Actual source code: minsurf1.c
petsc-3.12.4 2020-02-04
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      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*/

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

     41: /* -------- User-defined Routines --------- */

     43: static PetscErrorCode MSA_BoundaryConditions(AppCtx *);
     44: static PetscErrorCode MSA_InitialPoint(AppCtx *, Vec);
     45: PetscErrorCode FormConstraints(Tao, Vec, Vec, void *);
     46: PetscErrorCode FormJacobian(Tao, Vec, Mat, Mat, void *);

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

     62:   /* Initialize PETSc, TAO */
     63:   PetscInitialize(&argc, &argv, (char *)0, help );if (ierr) return ierr;

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

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

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

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

     78:   /* Create appropriate vectors and matrices */
     79:   VecCreateSeq(MPI_COMM_SELF, N, &x);
     80:   VecDuplicate(x, &c);
     81:   MatCreateSeqAIJ(MPI_COMM_SELF, N, N, 7, NULL, &J);

     83:   /* The TAO code begins here */

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

     89:   /* Set data structure */
     90:   TaoSetInitialVector(tao, x);

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

     96:   /* Set the variable bounds */
     97:   MSA_BoundaryConditions(&user);

     99:   /* Set initial solution guess */
    100:   MSA_InitialPoint(&user, x);

    102:   /* Set Bounds on variables */
    103:   VecDuplicate(x, &xl);
    104:   VecDuplicate(x, &xu);
    105:   VecSet(xl, lb);
    106:   VecSet(xu, ub);
    107:   TaoSetVariableBounds(tao,xl,xu);

    109:   /* Check for any tao command line options */
    110:   TaoSetFromOptions(tao);

    112:   /* Solve the application */
    113:   TaoSolve(tao);

    115:   /* Free Tao data structures */
    116:   TaoDestroy(&tao);

    118:   /* Free PETSc data structures */
    119:   VecDestroy(&x);
    120:   VecDestroy(&xl);
    121:   VecDestroy(&xu);
    122:   VecDestroy(&c);
    123:   MatDestroy(&J);

    125:   /* Free user-created data structures */
    126:   PetscFree(user.bottom);
    127:   PetscFree(user.top);
    128:   PetscFree(user.left);
    129:   PetscFree(user.right);

    131:   PetscFinalize();
    132:   return ierr;
    133: }

    135: /* -------------------------------------------------------------------- */

    137: /*  FormConstraints - Evaluates gradient of f.

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

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

    160:   /* Initialize vector to zero */
    161:   VecSet(G, zero);

    163:   /* Get pointers to vector data */
    164:   VecGetArray(X, &x);
    165:   VecGetArray(G, &g);

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

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

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

    182:       if (j==0){ /* bottom side */
    183:         xb=user->bottom[i+1];
    184:         xrb = user->bottom[i+2];
    185:       } else {
    186:         xb = x[row-mx];
    187:       }

    189:       if (i+1 == mx){ /* right side */
    190:         xr=user->right[j+1];
    191:         xrb = user->right[j];
    192:       } else {
    193:         xr = x[row+1];
    194:       }

    196:       if (j+1==0+my){ /* top side */
    197:         xt=user->top[i+1];
    198:         xlt = user->top[i];
    199:       }else {
    200:         xt = x[row+mx];
    201:       }

    203:       if (i>0 && j+1<my){
    204:         xlt = x[row-1+mx];
    205:       }
    206:       if (j>0 && i+1<mx){
    207:         xrb = x[row+1-mx];
    208:       }

    210:       d1 = (xc-xl);
    211:       d2 = (xc-xr);
    212:       d3 = (xc-xt);
    213:       d4 = (xc-xb);
    214:       d5 = (xr-xrb);
    215:       d6 = (xrb-xb);
    216:       d7 = (xlt-xl);
    217:       d8 = (xt-xlt);

    219:       df1dxc = d1*hydhx;
    220:       df2dxc = ( d1*hydhx + d4*hxdhy );
    221:       df3dxc = d3*hxdhy;
    222:       df4dxc = ( d2*hydhx + d3*hxdhy );
    223:       df5dxc = d2*hydhx;
    224:       df6dxc = d4*hxdhy;

    226:       d1 /= hx;
    227:       d2 /= hx;
    228:       d3 /= hy;
    229:       d4 /= hy;
    230:       d5 /= hy;
    231:       d6 /= hx;
    232:       d7 /= hy;
    233:       d8 /= hx;

    235:       f1 = PetscSqrtScalar( 1.0 + d1*d1 + d7*d7);
    236:       f2 = PetscSqrtScalar( 1.0 + d1*d1 + d4*d4);
    237:       f3 = PetscSqrtScalar( 1.0 + d3*d3 + d8*d8);
    238:       f4 = PetscSqrtScalar( 1.0 + d3*d3 + d2*d2);
    239:       f5 = PetscSqrtScalar( 1.0 + d2*d2 + d5*d5);
    240:       f6 = PetscSqrtScalar( 1.0 + d4*d4 + d6*d6);

    242:       df1dxc /= f1;
    243:       df2dxc /= f2;
    244:       df3dxc /= f3;
    245:       df4dxc /= f4;
    246:       df5dxc /= f5;
    247:       df6dxc /= f6;

    249:       g[row] = (df1dxc+df2dxc+df3dxc+df4dxc+df5dxc+df6dxc )/2.0;
    250:     }
    251:   }

    253:   /* Restore vectors */
    254:   VecRestoreArray(X, &x);
    255:   VecRestoreArray(G, &g);
    256:   PetscLogFlops(67*mx*my);
    257:   return(0);
    258: }

    260: /* ------------------------------------------------------------------- */
    261: /*
    262:    FormJacobian - Evaluates Jacobian matrix.

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

    269:    Output Parameters:
    270: .  tH    - Jacobian matrix

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

    287:   /* Set various matrix options */
    288:   MatSetOption(H,MAT_IGNORE_OFF_PROC_ENTRIES,PETSC_TRUE);
    289:   MatAssembled(H,&assembled);
    290:   if (assembled){MatZeroEntries(H); }

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

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

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

    303:       /* Left side */
    304:       if (i==0){
    305:         xl  = user->left[j+1];
    306:         xlt = user->left[j+2];
    307:       } else {
    308:         xl = x[row-1];
    309:       }

    311:       if (j==0){
    312:         xb  = user->bottom[i+1];
    313:         xrb = user->bottom[i+2];
    314:       } else {
    315:         xb = x[row-mx];
    316:       }

    318:       if (i+1 == mx){
    319:         xr  = user->right[j+1];
    320:         xrb = user->right[j];
    321:       } else {
    322:         xr = x[row+1];
    323:       }

    325:       if (j+1==my){
    326:         xt  = user->top[i+1];
    327:         xlt = user->top[i];
    328:       }else {
    329:         xt = x[row+mx];
    330:       }

    332:       if (i>0 && j+1<my){
    333:         xlt = x[row-1+mx];
    334:       }
    335:       if (j>0 && i+1<mx){
    336:         xrb = x[row+1-mx];
    337:       }

    340:       d1 = (xc-xl)/hx;
    341:       d2 = (xc-xr)/hx;
    342:       d3 = (xc-xt)/hy;
    343:       d4 = (xc-xb)/hy;
    344:       d5 = (xrb-xr)/hy;
    345:       d6 = (xrb-xb)/hx;
    346:       d7 = (xlt-xl)/hy;
    347:       d8 = (xlt-xt)/hx;

    349:       f1 = PetscSqrtScalar( 1.0 + d1*d1 + d7*d7);
    350:       f2 = PetscSqrtScalar( 1.0 + d1*d1 + d4*d4);
    351:       f3 = PetscSqrtScalar( 1.0 + d3*d3 + d8*d8);
    352:       f4 = PetscSqrtScalar( 1.0 + d3*d3 + d2*d2);
    353:       f5 = PetscSqrtScalar( 1.0 + d2*d2 + d5*d5);
    354:       f6 = PetscSqrtScalar( 1.0 + d4*d4 + d6*d6);

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

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

    365:       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) +
    366:            (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);

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

    370:       k=0;
    371:       if (j>0){
    372:         v[k]=hb; col[k]=row - mx; k++;
    373:       }

    375:       if (j>0 && i < mx -1){
    376:         v[k]=hbr; col[k]=row - mx+1; k++;
    377:       }

    379:       if (i>0){
    380:         v[k]= hl; col[k]=row - 1; k++;
    381:       }

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

    385:       if (i < mx-1 ){
    386:         v[k]= hr; col[k]=row+1; k++;
    387:       }

    389:       if (i>0 && j < my-1 ){
    390:         v[k]= htl; col[k] = row+mx-1; k++;
    391:       }

    393:       if (j < my-1 ){
    394:         v[k]= ht; col[k] = row+mx; k++;
    395:       }

    397:       /*
    398:          Set matrix values using local numbering, which was defined
    399:          earlier, in the main routine.
    400:       */
    401:       MatSetValues(H,1,&row,k,col,v,INSERT_VALUES);
    402:     }
    403:   }

    405:   /* Restore vectors */
    406:   VecRestoreArrayRead(X,&x);

    408:   /* Assemble the matrix */
    409:   MatAssemblyBegin(H,MAT_FINAL_ASSEMBLY);
    410:   MatAssemblyEnd(H,MAT_FINAL_ASSEMBLY);
    411:   PetscLogFlops(199*mx*my);
    412:   return(0);
    413: }

    415: /* ------------------------------------------------------------------- */
    416: /*
    417:    MSA_BoundaryConditions -  Calculates the boundary conditions for
    418:    the region.

    420:    Input Parameter:
    421: .  user - user-defined application context

    423:    Output Parameter:
    424: .  user - user-defined application context
    425: */
    426: static PetscErrorCode MSA_BoundaryConditions(AppCtx * user)
    427: {
    428:   PetscErrorCode  ierr;
    429:   PetscInt        i,j,k,limit=0,maxits=5;
    430:   PetscInt        mx=user->mx,my=user->my;
    431:   PetscInt        bsize=0, lsize=0, tsize=0, rsize=0;
    432:   PetscReal       one=1.0, two=2.0, three=3.0, tol=1e-10;
    433:   PetscReal       fnorm,det,hx,hy,xt=0,yt=0;
    434:   PetscReal       u1,u2,nf1,nf2,njac11,njac12,njac21,njac22;
    435:   PetscReal       b=-0.5, t=0.5, l=-0.5, r=0.5;
    436:   PetscReal       *boundary;

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

    441:   PetscMalloc1(bsize, &user->bottom);
    442:   PetscMalloc1(tsize, &user->top);
    443:   PetscMalloc1(lsize, &user->left);
    444:   PetscMalloc1(rsize, &user->right);

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

    448:   for (j=0; j<4; j++){
    449:     if (j==0){
    450:       yt=b;
    451:       xt=l;
    452:       limit=bsize;
    453:       boundary=user->bottom;
    454:     } else if (j==1){
    455:       yt=t;
    456:       xt=l;
    457:       limit=tsize;
    458:       boundary=user->top;
    459:     } else if (j==2){
    460:       yt=b;
    461:       xt=l;
    462:       limit=lsize;
    463:       boundary=user->left;
    464:     } else { /* if  (j==3) */
    465:       yt=b;
    466:       xt=r;
    467:       limit=rsize;
    468:       boundary=user->right;
    469:     }

    471:     for (i=0; i<limit; i++){
    472:       u1=xt;
    473:       u2=-yt;
    474:       for (k=0; k<maxits; k++){
    475:         nf1=u1 + u1*u2*u2 - u1*u1*u1/three-xt;
    476:         nf2=-u2 - u1*u1*u2 + u2*u2*u2/three-yt;
    477:         fnorm=PetscSqrtScalar(nf1*nf1+nf2*nf2);
    478:         if (fnorm <= tol) break;
    479:         njac11=one+u2*u2-u1*u1;
    480:         njac12=two*u1*u2;
    481:         njac21=-two*u1*u2;
    482:         njac22=-one - u1*u1 + u2*u2;
    483:         det = njac11*njac22-njac21*njac12;
    484:         u1 = u1-(njac22*nf1-njac12*nf2)/det;
    485:         u2 = u2-(njac11*nf2-njac21*nf1)/det;
    486:       }

    488:       boundary[i]=u1*u1-u2*u2;
    489:       if (j==0 || j==1) {
    490:         xt=xt+hx;
    491:       } else { /* if (j==2 || j==3) */
    492:         yt=yt+hy;
    493:       }
    494:     }
    495:   }
    496:   return(0);
    497: }

    499: /* ------------------------------------------------------------------- */
    500: /*
    501:    MSA_InitialPoint - Calculates the initial guess in one of three ways.

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

    507:    Output Parameters:
    508: .  X - newly computed initial guess
    509: */
    510: static PetscErrorCode MSA_InitialPoint(AppCtx * user, Vec X)
    511: {
    513:   PetscInt       start=-1,i,j;
    514:   PetscScalar    zero=0.0;
    515:   PetscBool      flg;

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

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

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

    530:     /* Perform local computations */
    531:     for (j=0; j<my; j++){
    532:       for (i=0; i< mx; i++){
    533:         row=(j)*mx + (i);
    534:         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;
    535:       }
    536:     }

    538:     /* Restore vectors */
    539:     VecRestoreArray(X,&x);
    540:   }
    541:   return(0);
    542: }

    545: /*TEST

    547:    build:
    548:       requires: !complex

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

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

    558: TEST*/