Actual source code: plate2.c

  1: /*$Id: plate2.c 1.74 05/05/11 08:44:55-05:00 sarich@zorak.(none) $*/

  3: #include "petscda.h"
 4:  #include tao.h

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

 22: /*T 
 23:    Concepts: TAO - Solving a bound constrained minimization problem
 24:    Routines: TaoInitialize(); TaoFinalize();
 25:    Routines: TaoApplicationCreate(); TaoAppDestroy();
 26:    Routines: TaoAppSetObjectiveAndGradientRoutine(); TaoAppSetHessianRoutine(); 
 27:    Routines: TaoAppSetInitialSolutionVec(); TaoAppSetHessianMat();
 28:    Routines: TaoAppSetVariableBounds();
 29:    Routines: TaoCreate();  TaoDestroy(); 
 30:    Routines: TaoSetOptions();
 31:    Processors: n
 32: T*/


 35: /* 
 36:    User-defined application context - contains data needed by the 
 37:    application-provided call-back routines, FormFunctionGradient(),
 38:    FormHessian().
 39: */
 40: typedef struct {
 41:   /* problem parameters */
 42:   PetscReal      bheight;                  /* Height of plate under the surface */
 43:   int         mx, my;                   /* discretization in x, y directions */
 44:   int         bmx,bmy;                  /* Size of plate under the surface */
 45:   Vec         Bottom, Top, Left, Right; /* boundary values */
 46: 
 47:   /* Working space */
 48:   Vec         localX, localV;           /* ghosted local vector */
 49:   DA          da;                       /* distributed array data structure */
 50:   Mat         H;
 51: } AppCtx;

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

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

 63: int main( int argc, char **argv )
 64: {
 65:   int        info;                 /* used to check for functions returning nonzeros */
 66:   int        Nx, Ny;               /* number of processors in x- and y- directions */
 67:   int        m, N;                 /* number of local and global elements in vectors */
 68:   Vec        x;                    /* solution vector */
 69:   PetscTruth   flg;                /* A return variable when checking for user options */
 70:   TAO_SOLVER tao;                  /* TAO_SOLVER solver context */
 71:   TAO_APPLICATION plateapp;        /* The PETSc application */
 72:   TaoMethod  method = "tao_blmvm"; /* default minimization method */
 73:   double     ff,gnorm,cnorm;       /* iteration information */
 74:   int        iter;
 75:   ISLocalToGlobalMapping isltog;   /* local-to-global mapping object */
 76:   TaoTerminateReason reason;
 77:   AppCtx     user;                 /* user-defined work context */

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

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

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

 91:   user.bmx = user.mx/2; user.bmy = user.my/2;
 92:   info = PetscOptionsGetInt(PETSC_NULL,"-bmx",&user.bmx,&flg); CHKERRQ(info);
 93:   info = PetscOptionsGetInt(PETSC_NULL,"-bmy",&user.bmy,&flg); CHKERRQ(info);

 95:   PetscPrintf(PETSC_COMM_WORLD,"\n---- Minimum Surface Area With Plate Problem -----\n");
 96:   PetscPrintf(PETSC_COMM_WORLD,"mx:%d, my:%d, bmx:%d, bmy:%d, height:%4.2f\n",
 97:               user.mx,user.my,user.bmx,user.bmy,user.bheight);

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

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

105:   /*
106:      A two dimensional distributed array will help define this problem,
107:      which derives from an elliptic PDE on two dimensional domain.  From
108:      the distributed array, Create the vectors.
109:   */
110:   info = DACreate2d(MPI_COMM_WORLD,DA_NONPERIODIC,DA_STENCIL_BOX,user.mx,
111:                     user.my,Nx,Ny,1,1,PETSC_NULL,PETSC_NULL,&user.da); CHKERRQ(info);

113:   /*
114:      Extract global and local vectors from DA; The local vectors are
115:      used solely as work space for the evaluation of the function, 
116:      gradient, and Hessian.  Duplicate for remaining vectors that are 
117:      the same types.
118:   */
119:   info = DACreateGlobalVector(user.da,&x); CHKERRQ(info); /* Solution */
120:   info = DACreateLocalVector(user.da,&user.localX); CHKERRQ(info);
121:   info = VecDuplicate(user.localX,&user.localV); CHKERRQ(info);

123:   /* 
124:      Create a matrix data structure to store the Hessian.
125:      Here we (optionally) also
126:      associate the local numbering scheme with the matrix so that
127:      later we can use local indices for matrix assembly.  We could
128:      alternatively use global indices for matrix assembly.
129:   */
130:   info = VecGetLocalSize(x,&m); CHKERRQ(info);
131:   info = MatCreateMPIAIJ(MPI_COMM_WORLD,m,m,N,N,7,PETSC_NULL,
132:                          3,PETSC_NULL,&(user.H)); CHKERRQ(info);
133:   info = MatSetOption(user.H,MAT_SYMMETRIC); CHKERRQ(info);

135:   info = DAGetISLocalToGlobalMapping(user.da,&isltog); CHKERRQ(info);
136:   info = MatSetLocalToGlobalMapping(user.H,isltog); CHKERRQ(info);


139:   /* The TAO code begins here */

141:   /* 
142:      Create TAO solver and set desired solution method 
143:      The method must either be 'tao_tron' or 'tao_blmvm'
144:      If blmvm is used, then hessian function is not called.
145:   */
146:   info = TaoCreate(MPI_COMM_WORLD,method,&tao); CHKERRQ(info);
147:   info = TaoApplicationCreate(MPI_COMM_WORLD,&plateapp); CHKERRQ(info);

149:   /* Set initial solution guess; */
150:   info = MSA_BoundaryConditions(&user); CHKERRQ(info);
151:   info = MSA_InitialPoint(&user,x); CHKERRQ(info);
152:   info = TaoAppSetInitialSolutionVec(plateapp,x); CHKERRQ(info);

154:   /* Set routines for function, gradient and hessian evaluation */
155:   info = TaoAppSetObjectiveAndGradientRoutine(plateapp,FormFunctionGradient,(void*) &user);
156:   CHKERRQ(info);
157: 
158:   info = TaoAppSetHessianMat(plateapp,user.H,user.H); CHKERRQ(info);
159:   info = TaoAppSetHessianRoutine(plateapp,FormHessian,(void*)&user);
160:   CHKERRQ(info);

162:   /* Set Variable bounds */
163:   info = TaoAppSetVariableBoundsRoutine(plateapp,MSA_Plate,(void*)&user); CHKERRQ(info);

165:   /* Check for any tao command line options */
166:   info = TaoSetOptions(plateapp,tao); CHKERRQ(info);

168:   /* SOLVE THE APPLICATION */
169:   info = TaoSolveApplication(plateapp,tao);  CHKERRQ(info);

171:   /* Get information on termination */
172:   info = TaoGetSolutionStatus(tao,&iter,&ff,&gnorm,&cnorm,0,&reason); CHKERRQ(info);
173:   if (reason <= 0){
174:     PetscPrintf(MPI_COMM_WORLD,"Try a different TAO method, adjust some parameters, or check the function evaluation routines\n");
175:     PetscPrintf(MPI_COMM_WORLD,"Iteration: %d, f: %4.2e, Residual: %4.2e, Infeas: %4.2e\n",iter,ff,gnorm,cnorm);
176:   }
177:   /* Free TAO data structures */
178:   info = TaoDestroy(tao); CHKERRQ(info);
179:   info = TaoAppDestroy(plateapp); CHKERRQ(info);

181:   /* Free PETSc data structures */
182:   info = VecDestroy(x); CHKERRQ(info);
183:   info = MatDestroy(user.H); CHKERRQ(info);
184:   info = VecDestroy(user.localX); CHKERRQ(info);
185:   info = VecDestroy(user.localV); CHKERRQ(info);
186:   info = VecDestroy(user.Bottom); CHKERRQ(info);
187:   info = VecDestroy(user.Top); CHKERRQ(info);
188:   info = VecDestroy(user.Left); CHKERRQ(info);
189:   info = VecDestroy(user.Right); CHKERRQ(info);
190:   info = DADestroy(user.da); CHKERRQ(info);

192:   /* Finalize TAO and PETSc */
193:   PetscFinalize();
194:   TaoFinalize();
195: 
196:   return 0;
197: }

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

203:     Input Parameters:
204: .   taoapp     - the TAO_APPLICATION context
205: .   X      - input vector
206: .   userCtx - optional user-defined context, as set by TaoSetPetscFunctionGradient()
207:     
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: int FormFunctionGradient(TAO_APPLICATION taoapp, Vec X, double *fcn, Vec G,void *userCtx)
223: {
224:   AppCtx * user = (AppCtx *) userCtx;
225:   int    info,i,j,row;
226:   int    mx=user->mx, my=user->my;
227:   int    xs,xm,gxs,gxm,ys,ym,gys,gym;
228:   double ft=0;
229:   PetscReal zero=0.0;
230:   PetscReal hx=1.0/(mx+1),hy=1.0/(my+1), hydhx=hy/hx, hxdhy=hx/hy, area=0.5*hx*hy;
231:   PetscReal rhx=mx+1, rhy=my+1;
232:   PetscReal f1,f2,f3,f4,f5,f6,d1,d2,d3,d4,d5,d6,d7,d8,xc,xl,xr,xt,xb,xlt,xrb;
233:   PetscReal df1dxc,df2dxc,df3dxc,df4dxc,df5dxc,df6dxc;
234:   PetscReal *g, *x,*left,*right,*bottom,*top;
235:   Vec    localX = user->localX, localG = user->localV;

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

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

245:   /* Initialize vector to zero */
246:   info = VecSet(localG, zero); CHKERRQ(info);

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

256:   /* Compute function over the locally owned part of the mesh */
257:   for (j=ys; j<ys+ym; j++){
258:     for (i=xs; i< xs+xm; i++){
259:       row=(j-gys)*gxm + (i-gxs);
260: 
261:       xc = x[row];
262:       xlt=xrb=xl=xr=xb=xt=xc;
263: 
264:       if (i==0){ /* left side */
265:         xl= left[j-ys+1];
266:         xlt = left[j-ys+2];
267:       } else {
268:         xl = x[row-1];
269:       }

271:       if (j==0){ /* bottom side */
272:         xb=bottom[i-xs+1];
273:         xrb = bottom[i-xs+2];
274:       } else {
275:         xb = x[row-gxm];
276:       }
277: 
278:       if (i+1 == gxs+gxm){ /* right side */
279:         xr=right[j-ys+1];
280:         xrb = right[j-ys];
281:       } else {
282:         xr = x[row+1];
283:       }

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

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

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

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

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

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

340:       g[row] = (df1dxc+df2dxc+df3dxc+df4dxc+df5dxc+df6dxc ) * 0.5;
341: 
342:     }
343:   }


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

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

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

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

391: 
392:   /* Restore vectors */
393:   info = VecRestoreArray(localX,&x); CHKERRQ(info);
394:   info = VecRestoreArray(localG,&g); CHKERRQ(info);
395:   info = VecRestoreArray(user->Left,&left); CHKERRQ(info);
396:   info = VecRestoreArray(user->Top,&top); CHKERRQ(info);
397:   info = VecRestoreArray(user->Bottom,&bottom); CHKERRQ(info);
398:   info = VecRestoreArray(user->Right,&right); CHKERRQ(info);

400:   /* Scatter values to global vector */
401:   info = DALocalToGlobal(user->da,localG,INSERT_VALUES,G); CHKERRQ(info);

403:   info = PetscLogFlops(70*xm*ym); CHKERRQ(info);

405:   return 0;
406: }

408: /* ------------------------------------------------------------------- */
411: /*
412:    FormHessian - Evaluates Hessian matrix.

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

419:    Output Parameters:
420: .  A    - Hessian matrix
421: .  B    - optionally different preconditioning matrix
422: .  flag - flag indicating matrix structure

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

430:    Two methods are available for imposing this transformation
431:    when setting matrix entries:
432:      (A) MatSetValuesLocal(), using the local ordering (including
433:          ghost points!)
434:          - Do the following two steps once, before calling TaoSolve()
435:            - Use DAGetISLocalToGlobalMapping() to extract the
436:              local-to-global map from the DA
437:            - Associate this map with the matrix by calling
438:              MatSetLocalToGlobalMapping() 
439:          - Then set matrix entries using the local ordering
440:            by calling MatSetValuesLocal()
441:      (B) MatSetValues(), using the global ordering 
442:          - Use DAGetGlobalIndices() to extract the local-to-global map
443:          - Then apply this map explicitly yourself
444:          - Set matrix entries using the global ordering by calling
445:            MatSetValues()
446:    Option (A) seems cleaner/easier in many cases, and is the procedure
447:    used in this example.
448: */
449: int FormHessian(TAO_APPLICATION taoapp,Vec X,Mat *Hptr, Mat *Hpc, MatStructure *flag, void *ptr)
450: {
451:   int    info;
452:   AppCtx *user = (AppCtx *) ptr;
453:   Mat Hessian = *Hpc;
454:   int    i,j,k,row;
455:   int    mx=user->mx, my=user->my;
456:   int    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:   PetscTruth assembled;


467:   /* Set various matrix options */
468:   info = MatSetOption(Hessian,MAT_IGNORE_OFF_PROC_ENTRIES); CHKERRQ(info);
469:   info = MatSetOption(Hessian,MAT_COLUMNS_SORTED); CHKERRQ(info);
470:   info = MatSetOption(Hessian,MAT_ROWS_SORTED); CHKERRQ(info);

472:   /* Initialize matrix entries to zero */
473:   info = MatAssembled(Hessian,&assembled); CHKERRQ(info);
474:   if (assembled){info = MatZeroEntries(Hessian);  CHKERRQ(info);}

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

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

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

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

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

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

497:       row=(j-gys)*gxm + (i-gxs);
498: 
499:       xc = x[row];
500:       xlt=xrb=xl=xr=xb=xt=xc;

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

524:       if (j+1==gys+gym){
525:         xt=top[i-xs+1];
526:         xlt = top[i-xs];
527:       }else {
528:         xt = x[row+gxm];
529:       }

531:       if (i>gxs && j+1<gys+gym){
532:         xlt = x[row-1+gxm];
533:       }
534:       if (j>gys && i+1<gxs+gxm){
535:         xrb = x[row+1-gxm];
536:       }


539:       d1 = (xc-xl)*rhx;
540:       d2 = (xc-xr)*rhx;
541:       d3 = (xc-xt)*rhy;
542:       d4 = (xc-xb)*rhy;
543:       d5 = (xrb-xr)*rhy;
544:       d6 = (xrb-xb)*rhx;
545:       d7 = (xlt-xl)*rhy;
546:       d8 = (xlt-xt)*rhx;
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*=0.5; hr*=0.5; ht*=0.5; hb*=0.5; hbr*=0.5; htl*=0.5;  hc*=0.5;

575:       k=0;
576:       if (j>0){
577:         v[k]=hb; col[k]=row - gxm; k++;
578:       }
579: 
580:       if (j>0 && i < mx -1){
581:         v[k]=hbr; col[k]=row - gxm+1; k++;
582:       }
583: 
584:       if (i>0){
585:         v[k]= hl; col[k]=row - 1; k++;
586:       }
587: 
588:       v[k]= hc; col[k]=row; k++;
589: 
590:       if (i < mx-1 ){
591:         v[k]= hr; col[k]=row+1; k++;
592:       }
593: 
594:       if (i>0 && j < my-1 ){
595:         v[k]= htl; col[k] = row+gxm-1; k++;
596:       }
597: 
598:       if (j < my-1 ){
599:         v[k]= ht; col[k] = row+gxm; k++;
600:       }
601: 
602:       /* 
603:          Set matrix values using local numbering, which was defined
604:          earlier, in the main routine.
605:       */
606:       info = MatSetValuesLocal(Hessian,1,&row,k,col,v,INSERT_VALUES);
607:       CHKERRQ(info);
608: 
609:     }
610:   }
611: 
612:   /* Restore vectors */
613:   info = VecRestoreArray(localX,&x); CHKERRQ(info);
614:   info = VecRestoreArray(user->Left,&left); CHKERRQ(info);
615:   info = VecRestoreArray(user->Top,&top); CHKERRQ(info);
616:   info = VecRestoreArray(user->Bottom,&bottom); CHKERRQ(info);
617:   info = VecRestoreArray(user->Right,&right); CHKERRQ(info);

619:   /* Assemble the matrix */
620:   info = MatAssemblyBegin(Hessian,MAT_FINAL_ASSEMBLY); CHKERRQ(info);
621:   info = MatAssemblyEnd(Hessian,MAT_FINAL_ASSEMBLY); CHKERRQ(info);

623:   info = PetscLogFlops(199*xm*ym); CHKERRQ(info);
624:   return 0;
625: }

627: /* ------------------------------------------------------------------- */
630: /* 
631:    MSA_BoundaryConditions -  Calculates the boundary conditions for
632:    the region.

634:    Input Parameter:
635: .  user - user-defined application context

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

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


659:   bsize=xm+2;
660:   lsize=ym+2;
661:   rsize=ym+2;
662:   tsize=xm+2;

664:   info = VecCreateMPI(MPI_COMM_WORLD,bsize,PETSC_DECIDE,&Bottom); CHKERRQ(info);
665:   info = VecCreateMPI(MPI_COMM_WORLD,tsize,PETSC_DECIDE,&Top); CHKERRQ(info);
666:   info = VecCreateMPI(MPI_COMM_WORLD,lsize,PETSC_DECIDE,&Left); CHKERRQ(info);
667:   info = VecCreateMPI(MPI_COMM_WORLD,rsize,PETSC_DECIDE,&Right); CHKERRQ(info);

669:   user->Top=Top;
670:   user->Left=Left;
671:   user->Bottom=Bottom;
672:   user->Right=Right;

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

676:   for (j=0; j<4; j++){
677:     if (j==0){
678:       yt=b;
679:       xt=l+hx*xs;
680:       limit=bsize;
681:       VecGetArray(Bottom,&boundary);
682:     } else if (j==1){
683:       yt=t;
684:       xt=l+hx*xs;
685:       limit=tsize;
686:       VecGetArray(Top,&boundary);
687:     } else if (j==2){
688:       yt=b+hy*ys;
689:       xt=l;
690:       limit=lsize;
691:       VecGetArray(Left,&boundary);
692:     } else if (j==3){
693:       yt=b+hy*ys;
694:       xt=r;
695:       limit=rsize;
696:       VecGetArray(Right,&boundary);
697:     }

699:     for (i=0; i<limit; i++){
700:       u1=xt;
701:       u2=-yt;
702:       for (k=0; k<maxits; k++){
703:         nf1=u1 + u1*u2*u2 - u1*u1*u1/three-xt;
704:         nf2=-u2 - u1*u1*u2 + u2*u2*u2/three-yt;
705:         fnorm=sqrt(nf1*nf1+nf2*nf2);
706:         if (fnorm <= tol) break;
707:         njac11=one+u2*u2-u1*u1;
708:         njac12=two*u1*u2;
709:         njac21=-two*u1*u2;
710:         njac22=-one - u1*u1 + u2*u2;
711:         det = njac11*njac22-njac21*njac12;
712:         u1 = u1-(njac22*nf1-njac12*nf2)/det;
713:         u2 = u2-(njac11*nf2-njac21*nf1)/det;
714:       }

716:       boundary[i]=u1*u1-u2*u2;
717:       if (j==0 || j==1) {
718:         xt=xt+hx;
719:       } else if (j==2 || j==3){
720:         yt=yt+hy;
721:       }
722: 
723:     }
724: 
725:     if (j==0){
726:       info = VecRestoreArray(Bottom,&boundary); CHKERRQ(info);
727:     } else if (j==1){
728:       info = VecRestoreArray(Top,&boundary); CHKERRQ(info);
729:     } else if (j==2){
730:       info = VecRestoreArray(Left,&boundary); CHKERRQ(info);
731:     } else if (j==3){
732:       info = VecRestoreArray(Right,&boundary); CHKERRQ(info);
733:     }

735:   }

737:   /* Scale the boundary if desired */

739:   info = PetscOptionsGetReal(PETSC_NULL,"-bottom",&scl,&flg);
740:   CHKERRQ(info);
741:   if (flg){
742:     info = VecScale(Bottom, scl); CHKERRQ(info);
743:   }
744: 
745:   info = PetscOptionsGetReal(PETSC_NULL,"-top",&scl,&flg);
746:   CHKERRQ(info);
747:   if (flg){
748:     info = VecScale(Top, scl); CHKERRQ(info);
749:   }
750: 
751:   info = PetscOptionsGetReal(PETSC_NULL,"-right",&scl,&flg);
752:   CHKERRQ(info);
753:   if (flg){
754:     info = VecScale(Right, scl); CHKERRQ(info);
755:   }
756: 
757:   info = PetscOptionsGetReal(PETSC_NULL,"-left",&scl,&flg);
758:   CHKERRQ(info);
759:   if (flg){
760:     info = VecScale(Left, scl); CHKERRQ(info);
761:   }

763:   return 0;
764: }


767: /* ------------------------------------------------------------------- */
770: /* 
771:    MSA_Plate -  Calculates an obstacle for surface to stretch over.

773:    Input Parameter:
774: .  user - user-defined application context

776:    Output Parameter:
777: .  user - user-defined application context
778: */
779: static int MSA_Plate(TAO_APPLICATION taoapp, Vec XL,Vec XU,void *ctx){

781:   AppCtx *user=(AppCtx *)ctx;
782:   int i,j,row,info;
783:   int xs,ys,xm,ym;
784:   int mx=user->mx, my=user->my, bmy, bmx;
785:   PetscReal t1,t2,t3;
786:   PetscReal *xl, lb=TAO_NINFINITY, ub=TAO_INFINITY;
787:   PetscTruth cylinder;

789:   user->bmy = PetscMax(0,user->bmy);user->bmy = PetscMin(my,user->bmy);
790:   user->bmx = PetscMax(0,user->bmx);user->bmx = PetscMin(mx,user->bmx);
791:   bmy=user->bmy, bmx=user->bmx;

793:   info = DAGetCorners(user->da,&xs,&ys,PETSC_NULL,&xm,&ym,PETSC_NULL); CHKERRQ(info);

795:   info = VecSet(XL, lb); CHKERRQ(info);
796:   info = VecSet(XU, ub); CHKERRQ(info);

798:   info = VecGetArray(XL,&xl); CHKERRQ(info);

800:   info = PetscOptionsHasName(PETSC_NULL,"-cylinder",&cylinder); CHKERRQ(info);
801:   /* Compute the optional lower box */
802:   if (cylinder){
803:     for (i=xs; i< xs+xm; i++){
804:       for (j=ys; j<ys+ym; j++){
805:         row=(j-ys)*xm + (i-xs);
806:         t1=(2.0*i-mx)*bmy;
807:         t2=(2.0*j-my)*bmx;
808:         t3=bmx*bmx*bmy*bmy;
809:         if ( t1*t1 + t2*t2 <= t3 ){
810:           xl[row] = user->bheight;
811:         }
812:       }
813:     }
814:   } else {
815:     /* Compute the optional lower box */
816:     for (i=xs; i< xs+xm; i++){
817:       for (j=ys; j<ys+ym; j++){
818:         row=(j-ys)*xm + (i-xs);
819:         if (i>=(mx-bmx)/2 && i<mx-(mx-bmx)/2 &&
820:             j>=(my-bmy)/2 && j<my-(my-bmy)/2 ){
821:           xl[row] = user->bheight;
822:         }
823:       }
824:     }
825:   }
826:     info = VecRestoreArray(XL,&xl); CHKERRQ(info);

828:   return 0;
829: }


832: /* ------------------------------------------------------------------- */
835: /*
836:    MSA_InitialPoint - Calculates the initial guess in one of three ways. 

838:    Input Parameters:
839: .  user - user-defined application context
840: .  X - vector for initial guess

842:    Output Parameters:
843: .  X - newly computed initial guess
844: */
845: static int MSA_InitialPoint(AppCtx * user, Vec X)
846: {
847:   int      start=-1,i,j,info;
848:   PetscReal   zero=0.0;
849:   PetscTruth flg;

851:   info = PetscOptionsGetInt(PETSC_NULL,"-start",&start,&flg); CHKERRQ(info);

853:   if (flg && start==0){ /* The zero vector is reasonable */
854: 
855:     info = VecSet(X, zero); CHKERRQ(info);
856:     /* PetscLogInfo((user,"Min. Surface Area Problem: Start with 0 vector \n")); */

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

860:     PetscRandom rctx;  PetscReal np5=-0.5;

862:     info = PetscRandomCreate(MPI_COMM_WORLD,RANDOM_DEFAULT,&rctx);
863:     CHKERRQ(info);
864:     for (i=0; i<start; i++){
865:       info = VecSetRandom(X, rctx); CHKERRQ(info);
866:     }
867:     info = PetscRandomDestroy(rctx); CHKERRQ(info);
868:     info = VecShift(X, np5);
869:     /* PetscLogInfo((user,"Min. Surface Area Problem: Start with random vector %d\n",start)); */

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

873:     int    row,xs,xm,gxs,gxm,ys,ym,gys,gym;
874:     int    mx=user->mx,my=user->my;
875:     PetscReal *x,*left,*right,*bottom,*top;
876:     Vec    localX = user->localX;
877: 
878:     /* Get local mesh boundaries */
879:     info = DAGetCorners(user->da,&xs,&ys,PETSC_NULL,&xm,&ym,PETSC_NULL); CHKERRQ(info);
880:     info = DAGetGhostCorners(user->da,&gxs,&gys,PETSC_NULL,&gxm,&gym,PETSC_NULL); CHKERRQ(info);
881: 
882:     /* Get pointers to vector data */
883:     info = VecGetArray(user->Top,&top); CHKERRQ(info);
884:     info = VecGetArray(user->Bottom,&bottom); CHKERRQ(info);
885:     info = VecGetArray(user->Left,&left); CHKERRQ(info);
886:     info = VecGetArray(user->Right,&right); CHKERRQ(info);

888:     info = VecGetArray(localX,&x); CHKERRQ(info);
889:     /* Perform local computations */
890:     for (j=ys; j<ys+ym; j++){
891:       for (i=xs; i< xs+xm; i++){
892:         row=(j-gys)*gxm + (i-gxs);
893:         x[row] = ( (j+1)*bottom[i-xs+1]/my + (my-j+1)*top[i-xs+1]/(my+2)+
894:                    (i+1)*left[j-ys+1]/mx + (mx-i+1)*right[j-ys+1]/(mx+2))/2.0;
895:       }
896:     }
897: 
898:     /* Restore vectors */
899:     info = VecRestoreArray(localX,&x); CHKERRQ(info);

901:     info = VecRestoreArray(user->Left,&left); CHKERRQ(info);
902:     info = VecRestoreArray(user->Top,&top); CHKERRQ(info);
903:     info = VecRestoreArray(user->Bottom,&bottom); CHKERRQ(info);
904:     info = VecRestoreArray(user->Right,&right); CHKERRQ(info);
905: 
906:     /* Scatter values into global vector */
907:     info = DALocalToGlobal(user->da,localX,INSERT_VALUES,X); CHKERRQ(info);
908: 
909:   }
910:   return 0;
911: }