Actual source code: plate2.c

petsc-master 2018-02-19
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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: TaoDestroy();
 30:    Processors: n
 31: T*/




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

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

 54: /* -------- User-defined Routines --------- */

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

 62: /* For testing matrix free submatrices */
 63: PetscErrorCode MatrixFreeHessian(Tao,Vec,Mat, Mat,void*);
 64: 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:   Mat                    H_shell;                  /* to test matrix-free submatrices */
 76:   AppCtx                 user;                 /* user-defined work context */

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

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

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

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

 93:   PetscPrintf(PETSC_COMM_WORLD,"\n---- Minimum Surface Area With Plate Problem -----\n");
 94:   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);

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

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

102:   /*
103:      A two dimensional distributed array will help define this problem,
104:      which derives from an elliptic PDE on two dimensional domain.  From
105:      the distributed array, Create the vectors.
106:   */
107:   DMDACreate2d(MPI_COMM_WORLD,DM_BOUNDARY_NONE,DM_BOUNDARY_NONE,DMDA_STENCIL_BOX,user.mx,user.my,Nx,Ny,1,1,NULL,NULL,&user.dm);
108:   DMSetFromOptions(user.dm);
109:   DMSetUp(user.dm);
110:   /*
111:      Extract global and local vectors from DM; The local vectors are
112:      used solely as work space for the evaluation of the function,
113:      gradient, and Hessian.  Duplicate for remaining vectors that are
114:      the same types.
115:   */
116:   DMCreateGlobalVector(user.dm,&x); /* Solution */
117:   DMCreateLocalVector(user.dm,&user.localX);
118:   VecDuplicate(user.localX,&user.localV);

120:   VecDuplicate(x,&xl);
121:   VecDuplicate(x,&xu);

123:   /* The TAO code begins here */

125:   /*
126:      Create TAO solver and set desired solution method
127:      The method must either be TAOTRON or TAOBLMVM
128:      If TAOBLMVM is used, then hessian function is not called.
129:   */
130:   TaoCreate(PETSC_COMM_WORLD,&tao);
131:   TaoSetType(tao,TAOBLMVM);

133:   /* Set initial solution guess; */
134:   MSA_BoundaryConditions(&user);
135:   MSA_InitialPoint(&user,x);
136:   TaoSetInitialVector(tao,x);

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

141:   VecGetLocalSize(x,&m);
142:   MatCreateAIJ(MPI_COMM_WORLD,m,m,N,N,7,NULL,3,NULL,&(user.H));
143:   MatSetOption(user.H,MAT_SYMMETRIC,PETSC_TRUE);

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

157:   /* Set Variable bounds */
158:   MSA_Plate(xl,xu,(void*)&user);
159:   TaoSetVariableBounds(tao,xl,xu);

161:   /* Check for any tao command line options */
162:   TaoSetFromOptions(tao);

164:   /* SOLVE THE APPLICATION */
165:   TaoSolve(tao);

167:   TaoView(tao,PETSC_VIEWER_STDOUT_WORLD);

169:   /* Free TAO data structures */
170:   TaoDestroy(&tao);

172:   /* Free PETSc data structures */
173:   VecDestroy(&x);
174:   VecDestroy(&xl);
175:   VecDestroy(&xu);
176:   MatDestroy(&user.H);
177:   VecDestroy(&user.localX);
178:   VecDestroy(&user.localV);
179:   VecDestroy(&user.Bottom);
180:   VecDestroy(&user.Top);
181:   VecDestroy(&user.Left);
182:   VecDestroy(&user.Right);
183:   DMDestroy(&user.dm);
184:   if (flg) {
185:     MatDestroy(&H_shell);
186:   }
187:   PetscFinalize();
188:   return ierr;
189: }

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

193:     Input Parameters:
194: .   tao     - the Tao context
195: .   X      - input vector
196: .   userCtx - optional user-defined context, as set by TaoSetObjectiveAndGradientRoutine()

198:     Output Parameters:
199: .   fcn     - the function value
200: .   G      - vector containing the newly evaluated gradient

202:    Notes:
203:    In this case, we discretize the domain and Create triangles. The
204:    surface of each triangle is planar, whose surface area can be easily
205:    computed.  The total surface area is found by sweeping through the grid
206:    and computing the surface area of the two triangles that have their
207:    right angle at the grid point.  The diagonal line segments on the
208:    grid that define the triangles run from top left to lower right.
209:    The numbering of points starts at the lower left and runs left to
210:    right, then bottom to top.
211: */
212: PetscErrorCode FormFunctionGradient(Tao tao, Vec X, PetscReal *fcn, Vec G,void *userCtx)
213: {
214:   AppCtx         *user = (AppCtx *) userCtx;
216:   PetscInt       i,j,row;
217:   PetscInt       mx=user->mx, my=user->my;
218:   PetscInt       xs,xm,gxs,gxm,ys,ym,gys,gym;
219:   PetscReal      ft=0;
220:   PetscReal      zero=0.0;
221:   PetscReal      hx=1.0/(mx+1),hy=1.0/(my+1), hydhx=hy/hx, hxdhy=hx/hy, area=0.5*hx*hy;
222:   PetscReal      rhx=mx+1, rhy=my+1;
223:   PetscReal      f1,f2,f3,f4,f5,f6,d1,d2,d3,d4,d5,d6,d7,d8,xc,xl,xr,xt,xb,xlt,xrb;
224:   PetscReal      df1dxc,df2dxc,df3dxc,df4dxc,df5dxc,df6dxc;
225:   PetscReal      *g, *x,*left,*right,*bottom,*top;
226:   Vec            localX = user->localX, localG = user->localV;

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

232:   /* Scatter ghost points to local vector */
233:   DMGlobalToLocalBegin(user->dm,X,INSERT_VALUES,localX);
234:   DMGlobalToLocalEnd(user->dm,X,INSERT_VALUES,localX);

236:   /* Initialize vector to zero */
237:   VecSet(localG, zero);

239:   /* Get pointers to vector data */
240:   VecGetArray(localX,&x);
241:   VecGetArray(localG,&g);
242:   VecGetArray(user->Top,&top);
243:   VecGetArray(user->Bottom,&bottom);
244:   VecGetArray(user->Left,&left);
245:   VecGetArray(user->Right,&right);

247:   /* Compute function over the locally owned part of the mesh */
248:   for (j=ys; j<ys+ym; j++){
249:     for (i=xs; i< xs+xm; i++){
250:       row=(j-gys)*gxm + (i-gxs);

252:       xc = x[row];
253:       xlt=xrb=xl=xr=xb=xt=xc;

255:       if (i==0){ /* left side */
256:         xl= left[j-ys+1];
257:         xlt = left[j-ys+2];
258:       } else {
259:         xl = x[row-1];
260:       }

262:       if (j==0){ /* bottom side */
263:         xb=bottom[i-xs+1];
264:         xrb = bottom[i-xs+2];
265:       } else {
266:         xb = x[row-gxm];
267:       }

269:       if (i+1 == gxs+gxm){ /* right side */
270:         xr=right[j-ys+1];
271:         xrb = right[j-ys];
272:       } else {
273:         xr = x[row+1];
274:       }

276:       if (j+1==gys+gym){ /* top side */
277:         xt=top[i-xs+1];
278:         xlt = top[i-xs];
279:       }else {
280:         xt = x[row+gxm];
281:       }

283:       if (i>gxs && j+1<gys+gym){
284:         xlt = x[row-1+gxm];
285:       }
286:       if (j>gys && i+1<gxs+gxm){
287:         xrb = x[row+1-gxm];
288:       }

290:       d1 = (xc-xl);
291:       d2 = (xc-xr);
292:       d3 = (xc-xt);
293:       d4 = (xc-xb);
294:       d5 = (xr-xrb);
295:       d6 = (xrb-xb);
296:       d7 = (xlt-xl);
297:       d8 = (xt-xlt);

299:       df1dxc = d1*hydhx;
300:       df2dxc = ( d1*hydhx + d4*hxdhy );
301:       df3dxc = d3*hxdhy;
302:       df4dxc = ( d2*hydhx + d3*hxdhy );
303:       df5dxc = d2*hydhx;
304:       df6dxc = d4*hxdhy;

306:       d1 *= rhx;
307:       d2 *= rhx;
308:       d3 *= rhy;
309:       d4 *= rhy;
310:       d5 *= rhy;
311:       d6 *= rhx;
312:       d7 *= rhy;
313:       d8 *= rhx;

315:       f1 = PetscSqrtScalar( 1.0 + d1*d1 + d7*d7);
316:       f2 = PetscSqrtScalar( 1.0 + d1*d1 + d4*d4);
317:       f3 = PetscSqrtScalar( 1.0 + d3*d3 + d8*d8);
318:       f4 = PetscSqrtScalar( 1.0 + d3*d3 + d2*d2);
319:       f5 = PetscSqrtScalar( 1.0 + d2*d2 + d5*d5);
320:       f6 = PetscSqrtScalar( 1.0 + d4*d4 + d6*d6);

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

324:       df1dxc /= f1;
325:       df2dxc /= f2;
326:       df3dxc /= f3;
327:       df4dxc /= f4;
328:       df5dxc /= f5;
329:       df6dxc /= f6;

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

333:     }
334:   }


337:   /* Compute triangular areas along the border of the domain. */
338:   if (xs==0){ /* left side */
339:     for (j=ys; j<ys+ym; j++){
340:       d3=(left[j-ys+1] - left[j-ys+2])*rhy;
341:       d2=(left[j-ys+1] - x[(j-gys)*gxm])*rhx;
342:       ft = ft+PetscSqrtScalar( 1.0 + d3*d3 + d2*d2);
343:     }
344:   }
345:   if (ys==0){ /* bottom side */
346:     for (i=xs; i<xs+xm; i++){
347:       d2=(bottom[i+1-xs]-bottom[i-xs+2])*rhx;
348:       d3=(bottom[i-xs+1]-x[i-gxs])*rhy;
349:       ft = ft+PetscSqrtScalar( 1.0 + d3*d3 + d2*d2);
350:     }
351:   }

353:   if (xs+xm==mx){ /* right side */
354:     for (j=ys; j< ys+ym; j++){
355:       d1=(x[(j+1-gys)*gxm-1]-right[j-ys+1])*rhx;
356:       d4=(right[j-ys]-right[j-ys+1])*rhy;
357:       ft = ft+PetscSqrtScalar( 1.0 + d1*d1 + d4*d4);
358:     }
359:   }
360:   if (ys+ym==my){ /* top side */
361:     for (i=xs; i<xs+xm; i++){
362:       d1=(x[(gym-1)*gxm + i-gxs] - top[i-xs+1])*rhy;
363:       d4=(top[i-xs+1] - top[i-xs])*rhx;
364:       ft = ft+PetscSqrtScalar( 1.0 + d1*d1 + d4*d4);
365:     }
366:   }

368:   if (ys==0 && xs==0){
369:     d1=(left[0]-left[1])*rhy;
370:     d2=(bottom[0]-bottom[1])*rhx;
371:     ft +=PetscSqrtScalar( 1.0 + d1*d1 + d2*d2);
372:   }
373:   if (ys+ym == my && xs+xm == mx){
374:     d1=(right[ym+1] - right[ym])*rhy;
375:     d2=(top[xm+1] - top[xm])*rhx;
376:     ft +=PetscSqrtScalar( 1.0 + d1*d1 + d2*d2);
377:   }

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


383:   /* Restore vectors */
384:   VecRestoreArray(localX,&x);
385:   VecRestoreArray(localG,&g);
386:   VecRestoreArray(user->Left,&left);
387:   VecRestoreArray(user->Top,&top);
388:   VecRestoreArray(user->Bottom,&bottom);
389:   VecRestoreArray(user->Right,&right);

391:   /* Scatter values to global vector */
392:   DMLocalToGlobalBegin(user->dm,localG,INSERT_VALUES,G);
393:   DMLocalToGlobalEnd(user->dm,localG,INSERT_VALUES,G);

395:   PetscLogFlops(70*xm*ym);

397:   return 0;
398: }

400: /* ------------------------------------------------------------------- */
401: /*
402:    FormHessian - Evaluates Hessian matrix.

404:    Input Parameters:
405: .  tao  - the Tao context
406: .  x    - input vector
407: .  ptr  - optional user-defined context, as set by TaoSetHessianRoutine()

409:    Output Parameters:
410: .  A    - Hessian matrix
411: .  B    - optionally different preconditioning matrix

413:    Notes:
414:    Due to mesh point reordering with DMs, we must always work
415:    with the local mesh points, and then transform them to the new
416:    global numbering with the local-to-global mapping.  We cannot work
417:    directly with the global numbers for the original uniprocessor mesh!

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


455:   /* Set various matrix options */
456:   MatSetOption(Hessian,MAT_IGNORE_OFF_PROC_ENTRIES,PETSC_TRUE);

458:   /* Initialize matrix entries to zero */
459:   MatAssembled(Hessian,&assembled);
460:   if (assembled){MatZeroEntries(Hessian);}

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

466:   /* Scatter ghost points to local vector */
467:   DMGlobalToLocalBegin(user->dm,X,INSERT_VALUES,localX);
468:   DMGlobalToLocalEnd(user->dm,X,INSERT_VALUES,localX);

470:   /* Get pointers to vector data */
471:   VecGetArray(localX,&x);
472:   VecGetArray(user->Top,&top);
473:   VecGetArray(user->Bottom,&bottom);
474:   VecGetArray(user->Left,&left);
475:   VecGetArray(user->Right,&right);

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

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

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

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

485:       xc = x[row];
486:       xlt=xrb=xl=xr=xb=xt=xc;

488:       /* Left side */
489:       if (i==gxs){
490:         xl= left[j-ys+1];
491:         xlt = left[j-ys+2];
492:       } else {
493:         xl = x[row-1];
494:       }

496:       if (j==gys){
497:         xb=bottom[i-xs+1];
498:         xrb = bottom[i-xs+2];
499:       } else {
500:         xb = x[row-gxm];
501:       }

503:       if (i+1 == gxs+gxm){
504:         xr=right[j-ys+1];
505:         xrb = right[j-ys];
506:       } else {
507:         xr = x[row+1];
508:       }

510:       if (j+1==gys+gym){
511:         xt=top[i-xs+1];
512:         xlt = top[i-xs];
513:       }else {
514:         xt = x[row+gxm];
515:       }

517:       if (i>gxs && j+1<gys+gym){
518:         xlt = x[row-1+gxm];
519:       }
520:       if (j>gys && i+1<gxs+gxm){
521:         xrb = x[row+1-gxm];
522:       }


525:       d1 = (xc-xl)*rhx;
526:       d2 = (xc-xr)*rhx;
527:       d3 = (xc-xt)*rhy;
528:       d4 = (xc-xb)*rhy;
529:       d5 = (xrb-xr)*rhy;
530:       d6 = (xrb-xb)*rhx;
531:       d7 = (xlt-xl)*rhy;
532:       d8 = (xlt-xt)*rhx;

534:       f1 = PetscSqrtScalar( 1.0 + d1*d1 + d7*d7);
535:       f2 = PetscSqrtScalar( 1.0 + d1*d1 + d4*d4);
536:       f3 = PetscSqrtScalar( 1.0 + d3*d3 + d8*d8);
537:       f4 = PetscSqrtScalar( 1.0 + d3*d3 + d2*d2);
538:       f5 = PetscSqrtScalar( 1.0 + d2*d2 + d5*d5);
539:       f6 = PetscSqrtScalar( 1.0 + d4*d4 + d6*d6);


542:       hl = (-hydhx*(1.0+d7*d7)+d1*d7)/(f1*f1*f1)+
543:         (-hydhx*(1.0+d4*d4)+d1*d4)/(f2*f2*f2);
544:       hr = (-hydhx*(1.0+d5*d5)+d2*d5)/(f5*f5*f5)+
545:         (-hydhx*(1.0+d3*d3)+d2*d3)/(f4*f4*f4);
546:       ht = (-hxdhy*(1.0+d8*d8)+d3*d8)/(f3*f3*f3)+
547:         (-hxdhy*(1.0+d2*d2)+d2*d3)/(f4*f4*f4);
548:       hb = (-hxdhy*(1.0+d6*d6)+d4*d6)/(f6*f6*f6)+
549:         (-hxdhy*(1.0+d1*d1)+d1*d4)/(f2*f2*f2);

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

554:       hc = hydhx*(1.0+d7*d7)/(f1*f1*f1) + hxdhy*(1.0+d8*d8)/(f3*f3*f3) +
555:         hydhx*(1.0+d5*d5)/(f5*f5*f5) + hxdhy*(1.0+d6*d6)/(f6*f6*f6) +
556:         (hxdhy*(1.0+d1*d1)+hydhx*(1.0+d4*d4)-2*d1*d4)/(f2*f2*f2) +
557:         (hxdhy*(1.0+d2*d2)+hydhx*(1.0+d3*d3)-2*d2*d3)/(f4*f4*f4);

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

561:       k=0;
562:       if (j>0){
563:         v[k]=hb; col[k]=row - gxm; k++;
564:       }

566:       if (j>0 && i < mx -1){
567:         v[k]=hbr; col[k]=row - gxm+1; k++;
568:       }

570:       if (i>0){
571:         v[k]= hl; col[k]=row - 1; k++;
572:       }

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

576:       if (i < mx-1 ){
577:         v[k]= hr; col[k]=row+1; k++;
578:       }

580:       if (i>0 && j < my-1 ){
581:         v[k]= htl; col[k] = row+gxm-1; k++;
582:       }

584:       if (j < my-1 ){
585:         v[k]= ht; col[k] = row+gxm; k++;
586:       }

588:       /*
589:          Set matrix values using local numbering, which was defined
590:          earlier, in the main routine.
591:       */
592:       MatSetValuesLocal(Hessian,1,&row,k,col,v,INSERT_VALUES);

594:     }
595:   }

597:   /* Restore vectors */
598:   VecRestoreArray(localX,&x);
599:   VecRestoreArray(user->Left,&left);
600:   VecRestoreArray(user->Top,&top);
601:   VecRestoreArray(user->Bottom,&bottom);
602:   VecRestoreArray(user->Right,&right);

604:   /* Assemble the matrix */
605:   MatAssemblyBegin(Hessian,MAT_FINAL_ASSEMBLY);
606:   MatAssemblyEnd(Hessian,MAT_FINAL_ASSEMBLY);

608:   PetscLogFlops(199*xm*ym);
609:   return 0;
610: }

612: /* ------------------------------------------------------------------- */
613: /*
614:    MSA_BoundaryConditions -  Calculates the boundary conditions for
615:    the region.

617:    Input Parameter:
618: .  user - user-defined application context

620:    Output Parameter:
621: .  user - user-defined application context
622: */
623: static PetscErrorCode MSA_BoundaryConditions(AppCtx * user)
624: {
625:   int        ierr;
626:   PetscInt   i,j,k,maxits=5,limit=0;
627:   PetscInt   xs,ys,xm,ym,gxs,gys,gxm,gym;
628:   PetscInt   mx=user->mx,my=user->my;
629:   PetscInt   bsize=0, lsize=0, tsize=0, rsize=0;
630:   PetscReal  one=1.0, two=2.0, three=3.0, scl=1.0, tol=1e-10;
631:   PetscReal  fnorm,det,hx,hy,xt=0,yt=0;
632:   PetscReal  u1,u2,nf1,nf2,njac11,njac12,njac21,njac22;
633:   PetscReal  b=-0.5, t=0.5, l=-0.5, r=0.5;
634:   PetscReal  *boundary;
635:   PetscBool  flg;
636:   Vec        Bottom,Top,Right,Left;

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


643:   bsize=xm+2;
644:   lsize=ym+2;
645:   rsize=ym+2;
646:   tsize=xm+2;

648:   VecCreateMPI(MPI_COMM_WORLD,bsize,PETSC_DECIDE,&Bottom);
649:   VecCreateMPI(MPI_COMM_WORLD,tsize,PETSC_DECIDE,&Top);
650:   VecCreateMPI(MPI_COMM_WORLD,lsize,PETSC_DECIDE,&Left);
651:   VecCreateMPI(MPI_COMM_WORLD,rsize,PETSC_DECIDE,&Right);

653:   user->Top=Top;
654:   user->Left=Left;
655:   user->Bottom=Bottom;
656:   user->Right=Right;

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

660:   for (j=0; j<4; j++){
661:     if (j==0){
662:       yt=b;
663:       xt=l+hx*xs;
664:       limit=bsize;
665:       VecGetArray(Bottom,&boundary);
666:     } else if (j==1){
667:       yt=t;
668:       xt=l+hx*xs;
669:       limit=tsize;
670:       VecGetArray(Top,&boundary);
671:     } else if (j==2){
672:       yt=b+hy*ys;
673:       xt=l;
674:       limit=lsize;
675:       VecGetArray(Left,&boundary);
676:     } else if (j==3){
677:       yt=b+hy*ys;
678:       xt=r;
679:       limit=rsize;
680:       VecGetArray(Right,&boundary);
681:     }

683:     for (i=0; i<limit; i++){
684:       u1=xt;
685:       u2=-yt;
686:       for (k=0; k<maxits; k++){
687:         nf1=u1 + u1*u2*u2 - u1*u1*u1/three-xt;
688:         nf2=-u2 - u1*u1*u2 + u2*u2*u2/three-yt;
689:         fnorm=PetscSqrtScalar(nf1*nf1+nf2*nf2);
690:         if (fnorm <= tol) break;
691:         njac11=one+u2*u2-u1*u1;
692:         njac12=two*u1*u2;
693:         njac21=-two*u1*u2;
694:         njac22=-one - u1*u1 + u2*u2;
695:         det = njac11*njac22-njac21*njac12;
696:         u1 = u1-(njac22*nf1-njac12*nf2)/det;
697:         u2 = u2-(njac11*nf2-njac21*nf1)/det;
698:       }

700:       boundary[i]=u1*u1-u2*u2;
701:       if (j==0 || j==1) {
702:         xt=xt+hx;
703:       } else if (j==2 || j==3){
704:         yt=yt+hy;
705:       }
706:     }
707:     if (j==0){
708:       VecRestoreArray(Bottom,&boundary);
709:     } else if (j==1){
710:       VecRestoreArray(Top,&boundary);
711:     } else if (j==2){
712:       VecRestoreArray(Left,&boundary);
713:     } else if (j==3){
714:       VecRestoreArray(Right,&boundary);
715:     }
716:   }

718:   /* Scale the boundary if desired */

720:   PetscOptionsGetReal(NULL,NULL,"-bottom",&scl,&flg);
721:   if (flg){
722:     VecScale(Bottom, scl);
723:   }
724:   PetscOptionsGetReal(NULL,NULL,"-top",&scl,&flg);
725:   if (flg){
726:     VecScale(Top, scl);
727:   }
728:   PetscOptionsGetReal(NULL,NULL,"-right",&scl,&flg);
729:   if (flg){
730:     VecScale(Right, scl);
731:   }

733:   PetscOptionsGetReal(NULL,NULL,"-left",&scl,&flg);
734:   if (flg){
735:     VecScale(Left, scl);
736:   }
737:   return 0;
738: }


741: /* ------------------------------------------------------------------- */
742: /*
743:    MSA_Plate -  Calculates an obstacle for surface to stretch over.

745:    Input Parameter:
746: .  user - user-defined application context

748:    Output Parameter:
749: .  user - user-defined application context
750: */
751: static PetscErrorCode MSA_Plate(Vec XL,Vec XU,void *ctx){

753:   AppCtx         *user=(AppCtx *)ctx;
755:   PetscInt       i,j,row;
756:   PetscInt       xs,ys,xm,ym;
757:   PetscInt       mx=user->mx, my=user->my, bmy, bmx;
758:   PetscReal      t1,t2,t3;
759:   PetscReal      *xl, lb=PETSC_NINFINITY, ub=PETSC_INFINITY;
760:   PetscBool      cylinder;

762:   user->bmy = PetscMax(0,user->bmy);user->bmy = PetscMin(my,user->bmy);
763:   user->bmx = PetscMax(0,user->bmx);user->bmx = PetscMin(mx,user->bmx);
764:   bmy=user->bmy; bmx=user->bmx;

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

768:   VecSet(XL, lb);
769:   VecSet(XU, ub);

771:   VecGetArray(XL,&xl);

773:   PetscOptionsHasName(NULL,NULL,"-cylinder",&cylinder);
774:   /* Compute the optional lower box */
775:   if (cylinder){
776:     for (i=xs; i< xs+xm; i++){
777:       for (j=ys; j<ys+ym; j++){
778:         row=(j-ys)*xm + (i-xs);
779:         t1=(2.0*i-mx)*bmy;
780:         t2=(2.0*j-my)*bmx;
781:         t3=bmx*bmx*bmy*bmy;
782:         if ( t1*t1 + t2*t2 <= t3 ){
783:           xl[row] = user->bheight;
784:         }
785:       }
786:     }
787:   } else {
788:     /* Compute the optional lower box */
789:     for (i=xs; i< xs+xm; i++){
790:       for (j=ys; j<ys+ym; j++){
791:         row=(j-ys)*xm + (i-xs);
792:         if (i>=(mx-bmx)/2 && i<mx-(mx-bmx)/2 &&
793:             j>=(my-bmy)/2 && j<my-(my-bmy)/2 ){
794:           xl[row] = user->bheight;
795:         }
796:       }
797:     }
798:   }
799:     VecRestoreArray(XL,&xl);

801:   return 0;
802: }


805: /* ------------------------------------------------------------------- */
806: /*
807:    MSA_InitialPoint - Calculates the initial guess in one of three ways.

809:    Input Parameters:
810: .  user - user-defined application context
811: .  X - vector for initial guess

813:    Output Parameters:
814: .  X - newly computed initial guess
815: */
816: static PetscErrorCode MSA_InitialPoint(AppCtx * user, Vec X)
817: {
819:   PetscInt       start=-1,i,j;
820:   PetscReal      zero=0.0;
821:   PetscBool      flg;

823:   PetscOptionsGetInt(NULL,NULL,"-start",&start,&flg);
824:   if (flg && start==0){ /* The zero vector is reasonable */
825:     VecSet(X, zero);
826:   } else if (flg && start>0){ /* Try a random start between -0.5 and 0.5 */
827:     PetscRandom rctx;  PetscReal np5=-0.5;

829:     PetscRandomCreate(MPI_COMM_WORLD,&rctx);
830:     for (i=0; i<start; i++){
831:       VecSetRandom(X, rctx);
832:     }
833:     PetscRandomDestroy(&rctx);
834:     VecShift(X, np5);

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

838:     PetscInt row,xs,xm,gxs,gxm,ys,ym,gys,gym;
839:     PetscInt mx=user->mx,my=user->my;
840:     PetscReal *x,*left,*right,*bottom,*top;
841:     Vec    localX = user->localX;

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

847:     /* Get pointers to vector data */
848:     VecGetArray(user->Top,&top);
849:     VecGetArray(user->Bottom,&bottom);
850:     VecGetArray(user->Left,&left);
851:     VecGetArray(user->Right,&right);

853:     VecGetArray(localX,&x);
854:     /* Perform local computations */
855:     for (j=ys; j<ys+ym; j++){
856:       for (i=xs; i< xs+xm; i++){
857:         row=(j-gys)*gxm + (i-gxs);
858:         x[row] = ( (j+1)*bottom[i-xs+1]/my + (my-j+1)*top[i-xs+1]/(my+2)+
859:                    (i+1)*left[j-ys+1]/mx + (mx-i+1)*right[j-ys+1]/(mx+2))/2.0;
860:       }
861:     }

863:     /* Restore vectors */
864:     VecRestoreArray(localX,&x);

866:     VecRestoreArray(user->Left,&left);
867:     VecRestoreArray(user->Top,&top);
868:     VecRestoreArray(user->Bottom,&bottom);
869:     VecRestoreArray(user->Right,&right);

871:     /* Scatter values into global vector */
872:     DMLocalToGlobalBegin(user->dm,localX,INSERT_VALUES,X);
873:     DMLocalToGlobalEnd(user->dm,localX,INSERT_VALUES,X);

875:   }
876:   return 0;
877: }

879: /* For testing matrix free submatrices */
880: PetscErrorCode MatrixFreeHessian(Tao tao, Vec x, Mat H, Mat Hpre, void *ptr)
881: {
883:   AppCtx         *user = (AppCtx*)ptr;
885:   FormHessian(tao,x,user->H,user->H,ptr);
886:   return(0);
887: }
888: PetscErrorCode MyMatMult(Mat H_shell, Vec X, Vec Y)
889: {
891:   void           *ptr;
892:   AppCtx         *user;
894:   MatShellGetContext(H_shell,&ptr);
895:   user = (AppCtx*)ptr;
896:   MatMult(user->H,X,Y);
897:   return(0);
898: }


901: /*TEST

903:    build:
904:       requires: !complex

906:    test:
907:       args: -tao_smonitor -mx 8 -my 6 -bmx 3 -bmy 3 -bheight 0.2 -tao_type tron
908:       requires: !single

910:    test:
911:       suffix: 2
912:       nsize: 2
913:       args: -tao_smonitor -mx 8 -my 8 -bmx 2 -bmy 5 -bheight 0.3 -tao_type blmvm -tao_gttol 1.e-5
914:       requires: !single

916:    test:
917:       suffix: 3
918:       nsize: 3
919:       args: -tao_smonitor -mx 8 -my 12 -bmx 4 -bmy 10 -bheight 0.1 -tao_type tron -tao_gttol 1.e-5
920:       requires: !single

922:    test:
923:       suffix: 4
924:       nsize: 3
925:       args: -tao_smonitor -mx 8 -my 12 -bmx 4 -bmy 10 -bheight 0.1 -tao_subset_type mask -tao_type tron -tao_gttol 1.e-5
926:       requires: !single

928:    test:
929:       suffix: 5
930:       nsize: 3
931:       args: -tao_smonitor -mx 8 -my 12 -bmx 4 -bmy 10 -bheight 0.1 -tao_subset_type matrixfree -matrixfree -pc_type none -tao_type tron -tao_gttol 1.e-5
932:       requires: !single

934:    test:
935:       suffix: 6
936:       nsize: 3
937:       args: -tao_smonitor -mx 8 -my 12 -bmx 4 -bmy 10 -bheight 0.1 -tao_subset_type matrixfree -matrixfree -tao_type blmvm -tao_gttol 1.e-5
938:       requires: !single

940:    test:
941:       suffix: 7
942:       nsize: 3
943:       args: -tao_smonitor -mx 8 -my 12 -bmx 4 -bmy 10 -bheight 0.1 -tao_subset_type matrixfree -pc_type none -tao_type gpcg -tao_gttol 1.e-5
944:       requires: !single

946: TEST*/