Actual source code: minsurf2.c

  1: /*$Id$*/

  3: /* Program usage: mpirun -np <proc> minsurf2 [-help] [all TAO options] */

  5: /*
  6:   Include "tao.h" so we can use TAO solvers.
  7:   petscda.h for distributed array
  8: */
  9: #include "petscda.h"
 10: #include "tao.h"

 12: static  char help[] = 
 13: "This example demonstrates use of the TAO package to \n\
 14: solve an unconstrained minimization problem.  This example is based on a \n\
 15: problem from the MINPACK-2 test suite.  Given a rectangular 2-D domain and \n\
 16: boundary values along the edges of the domain, the objective is to find the\n\
 17: surface with the minimal area that satisfies the boundary conditions.\n\
 18: The command line options are:\n\
 19:   -mx <xg>, where <xg> = number of grid points in the 1st coordinate direction\n\
 20:   -my <yg>, where <yg> = number of grid points in the 2nd coordinate direction\n\
 21:   -start <st>, where <st> =0 for zero vector, <st> >0 for random start, and <st> <0 \n\
 22:                for an average of the boundary conditions\n\n";

 24: /*T
 25:    Concepts: TAO - Solving an unconstrained minimization problem
 26:    Routines: TaoInitialize(); TaoFinalize();
 27:    Routines: TaoCreate(); TaoDestroy();
 28:    Routines: TaoApplicationCreate(); TaoAppDestroy();
 29:    Routines: TaoAppSetInitialSolutionVec();
 30:    Routines: TaoAppSetObjectiveAndGradientRoutine();
 31:    Routines: TaoAppSetHessianMat(); TaoAppSetHessianRoutine();
 32:    Routines: TaoSetOptions();
 33:    Routines: TaoAppGetKSP(); TaoSolveApplication();
 34:    Routines: TaoAppSetMonitor(); TaoView();
 35:    Routines: TaoAppGetSolutionVec();
 36:    Processors: 1
 37: T*/

 39: /* 
 40:    User-defined application context - contains data needed by the 
 41:    application-provided call-back routines, FormFunctionGradient() 
 42:    and FormHessian().
 43: */
 44: typedef struct {
 45:   int         mx, my;                 /* discretization in x, y directions */
 46:   double      *bottom, *top, *left, *right;             /* boundary values */
 47:   DA          da;                      /* distributed array data structure */
 48:   Mat         H;                       /* Hessian */
 49:   ISColoring  iscoloring;
 50: } AppCtx;


 53: /* -------- User-defined Routines --------- */

 55: static int MSA_BoundaryConditions(AppCtx*);
 56: static int MSA_InitialPoint(AppCtx*,Vec);
 57: int QuadraticH(AppCtx*,Vec,Mat);
 58: int FormFunctionGradient(TAO_APPLICATION,Vec,double *,Vec,void*);
 59: int FormGradient(TAO_APPLICATION,Vec,Vec,void*);
 60: int FormHessian(TAO_APPLICATION,Vec,Mat*,Mat*,MatStructure *,void*);
 61: int My_Monitor(TAO_APPLICATION, void *);

 65: int main( int argc, char **argv )
 66: {
 67:   int             info;                /* used to check for functions returning nonzeros */
 68:   int             Nx, Ny;              /* number of processors in x- and y- directions */
 69:   int             iter;                /* iteration information */
 70:   double          ff,gnorm;
 71:   Vec             x;                   /* solution, gradient vectors */
 72:   PetscTruth      flg, viewmat;        /* flags */
 73:   PetscTruth      fddefault, fdcoloring;   /* flags */
 74:   KSP             ksp;                 /* Krylov subspace method */
 75:   TaoMethod       method = "tao_nls";  /* minimization method */
 76:   TaoTerminateReason reason;           
 77:   TAO_SOLVER      tao;                 /* TAO_SOLVER solver context */
 78:   TAO_APPLICATION minsurfapp;          /* The PETSc application */
 79:   AppCtx          user;                /* user-defined work context */

 81:   /* Initialize TAO */
 82:   PetscInitialize( &argc, &argv,(char *)0,help );
 83:   TaoInitialize( &argc, &argv,(char *)0,help );

 85:   /* Specify dimension of the problem */
 86:   user.mx = 10; user.my = 10;

 88:   /* Check for any command line arguments that override defaults */
 89:   info = PetscOptionsGetInt(PETSC_NULL,"-mx",&user.mx,&flg); CHKERRQ(info);
 90:   info = PetscOptionsGetInt(PETSC_NULL,"-my",&user.my,&flg); CHKERRQ(info);

 92:   PetscPrintf(MPI_COMM_WORLD,"\n---- Minimum Surface Area Problem -----\n");
 93:   PetscPrintf(MPI_COMM_WORLD,"mx: %d     my: %d   \n\n",user.mx,user.my);


 96:   /* Let PETSc determine the vector distribution */
 97:   Nx = PETSC_DECIDE; Ny = PETSC_DECIDE;

 99:   /* Create distributed array (DA) to manage parallel grid and vectors  */
100:   info = DACreate2d(PETSC_COMM_WORLD,DA_NONPERIODIC,DA_STENCIL_BOX,user.mx,
101:                     user.my,Nx,Ny,1,1,PETSC_NULL,PETSC_NULL,&user.da); CHKERRQ(info);
102:   

104:   /* Create TAO solver and set desired solution method. Create an TAO application structure */
105:   info = TaoCreate(PETSC_COMM_WORLD,method,&tao); CHKERRQ(info);
106:   info = TaoApplicationCreate(PETSC_COMM_WORLD,&minsurfapp); CHKERRQ(info);

108:   /*
109:      Extract global vector from DA for the vector of variables --  PETSC routine
110:      Compute the initial solution                              --  application specific, see below
111:      Set this vector for use by TAO                            --  TAO routine
112:   */
113:   info = DACreateGlobalVector(user.da,&x); CHKERRQ(info);
114:   info = MSA_BoundaryConditions(&user); CHKERRQ(info);         
115:   info = MSA_InitialPoint(&user,x); CHKERRQ(info);
116:   info = TaoAppSetInitialSolutionVec(minsurfapp,x); CHKERRQ(info);

118:   /* 
119:      Initialize the Application context for use in function evaluations  --  application specific, see below.
120:      Set routines for function and gradient evaluation 
121:   */
122:   info = TaoAppSetObjectiveAndGradientRoutine(minsurfapp,FormFunctionGradient,(void *)&user); CHKERRQ(info);

124:   /* 
125:      Given the command line arguments, calculate the hessian with either the user-
126:      provided function FormHessian, or the default finite-difference driven Hessian
127:      functions 
128:   */
129:   info = PetscOptionsHasName(PETSC_NULL,"-tao_fddefault",&fddefault);CHKERRQ(info);
130:   info = PetscOptionsHasName(PETSC_NULL,"-tao_fdcoloring",&fdcoloring);CHKERRQ(info);

132:   if (fdcoloring) {
133:     info = DAGetColoring(user.da,IS_COLORING_GLOBAL,&(user.iscoloring));
134:     CHKERRQ(info);
135:     info = TaoAppSetColoring(minsurfapp, user.iscoloring); CHKERRQ(info);
136:     info = TaoAppSetHessianRoutine(minsurfapp,TaoAppDefaultComputeHessianColor,(void *)&user); CHKERRQ(info);
137:   } else if (fddefault){
138:     info = TaoAppSetHessianRoutine(minsurfapp,TaoAppDefaultComputeHessian,(void *)&user); CHKERRQ(info);
139:   } else {
140:     info = TaoAppSetHessianRoutine(minsurfapp,FormHessian,(void *)&user); CHKERRQ(info);
141:   }

143:   /* 
144:      Create a matrix data structure to store the Hessian and set 
145:      the Hessian evalution routine.
146:      Set the matrix structure to be used for Hessian evalutions
147:   */
148:   info = DAGetMatrix(user.da,MATAIJ,&user.H);CHKERRQ(info);
149:   info = MatSetOption(user.H,MAT_SYMMETRIC); CHKERRQ(info);

151:   info = TaoAppSetHessianMat(minsurfapp,user.H,user.H); CHKERRQ(info);

153:   /* 
154:      If my_monitor option is in command line, then use the user-provided
155:      monitoring function
156:   */
157:   info = PetscOptionsHasName(PETSC_NULL,"-my_monitor",&viewmat); CHKERRQ(info);
158:   if (viewmat){
159:     info = TaoAppSetMonitor(minsurfapp,My_Monitor,TAO_NULL); CHKERRQ(info);
160:   }

162:   /* Check for any tao command line options */
163:   info = TaoSetOptions(minsurfapp,tao); CHKERRQ(info);

165:   /* Limit the number of iterations in the KSP linear solver */
166:   info = TaoAppGetKSP(minsurfapp,&ksp); CHKERRQ(info);
167:   if (ksp) {                                              /* Modify the PETSc KSP structure */
168:     info = KSPSetTolerances(ksp,PETSC_DEFAULT,PETSC_DEFAULT,PETSC_DEFAULT,user.mx*user.my);
169:     CHKERRQ(info);
170:   }

172:   /* SOLVE THE APPLICATION */
173:   info = TaoSolveApplication(minsurfapp,tao);  CHKERRQ(info);

175:   /* Get information on termination */
176:   info = TaoGetSolutionStatus(tao,&iter,&ff,&gnorm,0,0,&reason); CHKERRQ(info);
177:   if (reason <= 0 ){
178:     PetscPrintf(MPI_COMM_WORLD,"Try a different TAO method, adjust some parameters, or check the function evaluation routines\n");
179:     PetscPrintf(MPI_COMM_WORLD," Iterations: %d,  Function Value: %4.2e, Residual: %4.2e \n",iter,ff,gnorm);
180:   }

182:   /* 
183:      To View TAO solver information use
184:       info = TaoView(tao); CHKERRQ(info); 
185:   */

187:   /* Free TAO data structures */
188:   info = TaoDestroy(tao); CHKERRQ(info);
189:   info = TaoAppDestroy(minsurfapp); CHKERRQ(info);

191:   /* Free PETSc data structures */
192:   info = VecDestroy(x); CHKERRQ(info);
193:   info = MatDestroy(user.H); CHKERRQ(info);
194:   if (fdcoloring) {
195:     info = ISColoringDestroy(user.iscoloring);CHKERRQ(info);
196:   }
197:   info = PetscFree(user.bottom); CHKERRQ(info);
198:   info = PetscFree(user.top); CHKERRQ(info);
199:   info = PetscFree(user.left); CHKERRQ(info);
200:   info = PetscFree(user.right); CHKERRQ(info);
201:   info = DADestroy(user.da); CHKERRQ(info);

203:   /* Finalize TAO */
204:   TaoFinalize();
205:   PetscFinalize();
206:   
207:   return 0;
208: }

212: int FormGradient(TAO_APPLICATION taoapp, Vec X, Vec G,void *userCtx){
213:   int info;
214:   double fcn;
215:   TaoFunctionBegin;
216:   info = FormFunctionGradient(taoapp,X,&fcn,G,userCtx);CHKERRQ(info);
217:   TaoFunctionReturn(0);
218: }

220: /* -------------------------------------------------------------------- */
223: /*  FormFunctionGradient - Evaluates the function and corresponding gradient.

225:     Input Parameters:
226: .   taoapp     - the TAO_APPLICATION context
227: .   XX      - input vector
228: .   userCtx - optional user-defined context, as set by TaoSetFunctionGradient()
229:     
230:     Output Parameters:
231: .   fcn     - the newly evaluated function
232: .   GG       - vector containing the newly evaluated gradient
233: */
234: int FormFunctionGradient(TAO_APPLICATION taoapp, Vec X, double *fcn,Vec G,void *userCtx){

236:   AppCtx * user = (AppCtx *) userCtx;
237:   int    info,i,j;
238:   int    mx=user->mx, my=user->my;
239:   int    xs,xm,gxs,gxm,ys,ym,gys,gym;
240:   double ft=0;
241:   double hx=1.0/(mx+1),hy=1.0/(my+1), hydhx=hy/hx, hxdhy=hx/hy, area=0.5*hx*hy;
242:   double rhx=mx+1, rhy=my+1;
243:   double f1,f2,f3,f4,f5,f6,d1,d2,d3,d4,d5,d6,d7,d8,xc,xl,xr,xt,xb,xlt,xrb;
244:   double df1dxc,df2dxc,df3dxc,df4dxc,df5dxc,df6dxc;
245:   PetscScalar **g, **x;
246:   Vec    localX;

248:   /* Get local mesh boundaries */
249:   info = DAGetLocalVector(user->da,&localX);CHKERRQ(info);

251:   info = DAGetCorners(user->da,&xs,&ys,PETSC_NULL,&xm,&ym,PETSC_NULL); CHKERRQ(info);
252:   info = DAGetGhostCorners(user->da,&gxs,&gys,PETSC_NULL,&gxm,&gym,PETSC_NULL); CHKERRQ(info);

254:   /* Scatter ghost points to local vector */
255:   info = DAGlobalToLocalBegin(user->da,X,INSERT_VALUES,localX); CHKERRQ(info);
256:   info = DAGlobalToLocalEnd(user->da,X,INSERT_VALUES,localX); CHKERRQ(info);

258:   /* Get pointers to vector data */
259:   info = DAVecGetArray(user->da,localX,(void**)&x);
260:   info = DAVecGetArray(user->da,G,(void**)&g);

262:   /* Compute function and gradient over the locally owned part of the mesh */
263:   for (j=ys; j<ys+ym; j++){
264:     for (i=xs; i< xs+xm; i++){
265:       
266:       xc = x[j][i];
267:       xlt=xrb=xl=xr=xb=xt=xc;
268:       
269:       if (i==0){ /* left side */
270:         xl= user->left[j-ys+1];
271:         xlt = user->left[j-ys+2];
272:       } else {
273:         xl = x[j][i-1];
274:       }

276:       if (j==0){ /* bottom side */
277:         xb=user->bottom[i-xs+1];
278:         xrb =user->bottom[i-xs+2];
279:       } else {
280:         xb = x[j-1][i];
281:       }
282:       
283:       if (i+1 == gxs+gxm){ /* right side */
284:         xr=user->right[j-ys+1];
285:         xrb = user->right[j-ys];
286:       } else {
287:         xr = x[j][i+1];
288:       }

290:       if (j+1==gys+gym){ /* top side */
291:         xt=user->top[i-xs+1];
292:         xlt = user->top[i-xs];
293:       }else {
294:         xt = x[j+1][i];
295:       }

297:       if (i>gxs && j+1<gys+gym){
298:         xlt = x[j+1][i-1];
299:       }
300:       if (j>gys && i+1<gxs+gxm){
301:         xrb = x[j-1][i+1];
302:       }

304:       d1 = (xc-xl);
305:       d2 = (xc-xr);
306:       d3 = (xc-xt);
307:       d4 = (xc-xb);
308:       d5 = (xr-xrb);
309:       d6 = (xrb-xb);
310:       d7 = (xlt-xl);
311:       d8 = (xt-xlt);
312:       
313:       df1dxc = d1*hydhx;
314:       df2dxc = ( d1*hydhx + d4*hxdhy );
315:       df3dxc = d3*hxdhy;
316:       df4dxc = ( d2*hydhx + d3*hxdhy );
317:       df5dxc = d2*hydhx;
318:       df6dxc = d4*hxdhy;

320:       d1 *= rhx;
321:       d2 *= rhx;
322:       d3 *= rhy;
323:       d4 *= rhy;
324:       d5 *= rhy;
325:       d6 *= rhx;
326:       d7 *= rhy;
327:       d8 *= rhx;

329:       f1 = sqrt( 1.0 + d1*d1 + d7*d7);
330:       f2 = sqrt( 1.0 + d1*d1 + d4*d4);
331:       f3 = sqrt( 1.0 + d3*d3 + d8*d8);
332:       f4 = sqrt( 1.0 + d3*d3 + d2*d2);
333:       f5 = sqrt( 1.0 + d2*d2 + d5*d5);
334:       f6 = sqrt( 1.0 + d4*d4 + d6*d6);
335:       
336:       f2 = sqrt( 1.0 + d1*d1 + d4*d4);
337:       f4 = sqrt( 1.0 + d3*d3 + d2*d2);

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

341:       df1dxc /= f1;
342:       df2dxc /= f2;
343:       df3dxc /= f3;
344:       df4dxc /= f4;
345:       df5dxc /= f5;
346:       df6dxc /= f6;

348:       g[j][i] = (df1dxc+df2dxc+df3dxc+df4dxc+df5dxc+df6dxc ) * 0.5;
349:       
350:     }
351:   }

353:   /* Compute triangular areas along the border of the domain. */
354:   if (xs==0){ /* left side */
355:     for (j=ys; j<ys+ym; j++){
356:       d3=(user->left[j-ys+1] - user->left[j-ys+2])*rhy;
357:       d2=(user->left[j-ys+1] - x[j][0]) *rhx;
358:       ft = ft+sqrt( 1.0 + d3*d3 + d2*d2);
359:     }
360:   }
361:   if (ys==0){ /* bottom side */
362:     for (i=xs; i<xs+xm; i++){
363:       d2=(user->bottom[i+1-xs]-user->bottom[i-xs+2])*rhx;
364:       d3=(user->bottom[i-xs+1]-x[0][i])*rhy;
365:       ft = ft+sqrt( 1.0 + d3*d3 + d2*d2);
366:     }
367:   }

369:   if (xs+xm==mx){ /* right side */
370:     for (j=ys; j< ys+ym; j++){
371:       d1=(x[j][mx-1] - user->right[j-ys+1])*rhx;
372:       d4=(user->right[j-ys]-user->right[j-ys+1])*rhy;
373:       ft = ft+sqrt( 1.0 + d1*d1 + d4*d4);
374:     }
375:   }
376:   if (ys+ym==my){ /* top side */
377:     for (i=xs; i<xs+xm; i++){
378:       d1=(x[my-1][i] - user->top[i-xs+1])*rhy;
379:       d4=(user->top[i-xs+1] - user->top[i-xs])*rhx;
380:       ft = ft+sqrt( 1.0 + d1*d1 + d4*d4);
381:     }
382:   }

384:   if (ys==0 && xs==0){
385:     d1=(user->left[0]-user->left[1])/hy;
386:     d2=(user->bottom[0]-user->bottom[1])*rhx;
387:     ft +=sqrt( 1.0 + d1*d1 + d2*d2);
388:   }
389:   if (ys+ym == my && xs+xm == mx){
390:     d1=(user->right[ym+1] - user->right[ym])*rhy;
391:     d2=(user->top[xm+1] - user->top[xm])*rhx;
392:     ft +=sqrt( 1.0 + d1*d1 + d2*d2);
393:   }

395:   ft=ft*area;
396:   info = MPI_Allreduce(&ft,fcn,1,MPI_DOUBLE,MPI_SUM,MPI_COMM_WORLD);CHKERRQ(info);

398:   /* Restore vectors */
399:   info = DAVecRestoreArray(user->da,localX,(void**)&x);
400:   info = DAVecRestoreArray(user->da,G,(void**)&g);

402:   /* Scatter values to global vector */
403:   info = DARestoreLocalVector(user->da,&localX); CHKERRQ(info);

405:   info = PetscLogFlops(67*xm*ym); CHKERRQ(info);

407:   return 0;
408: }

410: /* ------------------------------------------------------------------- */
413: /*
414:    FormHessian - Evaluates Hessian matrix.

416:    Input Parameters:
417: .  taoapp  - the TAO_APPLICATION context
418: .  x    - input vector
419: .  ptr  - optional user-defined context, as set by TaoSetHessian()

421:    Output Parameters:
422: .  H    - Hessian matrix
423: .  Hpre - optionally different preconditioning matrix
424: .  flg  - flag indicating matrix structure

426: */
427: int FormHessian(TAO_APPLICATION taoapp,Vec X,Mat *H, Mat *Hpre, MatStructure *flg, void *ptr)
428: { 
429:   int    info;
430:   AppCtx *user = (AppCtx *) ptr;

432:   /* Evaluate the Hessian entries*/
433:   info = QuadraticH(user,X,*H); CHKERRQ(info);


436:   /* Indicate that this matrix has the same sparsity pattern during
437:      successive iterations; setting this flag can save significant work
438:      in computing the preconditioner for some methods. */
439:   *flg=SAME_NONZERO_PATTERN;

441:   return 0;
442: }

444: /* ------------------------------------------------------------------- */
447: /*
448:    QuadraticH - Evaluates Hessian matrix.

450:    Input Parameters:
451: .  user - user-defined context, as set by TaoSetHessian()
452: .  X    - input vector

454:    Output Parameter:
455: .  H    - Hessian matrix
456: */
457: int QuadraticH(AppCtx *user, Vec X, Mat Hessian)
458: {
459:   int    i,j,k,info;
460:   int    mx=user->mx, my=user->my;
461:   int    xs,xm,gxs,gxm,ys,ym,gys,gym;
462:   double hx=1.0/(mx+1), hy=1.0/(my+1), hydhx=hy/hx, hxdhy=hx/hy;
463:   double f1,f2,f3,f4,f5,f6,d1,d2,d3,d4,d5,d6,d7,d8,xc,xl,xr,xt,xb,xlt,xrb;
464:   double hl,hr,ht,hb,hc,htl,hbr;
465:   PetscScalar **x, v[7];
466:   MatStencil col[7],row;
467:   Vec    localX;
468:   PetscTruth assembled;

470:   /* Get local mesh boundaries */
471:   info = DAGetLocalVector(user->da,&localX);CHKERRQ(info);

473:   info = DAGetCorners(user->da,&xs,&ys,PETSC_NULL,&xm,&ym,PETSC_NULL); CHKERRQ(info);
474:   info = DAGetGhostCorners(user->da,&gxs,&gys,PETSC_NULL,&gxm,&gym,PETSC_NULL); CHKERRQ(info);

476:   /* Scatter ghost points to local vector */
477:   info = DAGlobalToLocalBegin(user->da,X,INSERT_VALUES,localX); CHKERRQ(info);
478:   info = DAGlobalToLocalEnd(user->da,X,INSERT_VALUES,localX); CHKERRQ(info);

480:   /* Get pointers to vector data */
481:   info = DAVecGetArray(user->da,localX,(void**)&x);

483:   /* Initialize matrix entries to zero */
484:   info = MatAssembled(Hessian,&assembled); CHKERRQ(info);
485:   if (assembled){info = MatZeroEntries(Hessian);  CHKERRQ(info);}


488:   /* Set various matrix options */
489:   info = MatSetOption(Hessian,MAT_IGNORE_OFF_PROC_ENTRIES); CHKERRQ(info);
490:   info = MatSetOption(Hessian,MAT_COLUMNS_SORTED); CHKERRQ(info);
491:   info = MatSetOption(Hessian,MAT_ROWS_SORTED); CHKERRQ(info);

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

495:   for (j=ys; j<ys+ym; j++){
496:       
497:     for (i=xs; i< xs+xm; i++){

499:       xc = x[j][i];
500:       xlt=xrb=xl=xr=xb=xt=xc;

502:       /* Left side */
503:       if (i==0){
504:         xl  = user->left[j-ys+1];
505:         xlt = user->left[j-ys+2];
506:       } else {
507:         xl  = x[j][i-1];
508:       }
509:       
510:       if (j==0){
511:         xb  = user->bottom[i-xs+1];
512:         xrb = user->bottom[i-xs+2];
513:       } else {
514:         xb  = x[j-1][i];
515:       }
516:       
517:       if (i+1 == mx){
518:         xr  = user->right[j-ys+1];
519:         xrb = user->right[j-ys];
520:       } else {
521:         xr  = x[j][i+1];
522:       }

524:       if (j+1==my){
525:         xt  = user->top[i-xs+1];
526:         xlt = user->top[i-xs];
527:       }else {
528:         xt  = x[j+1][i];
529:       }

531:       if (i>0 && j+1<my){
532:         xlt = x[j+1][i-1];
533:       }
534:       if (j>0 && i+1<mx){
535:         xrb = x[j-1][i+1];
536:       }


539:       d1 = (xc-xl)/hx;
540:       d2 = (xc-xr)/hx;
541:       d3 = (xc-xt)/hy;
542:       d4 = (xc-xb)/hy;
543:       d5 = (xrb-xr)/hy;
544:       d6 = (xrb-xb)/hx;
545:       d7 = (xlt-xl)/hy;
546:       d8 = (xlt-xt)/hx;
547:       
548:       f1 = sqrt( 1.0 + d1*d1 + d7*d7);
549:       f2 = sqrt( 1.0 + d1*d1 + d4*d4);
550:       f3 = sqrt( 1.0 + d3*d3 + d8*d8);
551:       f4 = sqrt( 1.0 + d3*d3 + d2*d2);
552:       f5 = sqrt( 1.0 + d2*d2 + d5*d5);
553:       f6 = sqrt( 1.0 + d4*d4 + d6*d6);


556:       hl = (-hydhx*(1.0+d7*d7)+d1*d7)/(f1*f1*f1)+
557:         (-hydhx*(1.0+d4*d4)+d1*d4)/(f2*f2*f2);
558:       hr = (-hydhx*(1.0+d5*d5)+d2*d5)/(f5*f5*f5)+
559:         (-hydhx*(1.0+d3*d3)+d2*d3)/(f4*f4*f4);
560:       ht = (-hxdhy*(1.0+d8*d8)+d3*d8)/(f3*f3*f3)+
561:         (-hxdhy*(1.0+d2*d2)+d2*d3)/(f4*f4*f4);
562:       hb = (-hxdhy*(1.0+d6*d6)+d4*d6)/(f6*f6*f6)+
563:         (-hxdhy*(1.0+d1*d1)+d1*d4)/(f2*f2*f2);

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

568:       hc = hydhx*(1.0+d7*d7)/(f1*f1*f1) + hxdhy*(1.0+d8*d8)/(f3*f3*f3) +
569:         hydhx*(1.0+d5*d5)/(f5*f5*f5) + hxdhy*(1.0+d6*d6)/(f6*f6*f6) +
570:         (hxdhy*(1.0+d1*d1)+hydhx*(1.0+d4*d4)-2*d1*d4)/(f2*f2*f2) +
571:         (hxdhy*(1.0+d2*d2)+hydhx*(1.0+d3*d3)-2*d2*d3)/(f4*f4*f4);

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

575:       row.j = j; row.i = i;
576:       k=0;
577:       if (j>0){ 
578:         v[k]=hb;
579:         col[k].j = j - 1; col[k].i = i;
580:         k++;
581:       }
582:       
583:       if (j>0 && i < mx -1){
584:         v[k]=hbr;
585:         col[k].j = j - 1; col[k].i = i+1;
586:         k++;
587:       }
588:       
589:       if (i>0){
590:         v[k]= hl;
591:         col[k].j = j; col[k].i = i-1;
592:         k++;
593:       }
594:       
595:       v[k]= hc;
596:       col[k].j = j; col[k].i = i;
597:       k++;
598:       
599:       if (i < mx-1 ){
600:         v[k]= hr; 
601:         col[k].j = j; col[k].i = i+1;
602:         k++;
603:       }
604:       
605:       if (i>0 && j < my-1 ){
606:         v[k]= htl;
607:         col[k].j = j+1; col[k].i = i-1;
608:         k++;
609:       }
610:       
611:       if (j < my-1 ){
612:         v[k]= ht; 
613:         col[k].j = j+1; col[k].i = i;
614:         k++;
615:       }
616:       
617:       /* 
618:          Set matrix values using local numbering, which was defined
619:          earlier, in the main routine.
620:       */
621:       info = MatSetValuesStencil(Hessian,1,&row,k,col,v,INSERT_VALUES);
622:       CHKERRQ(info);
623:       
624:     }
625:   }
626:   
627:   /* Restore vectors */
628:   info = DAVecRestoreArray(user->da,localX,(void**)&x);

630:   info = DARestoreLocalVector(user->da,&localX); CHKERRQ(info);

632:   /* Assemble the matrix */
633:   info = MatAssemblyBegin(Hessian,MAT_FINAL_ASSEMBLY); CHKERRQ(info);
634:   info = MatAssemblyEnd(Hessian,MAT_FINAL_ASSEMBLY); CHKERRQ(info);

636:   info = PetscLogFlops(199*xm*ym); CHKERRQ(info);
637:   return 0;
638: }

640: /* ------------------------------------------------------------------- */
643: /* 
644:    MSA_BoundaryConditions -  Calculates the boundary conditions for
645:    the region.

647:    Input Parameter:
648: .  user - user-defined application context

650:    Output Parameter:
651: .  user - user-defined application context
652: */
653: static int MSA_BoundaryConditions(AppCtx * user)
654: {
655:   int        i,j,k,limit=0,info,maxits=5;
656:   int        xs,ys,xm,ym,gxs,gys,gxm,gym;
657:   int        mx=user->mx,my=user->my;
658:   int        bsize=0, lsize=0, tsize=0, rsize=0;
659:   double     one=1.0, two=2.0, three=3.0, tol=1e-10;
660:   double     fnorm,det,hx,hy,xt=0,yt=0;
661:   double     u1,u2,nf1,nf2,njac11,njac12,njac21,njac22;
662:   double     b=-0.5, t=0.5, l=-0.5, r=0.5;
663:   double     *boundary;
664:   PetscTruth   flg;

666:   /* Get local mesh boundaries */
667:   info = DAGetCorners(user->da,&xs,&ys,PETSC_NULL,&xm,&ym,PETSC_NULL); CHKERRQ(info);
668:   info = DAGetGhostCorners(user->da,&gxs,&gys,PETSC_NULL,&gxm,&gym,PETSC_NULL); CHKERRQ(info);

670:   bsize=xm+2;
671:   lsize=ym+2;
672:   rsize=ym+2;
673:   tsize=xm+2;

675:   info = PetscMalloc(bsize*sizeof(double),&user->bottom); CHKERRQ(info);
676:   info = PetscMalloc(tsize*sizeof(double),&user->top); CHKERRQ(info);
677:   info = PetscMalloc(lsize*sizeof(double),&user->left); CHKERRQ(info);
678:   info = PetscMalloc(rsize*sizeof(double),&user->right); CHKERRQ(info);

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

682:   for (j=0; j<4; j++){
683:     if (j==0){
684:       yt=b;
685:       xt=l+hx*xs;
686:       limit=bsize;
687:       boundary=user->bottom;
688:     } else if (j==1){
689:       yt=t;
690:       xt=l+hx*xs;
691:       limit=tsize;
692:       boundary=user->top;
693:     } else if (j==2){
694:       yt=b+hy*ys;
695:       xt=l;
696:       limit=lsize;
697:       boundary=user->left;
698:     } else { //if (j==3)
699:       yt=b+hy*ys;
700:       xt=r;
701:       limit=rsize;
702:       boundary=user->right;
703:     }

705:     for (i=0; i<limit; i++){
706:       u1=xt;
707:       u2=-yt;
708:       for (k=0; k<maxits; k++){
709:         nf1=u1 + u1*u2*u2 - u1*u1*u1/three-xt;
710:         nf2=-u2 - u1*u1*u2 + u2*u2*u2/three-yt;
711:         fnorm=sqrt(nf1*nf1+nf2*nf2);
712:         if (fnorm <= tol) break;
713:         njac11=one+u2*u2-u1*u1;
714:         njac12=two*u1*u2;
715:         njac21=-two*u1*u2;
716:         njac22=-one - u1*u1 + u2*u2;
717:         det = njac11*njac22-njac21*njac12;
718:         u1 = u1-(njac22*nf1-njac12*nf2)/det;
719:         u2 = u2-(njac11*nf2-njac21*nf1)/det;
720:       }

722:       boundary[i]=u1*u1-u2*u2;
723:       if (j==0 || j==1) {
724:         xt=xt+hx;
725:       } else { // if (j==2 || j==3)
726:         yt=yt+hy;
727:       }
728:       
729:     }

731:   }

733:   /* Scale the boundary if desired */
734:   if (1==1){ 
735:     PetscReal scl = 1.0;

737:     info = PetscOptionsGetReal(PETSC_NULL,"-bottom",&scl,&flg); 
738:     CHKERRQ(info);
739:     if (flg){
740:       for (i=0;i<bsize;i++) user->bottom[i]*=scl;
741:     }

743:     info = PetscOptionsGetReal(PETSC_NULL,"-top",&scl,&flg); 
744:     CHKERRQ(info);
745:     if (flg){
746:       for (i=0;i<tsize;i++) user->top[i]*=scl;
747:     }

749:     info = PetscOptionsGetReal(PETSC_NULL,"-right",&scl,&flg); 
750:     CHKERRQ(info);
751:     if (flg){
752:       for (i=0;i<rsize;i++) user->right[i]*=scl;
753:     }

755:     info = PetscOptionsGetReal(PETSC_NULL,"-left",&scl,&flg); 
756:     CHKERRQ(info);
757:     if (flg){
758:       for (i=0;i<lsize;i++) user->left[i]*=scl;
759:     }
760:   }
761:   
762:   return 0;
763: }

765: /* ------------------------------------------------------------------- */
768: /*
769:    MSA_InitialPoint - Calculates the initial guess in one of three ways. 

771:    Input Parameters:
772: .  user - user-defined application context
773: .  X - vector for initial guess

775:    Output Parameters:
776: .  X - newly computed initial guess
777: */
778: static int MSA_InitialPoint(AppCtx * user, Vec X)
779: {
780:   int      start2=-1,i,j,info;
781:   PetscReal   start1=0;
782:   PetscTruth flg1,flg2;

784:   info = PetscOptionsGetReal(PETSC_NULL,"-start",&start1,&flg1); CHKERRQ(info);
785:   info = PetscOptionsGetInt(PETSC_NULL,"-random",&start2,&flg2); CHKERRQ(info);

787:   if (flg1){ /* The zero vector is reasonable */
788:  
789:     info = VecSet(X, start1); CHKERRQ(info);

791:   } else if (flg2 && start2>0){ /* Try a random start between -0.5 and 0.5 */

793:     PetscRandom rctx;  PetscScalar np5=-0.5;

795:     info = PetscRandomCreate(PETSC_COMM_WORLD,&rctx);
796:     CHKERRQ(info);
797:     for (i=0; i<start2; i++){
798:       info = VecSetRandom(X, rctx); CHKERRQ(info);
799:     }
800:     info = PetscRandomDestroy(rctx); CHKERRQ(info);
801:     info = VecShift(X, np5); CHKERRQ(info);

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

805:     int    xs,xm,ys,ym;
806:     int    mx=user->mx,my=user->my;
807:     PetscScalar **x;
808:     
809:     /* Get local mesh boundaries */
810:     info = DAGetCorners(user->da,&xs,&ys,PETSC_NULL,&xm,&ym,PETSC_NULL); CHKERRQ(info);
811:     
812:     /* Get pointers to vector data */
813:     info = DAVecGetArray(user->da,X,(void**)&x);

815:     /* Perform local computations */    
816:     for (j=ys; j<ys+ym; j++){
817:       for (i=xs; i< xs+xm; i++){
818:         x[j][i] = ( ((j+1)*user->bottom[i-xs+1]+(my-j+1)*user->top[i-xs+1])/(my+2)+
819:                    ((i+1)*user->left[j-ys+1]+(mx-i+1)*user->right[j-ys+1])/(mx+2))/2.0; 
820:       }
821:     }
822:     
823:     /* Restore vectors */
824:     info = DAVecRestoreArray(user->da,X,(void**)&x);  CHKERRQ(info);

826:     info = PetscLogFlops(9*xm*ym); CHKERRQ(info);
827:     
828:   }
829:   return 0;
830: }

832: /*-----------------------------------------------------------------------*/
835: int My_Monitor(TAO_APPLICATION minsurfapp, void *ctx){
836:   int info;
837:   Vec X;

839:   info = TaoAppGetSolutionVec(minsurfapp,&X); CHKERRQ(info);
840:   info = VecView(X,PETSC_VIEWER_STDOUT_WORLD); CHKERRQ(info);
841:   return 0;
842: }