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: /*
 21:    User-defined application context - contains data needed by the
 22:    application-provided call-back routines, FormFunctionGradient(),
 23:    FormHessian().
 24: */
 25: typedef struct {
 26:   PetscInt  mx, my;
 27:   PetscReal *bottom, *top, *left, *right;
 28: } AppCtx;

 30: /* -------- User-defined Routines --------- */

 32: static PetscErrorCode MSA_BoundaryConditions(AppCtx *);
 33: static PetscErrorCode MSA_InitialPoint(AppCtx *, Vec);
 34: PetscErrorCode FormConstraints(Tao, Vec, Vec, void *);
 35: PetscErrorCode FormJacobian(Tao, Vec, Mat, Mat, void *);

 37: int main(int argc, char **argv)
 38: {
 39:   Vec            x;                 /* solution vector */
 40:   Vec            c;                 /* Constraints function vector */
 41:   Vec            xl,xu;             /* Bounds on the variables */
 42:   PetscBool      flg;               /* A return variable when checking for user options */
 43:   Tao            tao;               /* TAO solver context */
 44:   Mat            J;                 /* Jacobian matrix */
 45:   PetscInt       N;                 /* Number of elements in vector */
 46:   PetscScalar    lb =  PETSC_NINFINITY;      /* lower bound constant */
 47:   PetscScalar    ub =  PETSC_INFINITY;      /* upper bound constant */
 48:   AppCtx         user;                    /* user-defined work context */

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

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

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

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

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

 66:   /* Create appropriate vectors and matrices */
 67:   VecCreateSeq(MPI_COMM_SELF, N, &x);
 68:   VecDuplicate(x, &c);
 69:   MatCreateSeqAIJ(MPI_COMM_SELF, N, N, 7, NULL, &J);

 71:   /* The TAO code begins here */

 73:   /* Create TAO solver and set desired solution method */
 74:   TaoCreate(PETSC_COMM_SELF,&tao);
 75:   TaoSetType(tao,TAOSSILS);

 77:   /* Set data structure */
 78:   TaoSetSolution(tao, x);

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

 84:   /* Set the variable bounds */
 85:   MSA_BoundaryConditions(&user);

 87:   /* Set initial solution guess */
 88:   MSA_InitialPoint(&user, x);

 90:   /* Set Bounds on variables */
 91:   VecDuplicate(x, &xl);
 92:   VecDuplicate(x, &xu);
 93:   VecSet(xl, lb);
 94:   VecSet(xu, ub);
 95:   TaoSetVariableBounds(tao,xl,xu);

 97:   /* Check for any tao command line options */
 98:   TaoSetFromOptions(tao);

100:   /* Solve the application */
101:   TaoSolve(tao);

103:   /* Free Tao data structures */
104:   TaoDestroy(&tao);

106:   /* Free PETSc data structures */
107:   VecDestroy(&x);
108:   VecDestroy(&xl);
109:   VecDestroy(&xu);
110:   VecDestroy(&c);
111:   MatDestroy(&J);

113:   /* Free user-created data structures */
114:   PetscFree(user.bottom);
115:   PetscFree(user.top);
116:   PetscFree(user.left);
117:   PetscFree(user.right);

119:   PetscFinalize();
120:   return 0;
121: }

123: /* -------------------------------------------------------------------- */

125: /*  FormConstraints - Evaluates gradient of f.

127:     Input Parameters:
128: .   tao  - the TAO_APPLICATION context
129: .   X    - input vector
130: .   ptr  - optional user-defined context, as set by TaoSetConstraintsRoutine()

132:     Output Parameters:
133: .   G - vector containing the newly evaluated gradient
134: */
135: PetscErrorCode FormConstraints(Tao tao, Vec X, Vec G, void *ptr)
136: {
137:   AppCtx         *user = (AppCtx *) ptr;
138:   PetscInt       i,j,row;
139:   PetscInt       mx=user->mx, my=user->my;
140:   PetscReal      hx=1.0/(mx+1),hy=1.0/(my+1), hydhx=hy/hx, hxdhy=hx/hy;
141:   PetscReal      f1,f2,f3,f4,f5,f6,d1,d2,d3,d4,d5,d6,d7,d8,xc,xl,xr,xt,xb,xlt,xrb;
142:   PetscReal      df1dxc,df2dxc,df3dxc,df4dxc,df5dxc,df6dxc;
143:   PetscScalar    zero=0.0;
144:   PetscScalar    *g, *x;

146:   /* Initialize vector to zero */
147:   VecSet(G, zero);

149:   /* Get pointers to vector data */
150:   VecGetArray(X, &x);
151:   VecGetArray(G, &g);

153:   /* Compute function over the locally owned part of the mesh */
154:   for (j=0; j<my; j++) {
155:     for (i=0; i< mx; i++) {
156:       row= j*mx + i;

158:       xc = x[row];
159:       xlt=xrb=xl=xr=xb=xt=xc;

161:       if (i==0) { /* left side */
162:         xl= user->left[j+1];
163:         xlt = user->left[j+2];
164:       } else {
165:         xl = x[row-1];
166:       }

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

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

182:       if (j+1==0+my) { /* top side */
183:         xt=user->top[i+1];
184:         xlt = user->top[i];
185:       }else {
186:         xt = x[row+mx];
187:       }

189:       if (i>0 && j+1<my) {
190:         xlt = x[row-1+mx];
191:       }
192:       if (j>0 && i+1<mx) {
193:         xrb = x[row+1-mx];
194:       }

196:       d1 = (xc-xl);
197:       d2 = (xc-xr);
198:       d3 = (xc-xt);
199:       d4 = (xc-xb);
200:       d5 = (xr-xrb);
201:       d6 = (xrb-xb);
202:       d7 = (xlt-xl);
203:       d8 = (xt-xlt);

205:       df1dxc = d1*hydhx;
206:       df2dxc = (d1*hydhx + d4*hxdhy);
207:       df3dxc = d3*hxdhy;
208:       df4dxc = (d2*hydhx + d3*hxdhy);
209:       df5dxc = d2*hydhx;
210:       df6dxc = d4*hxdhy;

212:       d1 /= hx;
213:       d2 /= hx;
214:       d3 /= hy;
215:       d4 /= hy;
216:       d5 /= hy;
217:       d6 /= hx;
218:       d7 /= hy;
219:       d8 /= hx;

221:       f1 = PetscSqrtScalar(1.0 + d1*d1 + d7*d7);
222:       f2 = PetscSqrtScalar(1.0 + d1*d1 + d4*d4);
223:       f3 = PetscSqrtScalar(1.0 + d3*d3 + d8*d8);
224:       f4 = PetscSqrtScalar(1.0 + d3*d3 + d2*d2);
225:       f5 = PetscSqrtScalar(1.0 + d2*d2 + d5*d5);
226:       f6 = PetscSqrtScalar(1.0 + d4*d4 + d6*d6);

228:       df1dxc /= f1;
229:       df2dxc /= f2;
230:       df3dxc /= f3;
231:       df4dxc /= f4;
232:       df5dxc /= f5;
233:       df6dxc /= f6;

235:       g[row] = (df1dxc+df2dxc+df3dxc+df4dxc+df5dxc+df6dxc)/2.0;
236:     }
237:   }

239:   /* Restore vectors */
240:   VecRestoreArray(X, &x);
241:   VecRestoreArray(G, &g);
242:   PetscLogFlops(67*mx*my);
243:   return 0;
244: }

246: /* ------------------------------------------------------------------- */
247: /*
248:    FormJacobian - Evaluates Jacobian matrix.

250:    Input Parameters:
251: .  tao  - the TAO_APPLICATION context
252: .  X    - input vector
253: .  ptr  - optional user-defined context, as set by TaoSetJacobian()

255:    Output Parameters:
256: .  tH    - Jacobian matrix

258: */
259: PetscErrorCode FormJacobian(Tao tao, Vec X, Mat H, Mat tHPre, void *ptr)
260: {
261:   AppCtx            *user = (AppCtx *) ptr;
262:   PetscInt          i,j,k,row;
263:   PetscInt          mx=user->mx, my=user->my;
264:   PetscInt          col[7];
265:   PetscReal         hx=1.0/(mx+1), hy=1.0/(my+1), hydhx=hy/hx, hxdhy=hx/hy;
266:   PetscReal         f1,f2,f3,f4,f5,f6,d1,d2,d3,d4,d5,d6,d7,d8,xc,xl,xr,xt,xb,xlt,xrb;
267:   PetscReal         hl,hr,ht,hb,hc,htl,hbr;
268:   const PetscScalar *x;
269:   PetscScalar       v[7];
270:   PetscBool         assembled;

272:   /* Set various matrix options */
273:   MatSetOption(H,MAT_IGNORE_OFF_PROC_ENTRIES,PETSC_TRUE);
274:   MatAssembled(H,&assembled);
275:   if (assembled) MatZeroEntries(H);

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

280:   /* Compute Jacobian over the locally owned part of the mesh */
281:   for (i=0; i< mx; i++) {
282:     for (j=0; j<my; j++) {
283:       row= j*mx + i;

285:       xc = x[row];
286:       xlt=xrb=xl=xr=xb=xt=xc;

288:       /* Left side */
289:       if (i==0) {
290:         xl  = user->left[j+1];
291:         xlt = user->left[j+2];
292:       } else {
293:         xl = x[row-1];
294:       }

296:       if (j==0) {
297:         xb  = user->bottom[i+1];
298:         xrb = user->bottom[i+2];
299:       } else {
300:         xb = x[row-mx];
301:       }

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

310:       if (j+1==my) {
311:         xt  = user->top[i+1];
312:         xlt = user->top[i];
313:       }else {
314:         xt = x[row+mx];
315:       }

317:       if (i>0 && j+1<my) {
318:         xlt = x[row-1+mx];
319:       }
320:       if (j>0 && i+1<mx) {
321:         xrb = x[row+1-mx];
322:       }

324:       d1 = (xc-xl)/hx;
325:       d2 = (xc-xr)/hx;
326:       d3 = (xc-xt)/hy;
327:       d4 = (xc-xb)/hy;
328:       d5 = (xrb-xr)/hy;
329:       d6 = (xrb-xb)/hx;
330:       d7 = (xlt-xl)/hy;
331:       d8 = (xlt-xt)/hx;

333:       f1 = PetscSqrtScalar(1.0 + d1*d1 + d7*d7);
334:       f2 = PetscSqrtScalar(1.0 + d1*d1 + d4*d4);
335:       f3 = PetscSqrtScalar(1.0 + d3*d3 + d8*d8);
336:       f4 = PetscSqrtScalar(1.0 + d3*d3 + d2*d2);
337:       f5 = PetscSqrtScalar(1.0 + d2*d2 + d5*d5);
338:       f6 = PetscSqrtScalar(1.0 + d4*d4 + d6*d6);

340:       hl = (-hydhx*(1.0+d7*d7)+d1*d7)/(f1*f1*f1)+(-hydhx*(1.0+d4*d4)+d1*d4)/(f2*f2*f2);
341:       hr = (-hydhx*(1.0+d5*d5)+d2*d5)/(f5*f5*f5)+(-hydhx*(1.0+d3*d3)+d2*d3)/(f4*f4*f4);
342:       ht = (-hxdhy*(1.0+d8*d8)+d3*d8)/(f3*f3*f3)+(-hxdhy*(1.0+d2*d2)+d2*d3)/(f4*f4*f4);
343:       hb = (-hxdhy*(1.0+d6*d6)+d4*d6)/(f6*f6*f6)+(-hxdhy*(1.0+d1*d1)+d1*d4)/(f2*f2*f2);

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

348:       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) +
349:            (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);

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

353:       k=0;
354:       if (j>0) {
355:         v[k]=hb; col[k]=row - mx; k++;
356:       }

358:       if (j>0 && i < mx -1) {
359:         v[k]=hbr; col[k]=row - mx+1; k++;
360:       }

362:       if (i>0) {
363:         v[k]= hl; col[k]=row - 1; k++;
364:       }

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

368:       if (i < mx-1) {
369:         v[k]= hr; col[k]=row+1; k++;
370:       }

372:       if (i>0 && j < my-1) {
373:         v[k]= htl; col[k] = row+mx-1; k++;
374:       }

376:       if (j < my-1) {
377:         v[k]= ht; col[k] = row+mx; k++;
378:       }

380:       /*
381:          Set matrix values using local numbering, which was defined
382:          earlier, in the main routine.
383:       */
384:       MatSetValues(H,1,&row,k,col,v,INSERT_VALUES);
385:     }
386:   }

388:   /* Restore vectors */
389:   VecRestoreArrayRead(X,&x);

391:   /* Assemble the matrix */
392:   MatAssemblyBegin(H,MAT_FINAL_ASSEMBLY);
393:   MatAssemblyEnd(H,MAT_FINAL_ASSEMBLY);
394:   PetscLogFlops(199*mx*my);
395:   return 0;
396: }

398: /* ------------------------------------------------------------------- */
399: /*
400:    MSA_BoundaryConditions -  Calculates the boundary conditions for
401:    the region.

403:    Input Parameter:
404: .  user - user-defined application context

406:    Output Parameter:
407: .  user - user-defined application context
408: */
409: static PetscErrorCode MSA_BoundaryConditions(AppCtx * user)
410: {
411:   PetscInt        i,j,k,limit=0,maxits=5;
412:   PetscInt        mx=user->mx,my=user->my;
413:   PetscInt        bsize=0, lsize=0, tsize=0, rsize=0;
414:   PetscReal       one=1.0, two=2.0, three=3.0, tol=1e-10;
415:   PetscReal       fnorm,det,hx,hy,xt=0,yt=0;
416:   PetscReal       u1,u2,nf1,nf2,njac11,njac12,njac21,njac22;
417:   PetscReal       b=-0.5, t=0.5, l=-0.5, r=0.5;
418:   PetscReal       *boundary;

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

422:   PetscMalloc1(bsize, &user->bottom);
423:   PetscMalloc1(tsize, &user->top);
424:   PetscMalloc1(lsize, &user->left);
425:   PetscMalloc1(rsize, &user->right);

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

429:   for (j=0; j<4; j++) {
430:     if (j==0) {
431:       yt=b;
432:       xt=l;
433:       limit=bsize;
434:       boundary=user->bottom;
435:     } else if (j==1) {
436:       yt=t;
437:       xt=l;
438:       limit=tsize;
439:       boundary=user->top;
440:     } else if (j==2) {
441:       yt=b;
442:       xt=l;
443:       limit=lsize;
444:       boundary=user->left;
445:     } else { /* if  (j==3) */
446:       yt=b;
447:       xt=r;
448:       limit=rsize;
449:       boundary=user->right;
450:     }

452:     for (i=0; i<limit; i++) {
453:       u1=xt;
454:       u2=-yt;
455:       for (k=0; k<maxits; k++) {
456:         nf1=u1 + u1*u2*u2 - u1*u1*u1/three-xt;
457:         nf2=-u2 - u1*u1*u2 + u2*u2*u2/three-yt;
458:         fnorm=PetscSqrtScalar(nf1*nf1+nf2*nf2);
459:         if (fnorm <= tol) break;
460:         njac11=one+u2*u2-u1*u1;
461:         njac12=two*u1*u2;
462:         njac21=-two*u1*u2;
463:         njac22=-one - u1*u1 + u2*u2;
464:         det = njac11*njac22-njac21*njac12;
465:         u1 = u1-(njac22*nf1-njac12*nf2)/det;
466:         u2 = u2-(njac11*nf2-njac21*nf1)/det;
467:       }

469:       boundary[i]=u1*u1-u2*u2;
470:       if (j==0 || j==1) {
471:         xt=xt+hx;
472:       } else { /* if (j==2 || j==3) */
473:         yt=yt+hy;
474:       }
475:     }
476:   }
477:   return 0;
478: }

480: /* ------------------------------------------------------------------- */
481: /*
482:    MSA_InitialPoint - Calculates the initial guess in one of three ways.

484:    Input Parameters:
485: .  user - user-defined application context
486: .  X - vector for initial guess

488:    Output Parameters:
489: .  X - newly computed initial guess
490: */
491: static PetscErrorCode MSA_InitialPoint(AppCtx * user, Vec X)
492: {
493:   PetscInt       start=-1,i,j;
494:   PetscScalar    zero=0.0;
495:   PetscBool      flg;

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

499:   if (flg && start==0) { /* The zero vector is reasonable */
500:     VecSet(X, zero);
501:   } else { /* Take an average of the boundary conditions */
502:     PetscInt    row;
503:     PetscInt    mx=user->mx,my=user->my;
504:     PetscScalar *x;

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

509:     /* Perform local computations */
510:     for (j=0; j<my; j++) {
511:       for (i=0; i< mx; i++) {
512:         row=(j)*mx + (i);
513:         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;
514:       }
515:     }

517:     /* Restore vectors */
518:     VecRestoreArray(X,&x);
519:   }
520:   return 0;
521: }

523: /*TEST

525:    build:
526:       requires: !complex

528:    test:
529:       args: -tao_monitor -tao_view -tao_type ssils -tao_gttol 1.e-5
530:       requires: !single

532:    test:
533:       suffix: 2
534:       args: -tao_monitor -tao_view -tao_type ssfls -tao_gttol 1.e-5

536: TEST*/