Actual source code: plate2.c

petsc-dev 2014-07-25
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  1: #include <petscdmda.h>
  2: #include <petsctao.h>

  4: static  char help[] =
  5: "This example demonstrates use of the TAO package to \n\
  6: solve an unconstrained minimization problem.  This example is based on a \n\
  7: problem from the MINPACK-2 test suite.  Given a rectangular 2-D domain, \n\
  8: boundary values along the edges of the domain, and a plate represented by \n\
  9: lower boundary conditions, the objective is to find the\n\
 10: surface with the minimal area that satisfies the boundary conditions.\n\
 11: The command line options are:\n\
 12:   -mx <xg>, where <xg> = number of grid points in the 1st coordinate direction\n\
 13:   -my <yg>, where <yg> = number of grid points in the 2nd coordinate direction\n\
 14:   -bmx <bxg>, where <bxg> = number of grid points under plate in 1st direction\n\
 15:   -bmy <byg>, where <byg> = number of grid points under plate in 2nd direction\n\
 16:   -bheight <ht>, where <ht> = height of the plate\n\
 17:   -start <st>, where <st> =0 for zero vector, <st> >0 for random start, and <st> <0 \n\
 18:                for an average of the boundary conditions\n\n";

 20: /*T
 21:    Concepts: TAO^Solving a bound constrained minimization problem
 22:    Routines: TaoCreate();
 23:    Routines: TaoSetType(); TaoSetObjectiveAndGradientRoutine();
 24:    Routines: TaoSetHessianRoutine();
 25:    Routines: TaoSetInitialVector();
 26:    Routines: TaoSetVariableBounds();
 27:    Routines: TaoSetFromOptions();
 28:    Routines: TaoSolve(); TaoView();
 29:    Routines: TaoGetConvergedReason(); TaoDestroy();
 30:    Processors: n
 31: T*/


 34: /*
 35:    User-defined application context - contains data needed by the
 36:    application-provided call-back routines, FormFunctionGradient(),
 37:    FormHessian().
 38: */
 39: typedef struct {
 40:   /* problem parameters */
 41:   PetscReal      bheight;                  /* Height of plate under the surface */
 42:   PetscInt       mx, my;                   /* discretization in x, y directions */
 43:   PetscInt       bmx,bmy;                  /* Size of plate under the surface */
 44:   Vec            Bottom, Top, Left, Right; /* boundary values */

 46:   /* Working space */
 47:   Vec         localX, localV;           /* ghosted local vector */
 48:   DM          dm;                       /* distributed array data structure */
 49:   Mat         H;
 50: } AppCtx;

 52: /* -------- User-defined Routines --------- */

 54: static PetscErrorCode MSA_BoundaryConditions(AppCtx*);
 55: static PetscErrorCode MSA_InitialPoint(AppCtx*,Vec);
 56: static PetscErrorCode MSA_Plate(Vec,Vec,void*);
 57: PetscErrorCode FormFunctionGradient(Tao,Vec,PetscReal*,Vec,void*);
 58: PetscErrorCode FormHessian(Tao,Vec,Mat,Mat,void*);

 60: /* For testing matrix free submatrices */
 61: PetscErrorCode MatrixFreeHessian(Tao,Vec,Mat, Mat,void*);
 62: PetscErrorCode MyMatMult(Mat,Vec,Vec);

 66: int main( int argc, char **argv )
 67: {
 68:   PetscErrorCode         ierr;                 /* used to check for functions returning nonzeros */
 69:   PetscInt               Nx, Ny;               /* number of processors in x- and y- directions */
 70:   PetscInt               m, N;                 /* number of local and global elements in vectors */
 71:   Vec                    x,xl,xu;               /* solution vector  and bounds*/
 72:   PetscBool              flg;                /* A return variable when checking for user options */
 73:   Tao                    tao;                  /* Tao solver context */
 74:   ISLocalToGlobalMapping isltog;   /* local-to-global mapping object */
 75:   TaoConvergedReason     reason;
 76:   Mat                    H_shell;                  /* to test matrix-free submatrices */
 77:   AppCtx                 user;                 /* user-defined work context */

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

 82:   /* Specify default dimension of the problem */
 83:   user.mx = 10; user.my = 10; user.bheight=0.1;

 85:   /* Check for any command line arguments that override defaults */
 86:   PetscOptionsGetInt(NULL,"-mx",&user.mx,&flg);
 87:   PetscOptionsGetInt(NULL,"-my",&user.my,&flg);
 88:   PetscOptionsGetReal(NULL,"-bheight",&user.bheight,&flg);

 90:   user.bmx = user.mx/2; user.bmy = user.my/2;
 91:   PetscOptionsGetInt(NULL,"-bmx",&user.bmx,&flg);
 92:   PetscOptionsGetInt(NULL,"-bmy",&user.bmy,&flg);

 94:   PetscPrintf(PETSC_COMM_WORLD,"\n---- Minimum Surface Area With Plate Problem -----\n");
 95:   PetscPrintf(PETSC_COMM_WORLD,"mx:%D, my:%D, bmx:%D, bmy:%D, height:%g\n",user.mx,user.my,user.bmx,user.bmy,(double)user.bheight);

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

100:   /* Let Petsc determine the dimensions of the local vectors */
101:   Nx = PETSC_DECIDE; Ny = PETSC_DECIDE;

103:   /*
104:      A two dimensional distributed array will help define this problem,
105:      which derives from an elliptic PDE on two dimensional domain.  From
106:      the distributed array, Create the vectors.
107:   */
108:   DMDACreate2d(MPI_COMM_WORLD,DM_BOUNDARY_NONE,DM_BOUNDARY_NONE,
109:                       DMDA_STENCIL_BOX,user.mx,user.my,Nx,Ny,1,1,
110:                       NULL,NULL,&user.dm);

112:   /*
113:      Extract global and local vectors from DM; The local vectors are
114:      used solely as work space for the evaluation of the function,
115:      gradient, and Hessian.  Duplicate for remaining vectors that are
116:      the same types.
117:   */
118:   DMCreateGlobalVector(user.dm,&x); /* Solution */
119:   DMCreateLocalVector(user.dm,&user.localX);
120:   VecDuplicate(user.localX,&user.localV);

122:   VecDuplicate(x,&xl);
123:   VecDuplicate(x,&xu);



127:   /* The TAO code begins here */

129:   /*
130:      Create TAO solver and set desired solution method
131:      The method must either be TAOTRON or TAOBLMVM
132:      If TAOBLMVM is used, then hessian function is not called.
133:   */
134:   TaoCreate(PETSC_COMM_WORLD,&tao);
135:   TaoSetType(tao,TAOBLMVM);

137:   /* Set initial solution guess; */
138:   MSA_BoundaryConditions(&user);
139:   MSA_InitialPoint(&user,x);
140:   TaoSetInitialVector(tao,x);

142:   /* Set routines for function, gradient and hessian evaluation */
143:   TaoSetObjectiveAndGradientRoutine(tao,FormFunctionGradient,(void*) &user);

145:   VecGetLocalSize(x,&m);
146:   MatCreateAIJ(MPI_COMM_WORLD,m,m,N,N,7,NULL,3,NULL,&(user.H));
147:   MatSetOption(user.H,MAT_SYMMETRIC,PETSC_TRUE);

149:   DMGetLocalToGlobalMapping(user.dm,&isltog);
150:   MatSetLocalToGlobalMapping(user.H,isltog,isltog);
151:   PetscOptionsHasName(NULL,"-matrixfree",&flg);
152:   if (flg) {
153:       MatCreateShell(PETSC_COMM_WORLD,m,m,N,N,(void*)&user,&H_shell);
154:       MatShellSetOperation(H_shell,MATOP_MULT,(void(*)(void))MyMatMult);
155:       MatSetOption(H_shell,MAT_SYMMETRIC,PETSC_TRUE);
156:       TaoSetHessianRoutine(tao,H_shell,H_shell,MatrixFreeHessian,(void*)&user);
157:   } else {
158:       TaoSetHessianRoutine(tao,user.H,user.H,FormHessian,(void*)&user);
159:   }

161:   /* Set Variable bounds */
162:   MSA_Plate(xl,xu,(void*)&user);
163:   TaoSetVariableBounds(tao,xl,xu);

165:   /* Check for any tao command line options */
166:   TaoSetFromOptions(tao);

168:   /* SOLVE THE APPLICATION */
169:   TaoSolve(tao);

171:   /* Get ierrrmation on converged */
172:   TaoGetConvergedReason(tao,&reason);
173:   TaoView(tao,PETSC_VIEWER_STDOUT_WORLD);
174:   if (reason <= 0){
175:     PetscPrintf(PETSC_COMM_WORLD,"Try a different TAO method, adjust some parameters, or check the function evaluation routines\n");
176:   }
177:   /* Free TAO data structures */
178:   TaoDestroy(&tao);

180:   /* Free PETSc data structures */
181:   VecDestroy(&x);
182:   VecDestroy(&xl);
183:   VecDestroy(&xu);
184:   MatDestroy(&user.H);
185:   VecDestroy(&user.localX);
186:   VecDestroy(&user.localV);
187:   VecDestroy(&user.Bottom);
188:   VecDestroy(&user.Top);
189:   VecDestroy(&user.Left);
190:   VecDestroy(&user.Right);
191:   DMDestroy(&user.dm);
192:   if (flg) {
193:     MatDestroy(&H_shell);
194:   }
195:   PetscFinalize();
196:   return 0;
197: }

201: /*  FormFunctionGradient - Evaluates f(x) and gradient g(x).

203:     Input Parameters:
204: .   tao     - the Tao context
205: .   X      - input vector
206: .   userCtx - optional user-defined context, as set by TaoSetObjectiveAndGradientRoutine()

208:     Output Parameters:
209: .   fcn     - the function value
210: .   G      - vector containing the newly evaluated gradient

212:    Notes:
213:    In this case, we discretize the domain and Create triangles. The
214:    surface of each triangle is planar, whose surface area can be easily
215:    computed.  The total surface area is found by sweeping through the grid
216:    and computing the surface area of the two triangles that have their
217:    right angle at the grid point.  The diagonal line segments on the
218:    grid that define the triangles run from top left to lower right.
219:    The numbering of points starts at the lower left and runs left to
220:    right, then bottom to top.
221: */
222: PetscErrorCode FormFunctionGradient(Tao tao, Vec X, PetscReal *fcn, Vec G,void *userCtx)
223: {
224:   AppCtx         *user = (AppCtx *) userCtx;
226:   PetscInt       i,j,row;
227:   PetscInt       mx=user->mx, my=user->my;
228:   PetscInt       xs,xm,gxs,gxm,ys,ym,gys,gym;
229:   PetscReal      ft=0;
230:   PetscReal      zero=0.0;
231:   PetscReal      hx=1.0/(mx+1),hy=1.0/(my+1), hydhx=hy/hx, hxdhy=hx/hy, area=0.5*hx*hy;
232:   PetscReal      rhx=mx+1, rhy=my+1;
233:   PetscReal      f1,f2,f3,f4,f5,f6,d1,d2,d3,d4,d5,d6,d7,d8,xc,xl,xr,xt,xb,xlt,xrb;
234:   PetscReal      df1dxc,df2dxc,df3dxc,df4dxc,df5dxc,df6dxc;
235:   PetscReal      *g, *x,*left,*right,*bottom,*top;
236:   Vec            localX = user->localX, localG = user->localV;

238:   /* Get local mesh boundaries */
239:   DMDAGetCorners(user->dm,&xs,&ys,NULL,&xm,&ym,NULL);
240:   DMDAGetGhostCorners(user->dm,&gxs,&gys,NULL,&gxm,&gym,NULL);

242:   /* Scatter ghost points to local vector */
243:   DMGlobalToLocalBegin(user->dm,X,INSERT_VALUES,localX);
244:   DMGlobalToLocalEnd(user->dm,X,INSERT_VALUES,localX);

246:   /* Initialize vector to zero */
247:   VecSet(localG, zero);

249:   /* Get pointers to vector data */
250:   VecGetArray(localX,&x);
251:   VecGetArray(localG,&g);
252:   VecGetArray(user->Top,&top);
253:   VecGetArray(user->Bottom,&bottom);
254:   VecGetArray(user->Left,&left);
255:   VecGetArray(user->Right,&right);

257:   /* Compute function over the locally owned part of the mesh */
258:   for (j=ys; j<ys+ym; j++){
259:     for (i=xs; i< xs+xm; i++){
260:       row=(j-gys)*gxm + (i-gxs);

262:       xc = x[row];
263:       xlt=xrb=xl=xr=xb=xt=xc;

265:       if (i==0){ /* left side */
266:         xl= left[j-ys+1];
267:         xlt = left[j-ys+2];
268:       } else {
269:         xl = x[row-1];
270:       }

272:       if (j==0){ /* bottom side */
273:         xb=bottom[i-xs+1];
274:         xrb = bottom[i-xs+2];
275:       } else {
276:         xb = x[row-gxm];
277:       }

279:       if (i+1 == gxs+gxm){ /* right side */
280:         xr=right[j-ys+1];
281:         xrb = right[j-ys];
282:       } else {
283:         xr = x[row+1];
284:       }

286:       if (j+1==gys+gym){ /* top side */
287:         xt=top[i-xs+1];
288:         xlt = top[i-xs];
289:       }else {
290:         xt = x[row+gxm];
291:       }

293:       if (i>gxs && j+1<gys+gym){
294:         xlt = x[row-1+gxm];
295:       }
296:       if (j>gys && i+1<gxs+gxm){
297:         xrb = x[row+1-gxm];
298:       }

300:       d1 = (xc-xl);
301:       d2 = (xc-xr);
302:       d3 = (xc-xt);
303:       d4 = (xc-xb);
304:       d5 = (xr-xrb);
305:       d6 = (xrb-xb);
306:       d7 = (xlt-xl);
307:       d8 = (xt-xlt);

309:       df1dxc = d1*hydhx;
310:       df2dxc = ( d1*hydhx + d4*hxdhy );
311:       df3dxc = d3*hxdhy;
312:       df4dxc = ( d2*hydhx + d3*hxdhy );
313:       df5dxc = d2*hydhx;
314:       df6dxc = d4*hxdhy;

316:       d1 *= rhx;
317:       d2 *= rhx;
318:       d3 *= rhy;
319:       d4 *= rhy;
320:       d5 *= rhy;
321:       d6 *= rhx;
322:       d7 *= rhy;
323:       d8 *= rhx;

325:       f1 = PetscSqrtScalar( 1.0 + d1*d1 + d7*d7);
326:       f2 = PetscSqrtScalar( 1.0 + d1*d1 + d4*d4);
327:       f3 = PetscSqrtScalar( 1.0 + d3*d3 + d8*d8);
328:       f4 = PetscSqrtScalar( 1.0 + d3*d3 + d2*d2);
329:       f5 = PetscSqrtScalar( 1.0 + d2*d2 + d5*d5);
330:       f6 = PetscSqrtScalar( 1.0 + d4*d4 + d6*d6);

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

334:       df1dxc /= f1;
335:       df2dxc /= f2;
336:       df3dxc /= f3;
337:       df4dxc /= f4;
338:       df5dxc /= f5;
339:       df6dxc /= f6;

341:       g[row] = (df1dxc+df2dxc+df3dxc+df4dxc+df5dxc+df6dxc ) * 0.5;

343:     }
344:   }


347:   /* Compute triangular areas along the border of the domain. */
348:   if (xs==0){ /* left side */
349:     for (j=ys; j<ys+ym; j++){
350:       d3=(left[j-ys+1] - left[j-ys+2])*rhy;
351:       d2=(left[j-ys+1] - x[(j-gys)*gxm])*rhx;
352:       ft = ft+PetscSqrtScalar( 1.0 + d3*d3 + d2*d2);
353:     }
354:   }
355:   if (ys==0){ /* bottom side */
356:     for (i=xs; i<xs+xm; i++){
357:       d2=(bottom[i+1-xs]-bottom[i-xs+2])*rhx;
358:       d3=(bottom[i-xs+1]-x[i-gxs])*rhy;
359:       ft = ft+PetscSqrtScalar( 1.0 + d3*d3 + d2*d2);
360:     }
361:   }

363:   if (xs+xm==mx){ /* right side */
364:     for (j=ys; j< ys+ym; j++){
365:       d1=(x[(j+1-gys)*gxm-1]-right[j-ys+1])*rhx;
366:       d4=(right[j-ys]-right[j-ys+1])*rhy;
367:       ft = ft+PetscSqrtScalar( 1.0 + d1*d1 + d4*d4);
368:     }
369:   }
370:   if (ys+ym==my){ /* top side */
371:     for (i=xs; i<xs+xm; i++){
372:       d1=(x[(gym-1)*gxm + i-gxs] - top[i-xs+1])*rhy;
373:       d4=(top[i-xs+1] - top[i-xs])*rhx;
374:       ft = ft+PetscSqrtScalar( 1.0 + d1*d1 + d4*d4);
375:     }
376:   }

378:   if (ys==0 && xs==0){
379:     d1=(left[0]-left[1])*rhy;
380:     d2=(bottom[0]-bottom[1])*rhx;
381:     ft +=PetscSqrtScalar( 1.0 + d1*d1 + d2*d2);
382:   }
383:   if (ys+ym == my && xs+xm == mx){
384:     d1=(right[ym+1] - right[ym])*rhy;
385:     d2=(top[xm+1] - top[xm])*rhx;
386:     ft +=PetscSqrtScalar( 1.0 + d1*d1 + d2*d2);
387:   }

389:   ft=ft*area;
390:   MPI_Allreduce(&ft,fcn,1,MPIU_REAL,MPIU_SUM,MPI_COMM_WORLD);


393:   /* Restore vectors */
394:   VecRestoreArray(localX,&x);
395:   VecRestoreArray(localG,&g);
396:   VecRestoreArray(user->Left,&left);
397:   VecRestoreArray(user->Top,&top);
398:   VecRestoreArray(user->Bottom,&bottom);
399:   VecRestoreArray(user->Right,&right);

401:   /* Scatter values to global vector */
402:   DMLocalToGlobalBegin(user->dm,localG,INSERT_VALUES,G);
403:   DMLocalToGlobalEnd(user->dm,localG,INSERT_VALUES,G);

405:   PetscLogFlops(70*xm*ym);

407:   return 0;
408: }

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

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

421:    Output Parameters:
422: .  A    - Hessian matrix
423: .  B    - optionally different preconditioning matrix

425:    Notes:
426:    Due to mesh point reordering with DMs, we must always work
427:    with the local mesh points, and then transform them to the new
428:    global numbering with the local-to-global mapping.  We cannot work
429:    directly with the global numbers for the original uniprocessor mesh!

431:    Two methods are available for imposing this transformation
432:    when setting matrix entries:
433:      (A) MatSetValuesLocal(), using the local ordering (including
434:          ghost points!)
435:          - Do the following two steps once, before calling TaoSolve()
436:            - Use DMGetISLocalToGlobalMapping() to extract the
437:              local-to-global map from the DM
438:            - Associate this map with the matrix by calling
439:              MatSetLocalToGlobalMapping()
440:          - Then set matrix entries using the local ordering
441:            by calling MatSetValuesLocal()
442:      (B) MatSetValues(), using the global ordering
443:          - Use DMGetGlobalIndices() to extract the local-to-global map
444:          - Then apply this map explicitly yourself
445:          - Set matrix entries using the global ordering by calling
446:            MatSetValues()
447:    Option (A) seems cleaner/easier in many cases, and is the procedure
448:    used in this example.
449: */
450: PetscErrorCode FormHessian(Tao tao,Vec X,Mat Hptr, Mat Hessian, void *ptr)
451: {
453:   AppCtx         *user = (AppCtx *) ptr;
454:   PetscInt       i,j,k,row;
455:   PetscInt       mx=user->mx, my=user->my;
456:   PetscInt       xs,xm,gxs,gxm,ys,ym,gys,gym,col[7];
457:   PetscReal      hx=1.0/(mx+1), hy=1.0/(my+1), hydhx=hy/hx, hxdhy=hx/hy;
458:   PetscReal      rhx=mx+1, rhy=my+1;
459:   PetscReal      f1,f2,f3,f4,f5,f6,d1,d2,d3,d4,d5,d6,d7,d8,xc,xl,xr,xt,xb,xlt,xrb;
460:   PetscReal      hl,hr,ht,hb,hc,htl,hbr;
461:   PetscReal      *x,*left,*right,*bottom,*top;
462:   PetscReal      v[7];
463:   Vec            localX = user->localX;
464:   PetscBool      assembled;


467:   /* Set various matrix options */
468:   MatSetOption(Hessian,MAT_IGNORE_OFF_PROC_ENTRIES,PETSC_TRUE);

470:   /* Initialize matrix entries to zero */
471:   MatAssembled(Hessian,&assembled);
472:   if (assembled){MatZeroEntries(Hessian);}

474:   /* Get local mesh boundaries */
475:   DMDAGetCorners(user->dm,&xs,&ys,NULL,&xm,&ym,NULL);
476:   DMDAGetGhostCorners(user->dm,&gxs,&gys,NULL,&gxm,&gym,NULL);

478:   /* Scatter ghost points to local vector */
479:   DMGlobalToLocalBegin(user->dm,X,INSERT_VALUES,localX);
480:   DMGlobalToLocalEnd(user->dm,X,INSERT_VALUES,localX);

482:   /* Get pointers to vector data */
483:   VecGetArray(localX,&x);
484:   VecGetArray(user->Top,&top);
485:   VecGetArray(user->Bottom,&bottom);
486:   VecGetArray(user->Left,&left);
487:   VecGetArray(user->Right,&right);

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

491:   for (i=xs; i< xs+xm; i++){

493:     for (j=ys; j<ys+ym; j++){

495:       row=(j-gys)*gxm + (i-gxs);

497:       xc = x[row];
498:       xlt=xrb=xl=xr=xb=xt=xc;

500:       /* Left side */
501:       if (i==gxs){
502:         xl= left[j-ys+1];
503:         xlt = left[j-ys+2];
504:       } else {
505:         xl = x[row-1];
506:       }

508:       if (j==gys){
509:         xb=bottom[i-xs+1];
510:         xrb = bottom[i-xs+2];
511:       } else {
512:         xb = x[row-gxm];
513:       }

515:       if (i+1 == gxs+gxm){
516:         xr=right[j-ys+1];
517:         xrb = right[j-ys];
518:       } else {
519:         xr = x[row+1];
520:       }

522:       if (j+1==gys+gym){
523:         xt=top[i-xs+1];
524:         xlt = top[i-xs];
525:       }else {
526:         xt = x[row+gxm];
527:       }

529:       if (i>gxs && j+1<gys+gym){
530:         xlt = x[row-1+gxm];
531:       }
532:       if (j>gys && i+1<gxs+gxm){
533:         xrb = x[row+1-gxm];
534:       }


537:       d1 = (xc-xl)*rhx;
538:       d2 = (xc-xr)*rhx;
539:       d3 = (xc-xt)*rhy;
540:       d4 = (xc-xb)*rhy;
541:       d5 = (xrb-xr)*rhy;
542:       d6 = (xrb-xb)*rhx;
543:       d7 = (xlt-xl)*rhy;
544:       d8 = (xlt-xt)*rhx;

546:       f1 = PetscSqrtScalar( 1.0 + d1*d1 + d7*d7);
547:       f2 = PetscSqrtScalar( 1.0 + d1*d1 + d4*d4);
548:       f3 = PetscSqrtScalar( 1.0 + d3*d3 + d8*d8);
549:       f4 = PetscSqrtScalar( 1.0 + d3*d3 + d2*d2);
550:       f5 = PetscSqrtScalar( 1.0 + d2*d2 + d5*d5);
551:       f6 = PetscSqrtScalar( 1.0 + d4*d4 + d6*d6);


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

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

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

571:       hl*=0.5; hr*=0.5; ht*=0.5; hb*=0.5; hbr*=0.5; htl*=0.5;  hc*=0.5;

573:       k=0;
574:       if (j>0){
575:         v[k]=hb; col[k]=row - gxm; k++;
576:       }

578:       if (j>0 && i < mx -1){
579:         v[k]=hbr; col[k]=row - gxm+1; k++;
580:       }

582:       if (i>0){
583:         v[k]= hl; col[k]=row - 1; k++;
584:       }

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

588:       if (i < mx-1 ){
589:         v[k]= hr; col[k]=row+1; k++;
590:       }

592:       if (i>0 && j < my-1 ){
593:         v[k]= htl; col[k] = row+gxm-1; k++;
594:       }

596:       if (j < my-1 ){
597:         v[k]= ht; col[k] = row+gxm; k++;
598:       }

600:       /*
601:          Set matrix values using local numbering, which was defined
602:          earlier, in the main routine.
603:       */
604:       MatSetValuesLocal(Hessian,1,&row,k,col,v,INSERT_VALUES);

606:     }
607:   }

609:   /* Restore vectors */
610:   VecRestoreArray(localX,&x);
611:   VecRestoreArray(user->Left,&left);
612:   VecRestoreArray(user->Top,&top);
613:   VecRestoreArray(user->Bottom,&bottom);
614:   VecRestoreArray(user->Right,&right);

616:   /* Assemble the matrix */
617:   MatAssemblyBegin(Hessian,MAT_FINAL_ASSEMBLY);
618:   MatAssemblyEnd(Hessian,MAT_FINAL_ASSEMBLY);

620:   PetscLogFlops(199*xm*ym);
621:   return 0;
622: }

624: /* ------------------------------------------------------------------- */
627: /*
628:    MSA_BoundaryConditions -  Calculates the boundary conditions for
629:    the region.

631:    Input Parameter:
632: .  user - user-defined application context

634:    Output Parameter:
635: .  user - user-defined application context
636: */
637: static PetscErrorCode MSA_BoundaryConditions(AppCtx * user)
638: {
639:   int        ierr;
640:   PetscInt   i,j,k,maxits=5,limit=0;
641:   PetscInt   xs,ys,xm,ym,gxs,gys,gxm,gym;
642:   PetscInt   mx=user->mx,my=user->my;
643:   PetscInt   bsize=0, lsize=0, tsize=0, rsize=0;
644:   PetscReal  one=1.0, two=2.0, three=3.0, scl=1.0, tol=1e-10;
645:   PetscReal  fnorm,det,hx,hy,xt=0,yt=0;
646:   PetscReal  u1,u2,nf1,nf2,njac11,njac12,njac21,njac22;
647:   PetscReal  b=-0.5, t=0.5, l=-0.5, r=0.5;
648:   PetscReal  *boundary;
649:   PetscBool  flg;
650:   Vec        Bottom,Top,Right,Left;

652:   /* Get local mesh boundaries */
653:   DMDAGetCorners(user->dm,&xs,&ys,NULL,&xm,&ym,NULL);
654:   DMDAGetGhostCorners(user->dm,&gxs,&gys,NULL,&gxm,&gym,NULL);


657:   bsize=xm+2;
658:   lsize=ym+2;
659:   rsize=ym+2;
660:   tsize=xm+2;

662:   VecCreateMPI(MPI_COMM_WORLD,bsize,PETSC_DECIDE,&Bottom);
663:   VecCreateMPI(MPI_COMM_WORLD,tsize,PETSC_DECIDE,&Top);
664:   VecCreateMPI(MPI_COMM_WORLD,lsize,PETSC_DECIDE,&Left);
665:   VecCreateMPI(MPI_COMM_WORLD,rsize,PETSC_DECIDE,&Right);

667:   user->Top=Top;
668:   user->Left=Left;
669:   user->Bottom=Bottom;
670:   user->Right=Right;

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

674:   for (j=0; j<4; j++){
675:     if (j==0){
676:       yt=b;
677:       xt=l+hx*xs;
678:       limit=bsize;
679:       VecGetArray(Bottom,&boundary);
680:     } else if (j==1){
681:       yt=t;
682:       xt=l+hx*xs;
683:       limit=tsize;
684:       VecGetArray(Top,&boundary);
685:     } else if (j==2){
686:       yt=b+hy*ys;
687:       xt=l;
688:       limit=lsize;
689:       VecGetArray(Left,&boundary);
690:     } else if (j==3){
691:       yt=b+hy*ys;
692:       xt=r;
693:       limit=rsize;
694:       VecGetArray(Right,&boundary);
695:     }

697:     for (i=0; i<limit; i++){
698:       u1=xt;
699:       u2=-yt;
700:       for (k=0; k<maxits; k++){
701:         nf1=u1 + u1*u2*u2 - u1*u1*u1/three-xt;
702:         nf2=-u2 - u1*u1*u2 + u2*u2*u2/three-yt;
703:         fnorm=PetscSqrtScalar(nf1*nf1+nf2*nf2);
704:         if (fnorm <= tol) break;
705:         njac11=one+u2*u2-u1*u1;
706:         njac12=two*u1*u2;
707:         njac21=-two*u1*u2;
708:         njac22=-one - u1*u1 + u2*u2;
709:         det = njac11*njac22-njac21*njac12;
710:         u1 = u1-(njac22*nf1-njac12*nf2)/det;
711:         u2 = u2-(njac11*nf2-njac21*nf1)/det;
712:       }

714:       boundary[i]=u1*u1-u2*u2;
715:       if (j==0 || j==1) {
716:         xt=xt+hx;
717:       } else if (j==2 || j==3){
718:         yt=yt+hy;
719:       }
720:     }
721:     if (j==0){
722:       VecRestoreArray(Bottom,&boundary);
723:     } else if (j==1){
724:       VecRestoreArray(Top,&boundary);
725:     } else if (j==2){
726:       VecRestoreArray(Left,&boundary);
727:     } else if (j==3){
728:       VecRestoreArray(Right,&boundary);
729:     }
730:   }

732:   /* Scale the boundary if desired */

734:   PetscOptionsGetReal(NULL,"-bottom",&scl,&flg);
735:   if (flg){
736:     VecScale(Bottom, scl);
737:   }
738:   PetscOptionsGetReal(NULL,"-top",&scl,&flg);
739:   if (flg){
740:     VecScale(Top, scl);
741:   }
742:   PetscOptionsGetReal(NULL,"-right",&scl,&flg);
743:   if (flg){
744:     VecScale(Right, scl);
745:   }

747:   PetscOptionsGetReal(NULL,"-left",&scl,&flg);
748:   if (flg){
749:     VecScale(Left, scl);
750:   }
751:   return 0;
752: }


755: /* ------------------------------------------------------------------- */
758: /*
759:    MSA_Plate -  Calculates an obstacle for surface to stretch over.

761:    Input Parameter:
762: .  user - user-defined application context

764:    Output Parameter:
765: .  user - user-defined application context
766: */
767: static PetscErrorCode MSA_Plate(Vec XL,Vec XU,void *ctx){

769:   AppCtx         *user=(AppCtx *)ctx;
771:   PetscInt       i,j,row;
772:   PetscInt       xs,ys,xm,ym;
773:   PetscInt       mx=user->mx, my=user->my, bmy, bmx;
774:   PetscReal      t1,t2,t3;
775:   PetscReal      *xl, lb=PETSC_NINFINITY, ub=PETSC_INFINITY;
776:   PetscBool      cylinder;

778:   user->bmy = PetscMax(0,user->bmy);user->bmy = PetscMin(my,user->bmy);
779:   user->bmx = PetscMax(0,user->bmx);user->bmx = PetscMin(mx,user->bmx);
780:   bmy=user->bmy, bmx=user->bmx;

782:   DMDAGetCorners(user->dm,&xs,&ys,NULL,&xm,&ym,NULL);

784:   VecSet(XL, lb);
785:   VecSet(XU, ub);

787:   VecGetArray(XL,&xl);

789:   PetscOptionsHasName(NULL,"-cylinder",&cylinder);
790:   /* Compute the optional lower box */
791:   if (cylinder){
792:     for (i=xs; i< xs+xm; i++){
793:       for (j=ys; j<ys+ym; j++){
794:         row=(j-ys)*xm + (i-xs);
795:         t1=(2.0*i-mx)*bmy;
796:         t2=(2.0*j-my)*bmx;
797:         t3=bmx*bmx*bmy*bmy;
798:         if ( t1*t1 + t2*t2 <= t3 ){
799:           xl[row] = user->bheight;
800:         }
801:       }
802:     }
803:   } else {
804:     /* Compute the optional lower box */
805:     for (i=xs; i< xs+xm; i++){
806:       for (j=ys; j<ys+ym; j++){
807:         row=(j-ys)*xm + (i-xs);
808:         if (i>=(mx-bmx)/2 && i<mx-(mx-bmx)/2 &&
809:             j>=(my-bmy)/2 && j<my-(my-bmy)/2 ){
810:           xl[row] = user->bheight;
811:         }
812:       }
813:     }
814:   }
815:     VecRestoreArray(XL,&xl);

817:   return 0;
818: }


821: /* ------------------------------------------------------------------- */
824: /*
825:    MSA_InitialPoint - Calculates the initial guess in one of three ways.

827:    Input Parameters:
828: .  user - user-defined application context
829: .  X - vector for initial guess

831:    Output Parameters:
832: .  X - newly computed initial guess
833: */
834: static PetscErrorCode MSA_InitialPoint(AppCtx * user, Vec X)
835: {
837:   PetscInt       start=-1,i,j;
838:   PetscReal      zero=0.0;
839:   PetscBool      flg;

841:   PetscOptionsGetInt(NULL,"-start",&start,&flg);
842:   if (flg && start==0){ /* The zero vector is reasonable */
843:     VecSet(X, zero);
844:   } else if (flg && start>0){ /* Try a random start between -0.5 and 0.5 */
845:     PetscRandom rctx;  PetscReal np5=-0.5;

847:     PetscRandomCreate(MPI_COMM_WORLD,&rctx);
848:     for (i=0; i<start; i++){
849:       VecSetRandom(X, rctx);
850:     }
851:     PetscRandomDestroy(&rctx);
852:     VecShift(X, np5);

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

856:     PetscInt row,xs,xm,gxs,gxm,ys,ym,gys,gym;
857:     PetscInt mx=user->mx,my=user->my;
858:     PetscReal *x,*left,*right,*bottom,*top;
859:     Vec    localX = user->localX;

861:     /* Get local mesh boundaries */
862:     DMDAGetCorners(user->dm,&xs,&ys,NULL,&xm,&ym,NULL);
863:     DMDAGetGhostCorners(user->dm,&gxs,&gys,NULL,&gxm,&gym,NULL);

865:     /* Get pointers to vector data */
866:     VecGetArray(user->Top,&top);
867:     VecGetArray(user->Bottom,&bottom);
868:     VecGetArray(user->Left,&left);
869:     VecGetArray(user->Right,&right);

871:     VecGetArray(localX,&x);
872:     /* Perform local computations */
873:     for (j=ys; j<ys+ym; j++){
874:       for (i=xs; i< xs+xm; i++){
875:         row=(j-gys)*gxm + (i-gxs);
876:         x[row] = ( (j+1)*bottom[i-xs+1]/my + (my-j+1)*top[i-xs+1]/(my+2)+
877:                    (i+1)*left[j-ys+1]/mx + (mx-i+1)*right[j-ys+1]/(mx+2))/2.0;
878:       }
879:     }

881:     /* Restore vectors */
882:     VecRestoreArray(localX,&x);

884:     VecRestoreArray(user->Left,&left);
885:     VecRestoreArray(user->Top,&top);
886:     VecRestoreArray(user->Bottom,&bottom);
887:     VecRestoreArray(user->Right,&right);

889:     /* Scatter values into global vector */
890:     DMLocalToGlobalBegin(user->dm,localX,INSERT_VALUES,X);
891:     DMLocalToGlobalEnd(user->dm,localX,INSERT_VALUES,X);

893:   }
894:   return 0;
895: }

897: /* For testing matrix free submatrices */
900: PetscErrorCode MatrixFreeHessian(Tao tao, Vec x, Mat H, Mat Hpre, void *ptr)
901: {
903:   AppCtx         *user = (AppCtx*)ptr;
905:   FormHessian(tao,x,user->H,user->H,ptr);
906:   return(0);
907: }
910: PetscErrorCode MyMatMult(Mat H_shell, Vec X, Vec Y)
911: {
913:   void           *ptr;
914:   AppCtx         *user;
916:   MatShellGetContext(H_shell,&ptr);
917:   user = (AppCtx*)ptr;
918:   MatMult(user->H,X,Y);
919:   return(0);
920: }