Actual source code: minsurf1.c

tao-2.1-p0 2012-07-24
  1: #include "tao.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: TaoInitialize(); TaoFinalize();
 23:    Routines: TaoCreate(); TaoDestroy();

 25:    Processors: 1
 26: T*/


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


 40: /* -------- User-defined Routines --------- */

 42: static PetscErrorCode MSA_BoundaryConditions(AppCtx *);
 43: static PetscErrorCode MSA_InitialPoint(AppCtx *, Vec);
 44: PetscErrorCode FormConstraints(TaoSolver, Vec, Vec, void *);
 45: PetscErrorCode FormJacobian(TaoSolver, Vec, Mat *, Mat*, MatStructure*,void *);

 49: int main(int argc, char **argv)
 50: {
 52:   Vec          x;                 /* solution vector */
 53:   Vec          c;                 /* Constraints function vector */
 54:   Vec          xl,xu;             /* Bounds on the variables */
 55:   PetscBool    flg;               /* A return variable when checking for user options */
 56:   TaoSolver    tao;               /* TAO solver context */
 57:   Mat          J;                 /* Jacobian matrix */
 58:   PetscInt     N;                 /* Number of elements in vector */
 59:   PetscScalar lb =  TAO_NINFINITY;      /* lower bound constant */
 60:   PetscScalar ub =  TAO_INFINITY;      /* upper bound constant */
 61:   AppCtx user;                    /* user-defined work context */

 63:   /* Initialize PETSc, TAO */
 64:   PetscInitialize(&argc, &argv, (char *)0, help );
 65:   TaoInitialize(&argc, &argv, (char *)0, help );

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

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

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

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


 82:   /* Create appropriate vectors and matrices */
 83:   VecCreateSeq(MPI_COMM_SELF, N, &x);
 84:   VecDuplicate(x, &c); 
 85:   MatCreateSeqAIJ(MPI_COMM_SELF, N, N, 7, PETSC_NULL, &J); 

 87:   /* The TAO code begins here */

 89:   /* Create TAO solver and set desired solution method */
 90:   TaoCreate(PETSC_COMM_SELF,&tao); 
 91:   TaoSetType(tao,"tao_ssils"); 

 93:   /* Set data structure */
 94:   TaoSetInitialVector(tao, x); 

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

100:   /* Set the variable bounds */
101:   MSA_BoundaryConditions(&user); 

103:   /* Set initial solution guess */
104:   MSA_InitialPoint(&user, x); 

106:   /* Set Bounds on variables */
107:   VecDuplicate(x, &xl); 
108:   VecDuplicate(x, &xu); 
109:   VecSet(xl, lb); 
110:   VecSet(xu, ub); 
111:   TaoSetVariableBounds(tao,xl,xu); 

113:   /* Check for any tao command line options */
114:   TaoSetFromOptions(tao); 

116:   /* Solve the application */
117:   TaoSolve(tao);  

119:   /* Free Tao data structures */
120:   TaoDestroy(&tao); 

122:   /* Free PETSc data structures */
123:   VecDestroy(&x); 
124:   VecDestroy(&xl); 
125:   VecDestroy(&xu); 
126:   VecDestroy(&c); 
127:   MatDestroy(&J); 

129:   /* Free user-created data structures */
130:   PetscFree(user.bottom);
131:   PetscFree(user.top);
132:   PetscFree(user.left);
133:   PetscFree(user.right);

135:   /* Finalize TAO and PETSc */
136:   PetscFinalize();
137:   TaoFinalize();

139:   return 0;
140: }

142: /* -------------------------------------------------------------------- */

146: /*  FormConstraints - Evaluates gradient of f.             

148:     Input Parameters:
149: .   tao  - the TAO_APPLICATION context
150: .   X    - input vector
151: .   ptr  - optional user-defined context, as set by TaoSetConstraintsRoutine()
152:     
153:     Output Parameters:
154: .   G - vector containing the newly evaluated gradient
155: */
156: PetscErrorCode FormConstraints(TaoSolver tao, Vec X, Vec G, void *ptr){
157:   AppCtx *user = (AppCtx *) ptr;
159:   PetscInt i,j,row;
160:   PetscInt mx=user->mx, my=user->my;
161:   PetscReal hx=1.0/(mx+1),hy=1.0/(my+1), hydhx=hy/hx, hxdhy=hx/hy;
162:   PetscReal f1,f2,f3,f4,f5,f6,d1,d2,d3,d4,d5,d6,d7,d8,xc,xl,xr,xt,xb,xlt,xrb;
163:   PetscReal df1dxc,df2dxc,df3dxc,df4dxc,df5dxc,df6dxc;
164:   PetscScalar zero=0.0;
165:   PetscScalar *g, *x;


169:   /* Initialize vector to zero */
170:   VecSet(G, zero); 

172:   /* Get pointers to vector data */
173:   VecGetArray(X, &x); 
174:   VecGetArray(G, &g); 

176:   /* Compute function over the locally owned part of the mesh */
177:   for (j=0; j<my; j++){
178:     for (i=0; i< mx; i++){
179:       row= j*mx + i;
180:       
181:       xc = x[row];
182:       xlt=xrb=xl=xr=xb=xt=xc;
183:       
184:       if (i==0){ /* left side */
185:         xl= user->left[j+1];
186:         xlt = user->left[j+2];
187:       } else {
188:         xl = x[row-1];
189:       }

191:       if (j==0){ /* bottom side */
192:         xb=user->bottom[i+1];
193:         xrb = user->bottom[i+2];
194:       } else {
195:         xb = x[row-mx];
196:       }
197:       
198:       if (i+1 == mx){ /* right side */
199:         xr=user->right[j+1];
200:         xrb = user->right[j];
201:       } else {
202:         xr = x[row+1];
203:       }

205:       if (j+1==0+my){ /* top side */
206:         xt=user->top[i+1];
207:         xlt = user->top[i];
208:       }else {
209:         xt = x[row+mx];
210:       }

212:       if (i>0 && j+1<my){
213:         xlt = x[row-1+mx];
214:       }
215:       if (j>0 && i+1<mx){
216:         xrb = x[row+1-mx];
217:       }

219:       d1 = (xc-xl);
220:       d2 = (xc-xr);
221:       d3 = (xc-xt);
222:       d4 = (xc-xb);
223:       d5 = (xr-xrb);
224:       d6 = (xrb-xb);
225:       d7 = (xlt-xl);
226:       d8 = (xt-xlt);
227:       
228:       df1dxc = d1*hydhx;
229:       df2dxc = ( d1*hydhx + d4*hxdhy );
230:       df3dxc = d3*hxdhy;
231:       df4dxc = ( d2*hydhx + d3*hxdhy );
232:       df5dxc = d2*hydhx;
233:       df6dxc = d4*hxdhy;

235:       d1 /= hx;
236:       d2 /= hx;
237:       d3 /= hy;
238:       d4 /= hy;
239:       d5 /= hy;
240:       d6 /= hx;
241:       d7 /= hy;
242:       d8 /= hx;

244:       f1 = PetscSqrtScalar( 1.0 + d1*d1 + d7*d7);
245:       f2 = PetscSqrtScalar( 1.0 + d1*d1 + d4*d4);
246:       f3 = PetscSqrtScalar( 1.0 + d3*d3 + d8*d8);
247:       f4 = PetscSqrtScalar( 1.0 + d3*d3 + d2*d2);
248:       f5 = PetscSqrtScalar( 1.0 + d2*d2 + d5*d5);
249:       f6 = PetscSqrtScalar( 1.0 + d4*d4 + d6*d6);
250:       
251:       df1dxc /= f1;
252:       df2dxc /= f2;
253:       df3dxc /= f3;
254:       df4dxc /= f4;
255:       df5dxc /= f5;
256:       df6dxc /= f6;

258:       g[row] = (df1dxc+df2dxc+df3dxc+df4dxc+df5dxc+df6dxc )/2.0;
259:       
260:     }
261:   }
262:   
263:   /* Restore vectors */
264:   VecRestoreArray(X, &x); 
265:   VecRestoreArray(G, &g); 
266:   PetscLogFlops(67*mx*my); 
267:   return(0);
268: }

270: /* ------------------------------------------------------------------- */
273: /*
274:    FormJacobian - Evaluates Jacobian matrix.

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

281:    Output Parameters:
282: .  tH    - Jacobian matrix

284: */
285: PetscErrorCode FormJacobian(TaoSolver tao, Vec X, Mat *tH, Mat* tHPre, MatStructure* flag, void *ptr)
286: { 
287:   AppCtx *user = (AppCtx *) ptr;
288:   Mat H = *tH;
290:   PetscInt    i,j,k,row;
291:   PetscInt    mx=user->mx, my=user->my;
292:   PetscInt    col[7];
293:   PetscReal hx=1.0/(mx+1), hy=1.0/(my+1), hydhx=hy/hx, hxdhy=hx/hy;
294:   PetscReal f1,f2,f3,f4,f5,f6,d1,d2,d3,d4,d5,d6,d7,d8,xc,xl,xr,xt,xb,xlt,xrb;
295:   PetscReal hl,hr,ht,hb,hc,htl,hbr;
296:   PetscScalar *x, v[7];
297:   PetscBool  assembled;

299:   /* Set various matrix options */
300:   MatSetOption(H,MAT_IGNORE_OFF_PROC_ENTRIES,PETSC_TRUE); 
301:   MatAssembled(H,&assembled); 
302:   if (assembled){MatZeroEntries(H);  }
303:   *flag=SAME_NONZERO_PATTERN;

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

308:   /* Compute Jacobian over the locally owned part of the mesh */
309:   for (i=0; i< mx; i++){
310:     for (j=0; j<my; j++){
311:       row= j*mx + i;
312:       
313:       xc = x[row]; 
314:       xlt=xrb=xl=xr=xb=xt=xc;

316:       /* Left side */
317:       if (i==0){
318:         xl= user->left[j+1];
319:         xlt = user->left[j+2];
320:       } else {
321:         xl = x[row-1];
322:       }
323:       
324:       if (j==0){
325:         xb=user->bottom[i+1];
326:         xrb = user->bottom[i+2];
327:       } else {
328:         xb = x[row-mx];
329:       }
330:       
331:       if (i+1 == mx){
332:         xr=user->right[j+1];
333:         xrb = user->right[j];
334:       } else {
335:         xr = x[row+1];
336:       }

338:       if (j+1==my){
339:         xt=user->top[i+1];
340:         xlt = user->top[i];
341:       }else {
342:         xt = x[row+mx];
343:       }

345:       if (i>0 && j+1<my){
346:         xlt = x[row-1+mx];
347:       }
348:       if (j>0 && i+1<mx){
349:         xrb = x[row+1-mx];
350:       }


353:       d1 = (xc-xl)/hx;
354:       d2 = (xc-xr)/hx;
355:       d3 = (xc-xt)/hy;
356:       d4 = (xc-xb)/hy;
357:       d5 = (xrb-xr)/hy;
358:       d6 = (xrb-xb)/hx;
359:       d7 = (xlt-xl)/hy;
360:       d8 = (xlt-xt)/hx;
361:       
362:       f1 = PetscSqrtScalar( 1.0 + d1*d1 + d7*d7);
363:       f2 = PetscSqrtScalar( 1.0 + d1*d1 + d4*d4);
364:       f3 = PetscSqrtScalar( 1.0 + d3*d3 + d8*d8);
365:       f4 = PetscSqrtScalar( 1.0 + d3*d3 + d2*d2);
366:       f5 = PetscSqrtScalar( 1.0 + d2*d2 + d5*d5);
367:       f6 = PetscSqrtScalar( 1.0 + d4*d4 + d6*d6);


370:       hl = (-hydhx*(1.0+d7*d7)+d1*d7)/(f1*f1*f1)+
371:         (-hydhx*(1.0+d4*d4)+d1*d4)/(f2*f2*f2);
372:       hr = (-hydhx*(1.0+d5*d5)+d2*d5)/(f5*f5*f5)+
373:         (-hydhx*(1.0+d3*d3)+d2*d3)/(f4*f4*f4);
374:       ht = (-hxdhy*(1.0+d8*d8)+d3*d8)/(f3*f3*f3)+
375:         (-hxdhy*(1.0+d2*d2)+d2*d3)/(f4*f4*f4);
376:       hb = (-hxdhy*(1.0+d6*d6)+d4*d6)/(f6*f6*f6)+
377:         (-hxdhy*(1.0+d1*d1)+d1*d4)/(f2*f2*f2);

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

382:       hc = hydhx*(1.0+d7*d7)/(f1*f1*f1) + hxdhy*(1.0+d8*d8)/(f3*f3*f3) +
383:         hydhx*(1.0+d5*d5)/(f5*f5*f5) + hxdhy*(1.0+d6*d6)/(f6*f6*f6) +
384:         (hxdhy*(1.0+d1*d1)+hydhx*(1.0+d4*d4)-2*d1*d4)/(f2*f2*f2) +
385:         (hxdhy*(1.0+d2*d2)+hydhx*(1.0+d3*d3)-2*d2*d3)/(f4*f4*f4);

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

389:       k=0;
390:       if (j>0){ 
391:         v[k]=hb; col[k]=row - mx; k++;
392:       }
393:       
394:       if (j>0 && i < mx -1){
395:         v[k]=hbr; col[k]=row - mx+1; k++;
396:       }
397:       
398:       if (i>0){
399:         v[k]= hl; col[k]=row - 1; k++;
400:       }
401:       
402:       v[k]= hc; col[k]=row; k++;
403:       
404:       if (i < mx-1 ){
405:         v[k]= hr; col[k]=row+1; k++;
406:       }
407:       
408:       if (i>0 && j < my-1 ){
409:         v[k]= htl; col[k] = row+mx-1; k++;
410:       }
411:       
412:       if (j < my-1 ){
413:         v[k]= ht; col[k] = row+mx; k++;
414:       }
415:       
416:       /* 
417:          Set matrix values using local numbering, which was defined
418:          earlier, in the main routine.
419:       */
420:       MatSetValues(H,1,&row,k,col,v,INSERT_VALUES); 
421:       
422:     }
423:   }
424:   
425:   /* Restore vectors */
426:   VecRestoreArray(X,&x); 

428:   /* Assemble the matrix */
429:   MatAssemblyBegin(H,MAT_FINAL_ASSEMBLY); 
430:   MatAssemblyEnd(H,MAT_FINAL_ASSEMBLY); 
431:   PetscLogFlops(199*mx*my); 
432:   return(0);
433: }

435: /* ------------------------------------------------------------------- */
438: /* 
439:    MSA_BoundaryConditions -  Calculates the boundary conditions for
440:    the region.

442:    Input Parameter:
443: .  user - user-defined application context

445:    Output Parameter:
446: .  user - user-defined application context
447: */
448: static PetscErrorCode MSA_BoundaryConditions(AppCtx * user)
449: {
450:   PetscErrorCode  ierr;
451:   PetscInt        i,j,k,limit=0,maxits=5;
452:   PetscInt        mx=user->mx,my=user->my;
453:   PetscInt        bsize=0, lsize=0, tsize=0, rsize=0;
454:   PetscReal     one=1.0, two=2.0, three=3.0, tol=1e-10;
455:   PetscReal     fnorm,det,hx,hy,xt=0,yt=0;
456:   PetscReal     u1,u2,nf1,nf2,njac11,njac12,njac21,njac22;
457:   PetscReal     b=-0.5, t=0.5, l=-0.5, r=0.5;
458:   PetscReal     *boundary;

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

463:   PetscMalloc(bsize*sizeof(PetscReal), &user->bottom);
464:   PetscMalloc(tsize*sizeof(PetscReal), &user->top);
465:   PetscMalloc(lsize*sizeof(PetscReal), &user->left);
466:   PetscMalloc(rsize*sizeof(PetscReal), &user->right);

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

470:   for (j=0; j<4; j++){
471:     if (j==0){
472:       yt=b;
473:       xt=l;
474:       limit=bsize;
475:       boundary=user->bottom;
476:     } else if (j==1){
477:       yt=t;
478:       xt=l;
479:       limit=tsize;
480:       boundary=user->top;
481:     } else if (j==2){
482:       yt=b;
483:       xt=l;
484:       limit=lsize;
485:       boundary=user->left;
486:     } else { /* if  (j==3) */
487:       yt=b;
488:       xt=r;
489:       limit=rsize;
490:       boundary=user->right;
491:     }

493:     for (i=0; i<limit; i++){
494:       u1=xt;
495:       u2=-yt;
496:       for (k=0; k<maxits; k++){
497:         nf1=u1 + u1*u2*u2 - u1*u1*u1/three-xt;
498:         nf2=-u2 - u1*u1*u2 + u2*u2*u2/three-yt;
499:         fnorm=PetscSqrtScalar(nf1*nf1+nf2*nf2);
500:         if (fnorm <= tol) break;
501:         njac11=one+u2*u2-u1*u1;
502:         njac12=two*u1*u2;
503:         njac21=-two*u1*u2;
504:         njac22=-one - u1*u1 + u2*u2;
505:         det = njac11*njac22-njac21*njac12;
506:         u1 = u1-(njac22*nf1-njac12*nf2)/det;
507:         u2 = u2-(njac11*nf2-njac21*nf1)/det;
508:       }

510:       boundary[i]=u1*u1-u2*u2;
511:       if (j==0 || j==1) {
512:         xt=xt+hx;
513:       } else { /* if (j==2 || j==3) */
514:         yt=yt+hy;
515:       }
516:     }
517:   }

519:   return(0);
520: }

522: /* ------------------------------------------------------------------- */
525: /*
526:    MSA_InitialPoint - Calculates the initial guess in one of three ways. 

528:    Input Parameters:
529: .  user - user-defined application context
530: .  X - vector for initial guess

532:    Output Parameters:
533: .  X - newly computed initial guess
534: */
535: static PetscErrorCode MSA_InitialPoint(AppCtx * user, Vec X)
536: {
538:   PetscInt      start=-1,i,j;
539:   PetscScalar   zero=0.0;
540:   PetscBool     flg;

543:   PetscOptionsGetInt(PETSC_NULL,"-start",&start,&flg); 

545:   if (flg && start==0){ /* The zero vector is reasonable */
546:  
547:     VecSet(X, zero); 
548:     /* PLogInfo(user,"Min. Surface Area Problem: Start with 0 vector \n"); */


551:   } else { /* Take an average of the boundary conditions */

553:     PetscInt    row;
554:     PetscInt    mx=user->mx,my=user->my;
555:     PetscScalar *x;
556:     
557:     /* Get pointers to vector data */
558:     VecGetArray(X,&x); 

560:     /* Perform local computations */    
561:     for (j=0; j<my; j++){
562:       for (i=0; i< mx; i++){
563:         row=(j)*mx + (i);
564:         x[row] = ( ((j+1)*user->bottom[i+1]+(my-j+1)*user->top[i+1])/(my+2)+
565:                    ((i+1)*user->left[j+1]+(mx-i+1)*user->right[j+1])/(mx+2))/2.0;
566:       }
567:     }
568:     
569:     /* Restore vectors */
570:     VecRestoreArray(X,&x); 
571:     
572:   }
573:   return(0);
574: }