set various matrix options */
    280:   MatSetOption(H,MAT_IGNORE_OFF_PROC_ENTRIES,PETSC_TRUE);
    281:   MatAssembled(H,&assembled);
    282:   if (assembled) MatZeroEntries(H);

    284:   /* Get pointers to vector data */
    285:   VecGetArrayRead(X, &x);

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

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

    295:       /* Left side */
    296:       if (i==0) {
    297:         xl  = user->left[j+1];
    298:         xlt = user->left[j+2];
    299:       } else {
    300:         xl = x[row-1];
    301:       }

    303:       if (j==0) {
    304:         xb  = user->bottom[i+1];
    305:         xrb = user->bottom[i+2];
    306:       } else {
    307:         xb = x[row-mx];
    308:       }

    310:       if (i+1 == mx) {
    311:         xr  = user->right[j+1];
    312:         xrb = user->right[j];
    313:       } else {
    314:         xr = x[row+1];
    315:       }

    317:       if (j+1==my) {
    318:         xt  = user->top[i+1];
    319:         xlt = user->top[i];
    320:       }else {
    321:         xt = x[row+mx];
    322:       }

    324:       if (i>0 && j+1<my) {
    325:         xlt = x[row-1+mx];
    326:       }
    327:       if (j>0 && i+1<mx) {
    328:         xrb = x[row+1-mx];
    329:       }

    331:       d1 = (xc-xl)/hx;
    332:       d2 = (xc-xr)/hx;
    333:       d3 = (xc-xt)/hy;
    334:       d4 = (xc-xb)/hy;
    335:       d5 = (xrb-xr)/hy;
    336:       d6 = (xrb-xb)/hx;
    337:       d7 = (xlt-xl)/hy;
    338:       d8 = (xlt-xt)/hx;

    340:       f1 = PetscSqrtScalar(1.0 + d1*d1 + d7*d7);
    341:       f2 = PetscSqrtScalar(1.0 + d1*d1 + d4*d4);
    342:       f3 = PetscSqrtScalar(1.0 + d3*d3 + d8*d8);
    343:       f4 = PetscSqrtScalar(1.0 + d3*d3 + d2*d2);
    344:       f5 = PetscSqrtScalar(1.0 + d2*d2 + d5*d5);
    345:       f6 = PetscSqrtScalar(1.0 + d4*d4 + d6*d6);

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

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

    355:       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) +
    356:            (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);

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

    360:       k=0;
    361:       if (j>0) {
    362:         v[k]=hb; col[k]=row - mx; k++;
    363:       }

    365:       if (j>0 && i < mx -1) {
    366:         v[k]=hbr; col[k]=row - mx+1; k++;
    367:       }

    369:       if (i>0) {
    370:         v[k]= hl; col[k]=row - 1; k++;
    371:       }

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

    375:       if (i < mx-1) {
    376:         v[k]= hr; col[k]=row+1; k++;
    377:       }

    379:       if (i>0 && j < my-1) {
    380:         v[k]= htl; col[k] = row+mx-1; k++;
    381:       }

    383:       if (j < my-1) {
    384:         v[k]= ht; col[k] = row+mx; k++;
    385:       }

    387:       /*
    388:          Set matrix values using local numbering, which was defined
    389:          earlier, in the main routine.
    390:       */
    391:       MatSetValues(H,1,&row,k,col,v,INSERT_VALUES);
    392:     }
    393:   }

    395:   /* Restore vectors */
    396:   VecRestoreArrayRead(X,&x);

    398:   /* Assemble the matrix */
    399:   MatAssemblyBegin(H,MAT_FINAL_ASSEMBLY);
    400:   MatAssemblyEnd(H,MAT_FINAL_ASSEMBLY);
    401:   PetscLogFlops(199*mx*my);
    402:   return 0;
    403: }

    405: /* ------------------------------------------------------------------- */
    406: /*
    407:    MSA_BoundaryConditions -  Calculates the boundary conditions for
    408:    the region.

    410:    Input Parameter:
    411: .  user - user-defined application context

    413:    Output Parameter:
    414: .  user - user-defined application context
    415: */
    416: static PetscErrorCode MSA_BoundaryConditions(AppCtx * user)
    417: {
    418:   PetscInt        i,j,k,limit=0,maxits=5;
    419:   PetscInt        mx=user->mx,my=user->my;
    420:   PetscInt        bsize=0, lsize=0, tsize=0, rsize=0;
    421:   PetscReal       one=1.0, two=2.0, three=3.0, tol=1e-10;
    422:   PetscReal       fnorm,det,hx,hy,xt=0,yt=0;
    423:   PetscReal       u1,u2,nf1,nf2,njac11,njac12,njac21,njac22;
    424:   PetscReal       b=-0.5, t=0.5, l=-0.5, r=0.5;
    425:   PetscReal       *boundary;

    427:   bsize=mx+2; lsize=my+2; rsize=my+2; tsize=mx+2;

    429:   PetscMalloc1(bsize, &user->bottom);
    430:   PetscMalloc1(tsize, &user->top);
    431:   PetscMalloc1(lsize, &user->left);
    432:   PetscMalloc1(rsize, &user->right);

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

    436:   for (j=0; j<4; j++) {
    437:     if (j==0) {
    438:       yt=b;
    439:       xt=l;
    440:       limit=bsize;
    441:       boundary=user->bottom;
    442:     } else if (j==1) {
    443:       yt=t;
    444:       xt=l;
    445:       limit=tsize;
    446:       boundary=user->top;
    447:     } else if (j==2) {
    448:       yt=b;
    449:       xt=l;
    450:       limit=lsize;
    451:       boundary=user->left;
    452:     } else { /* if  (j==3) */
    453:       yt=b;
    454:       xt=r;
    455:       limit=rsize;
    456:       boundary=user->right;
    457:     }

    459:     for (i=0; i<limit; i++) {
    460:       u1=xt;
    461:       u2=-yt;
    462:       for (k=0; k<maxits; k++) {
    463:         nf1=u1 + u1*u2*u2 - u1*u1*u1/three-xt;
    464:         nf2=-u2 - u1*u1*u2 + u2*u2*u2/three-yt;
    465:         fnorm=PetscSqrtScalar(nf1*nf1+nf2*nf2);
    466:         if (fnorm <= tol) break;
    467:         njac11=one+u2*u2-u1*u1;
    468:         njac12=two*u1*u2;
    469:         njac21=-two*u1*u2;
    470:         njac22=-one - u1*u1 + u2*u2;
    471:         det = njac11*njac22-njac21*njac12;
    472:         u1 = u1-(njac22*nf1-njac12*nf2)/det;
    473:         u2 = u2-(njac11*nf2-njac21*nf1)/det;
    474:       }

    476:       boundary[i]=u1*u1-u2*u2;
    477:       if (j==0 || j==1) {
    478:         xt=xt+hx;
    479:       } else { /* if (j==2 || j==3) */
    480:         yt=yt+hy;
    481:       }
    482:     }
    483:   }
    484:   return 0;
    485: }

    487: /* ------------------------------------------------------------------- */
    488: /*
    489:    MSA_InitialPoint - Calculates the initial guess in one of three ways.

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

    495:    Output Parameters:
    496: .  X - newly computed initial guess
    497: */
    498: static PetscErrorCode MSA_InitialPoint(AppCtx * user, Vec X)
    499: {
    500:   PetscInt       start=-1,i,j;
    501:   PetscScalar    zero=0.0;
    502:   PetscBool      flg;

    504:   PetscOptionsGetInt(NULL,NULL,"-start",&start,&flg);

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

    513:     /* Get pointers to vector data */
    514:     VecGetArray(X,&x);

    516:     /* Perform local computations */
    517:     for (j=0; j<my; j++) {
    518:       for (i=0; i< mx; i++) {
    519:         row=(j)*mx + (i);
    520:         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;
    521:       }
    522:     }

    524:     /* Restore vectors */
    525:     VecRestoreArray(X,&x);
    526:   }
    527:   return 0;
    528: }

    530: /*TEST

    532:    build:
    533:       requires: !complex

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

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

    543: TEST*/