Actual source code: elliptic.c

  1: #include <petsc/private/taoimpl.h>

  3: /*T
  4:    Concepts: TAO^Solving a system of nonlinear equations, nonlinear least squares
  5:    Routines: TaoCreate();
  6:    Routines: TaoSetType();
  7:    Routines: TaoSetInitialVector();
  8:    Routines: TaoSetObjectiveRoutine();
  9:    Routines: TaoSetGradientRoutine();
 10:    Routines: TaoSetConstraintsRoutine();
 11:    Routines: TaoSetJacobianStateRoutine();
 12:    Routines: TaoSetJacobianDesignRoutine();
 13:    Routines: TaoSetStateDesignIS();
 14:    Routines: TaoSetFromOptions();
 15:    Routines: TaoSolve();
 16:    Routines: TaoDestroy();
 17:    Processors: n
 18: T*/

 20: typedef struct {
 21:   PetscInt n; /* Number of total variables */
 22:   PetscInt m; /* Number of constraints */
 23:   PetscInt nstate;
 24:   PetscInt ndesign;
 25:   PetscInt mx; /* grid points in each direction */
 26:   PetscInt ns; /* Number of data samples (1<=ns<=8)
 27:                   Currently only ns=1 is supported */
 28:   PetscInt ndata; /* Number of data points per sample */
 29:   IS       s_is;
 30:   IS       d_is;

 32:   VecScatter state_scatter;
 33:   VecScatter design_scatter;
 34:   VecScatter *yi_scatter, *di_scatter;
 35:   Vec        suby,subq,subd;
 36:   Mat        Js,Jd,JsPrec,JsInv,JsBlock;

 38:   PetscReal alpha; /* Regularization parameter */
 39:   PetscReal beta; /* Weight attributed to ||u||^2 in regularization functional */
 40:   PetscReal noise; /* Amount of noise to add to data */
 41:   PetscReal *ones;
 42:   Mat       Q;
 43:   Mat       MQ;
 44:   Mat       L;

 46:   Mat Grad;
 47:   Mat Av,Avwork;
 48:   Mat Div, Divwork;
 49:   Mat DSG;
 50:   Mat Diag,Ones;

 52:   Vec q;
 53:   Vec ur; /* reference */

 55:   Vec d;
 56:   Vec dwork;

 58:   Vec x; /* super vec of y,u */

 60:   Vec y; /* state variables */
 61:   Vec ywork;

 63:   Vec ytrue;

 65:   Vec u; /* design variables */
 66:   Vec uwork;

 68:   Vec utrue;

 70:   Vec js_diag;

 72:   Vec c; /* constraint vector */
 73:   Vec cwork;

 75:   Vec lwork;
 76:   Vec S;
 77:   Vec Swork,Twork,Sdiag,Ywork;
 78:   Vec Av_u;

 80:   KSP solver;
 81:   PC  prec;

 83:   PetscReal tola,tolb,tolc,told;
 84:   PetscInt  ksp_its;
 85:   PetscInt  ksp_its_initial;
 86:   PetscLogStage stages[10];
 87:   PetscBool use_ptap;
 88:   PetscBool use_lrc;
 89: } AppCtx;

 91: PetscErrorCode FormFunction(Tao, Vec, PetscReal*, void*);
 92: PetscErrorCode FormGradient(Tao, Vec, Vec, void*);
 93: PetscErrorCode FormFunctionGradient(Tao, Vec, PetscReal*, Vec, void*);
 94: PetscErrorCode FormJacobianState(Tao, Vec, Mat, Mat, Mat, void*);
 95: PetscErrorCode FormJacobianDesign(Tao, Vec, Mat,void*);
 96: PetscErrorCode FormConstraints(Tao, Vec, Vec, void*);
 97: PetscErrorCode FormHessian(Tao, Vec, Mat, Mat, void*);
 98: PetscErrorCode Gather(Vec, Vec, VecScatter, Vec, VecScatter);
 99: PetscErrorCode Scatter(Vec, Vec, VecScatter, Vec, VecScatter);
100: PetscErrorCode EllipticInitialize(AppCtx*);
101: PetscErrorCode EllipticDestroy(AppCtx*);
102: PetscErrorCode EllipticMonitor(Tao, void*);

104: PetscErrorCode StateBlockMatMult(Mat,Vec,Vec);
105: PetscErrorCode StateMatMult(Mat,Vec,Vec);

107: PetscErrorCode StateInvMatMult(Mat,Vec,Vec);
108: PetscErrorCode DesignMatMult(Mat,Vec,Vec);
109: PetscErrorCode DesignMatMultTranspose(Mat,Vec,Vec);

111: PetscErrorCode QMatMult(Mat,Vec,Vec);
112: PetscErrorCode QMatMultTranspose(Mat,Vec,Vec);

114: static  char help[]="";

116: int main(int argc, char **argv)
117: {
118:   PetscErrorCode     ierr;
119:   Vec                x0;
120:   Tao                tao;
121:   AppCtx             user;
122:   PetscInt           ntests = 1;
123:   PetscInt           i;

125:   PetscInitialize(&argc, &argv, (char*)0,help);if (ierr) return ierr;
126:   user.mx = 8;
127:   PetscOptionsBegin(PETSC_COMM_WORLD,NULL,"elliptic example",NULL);
128:   PetscOptionsInt("-mx","Number of grid points in each direction","",user.mx,&user.mx,NULL);
129:   user.ns = 6;
130:   PetscOptionsInt("-ns","Number of data samples (1<=ns<=8)","",user.ns,&user.ns,NULL);
131:   user.ndata = 64;
132:   PetscOptionsInt("-ndata","Numbers of data points per sample","",user.ndata,&user.ndata,NULL);
133:   user.alpha = 0.1;
134:   PetscOptionsReal("-alpha","Regularization parameter","",user.alpha,&user.alpha,NULL);
135:   user.beta = 0.00001;
136:   PetscOptionsReal("-beta","Weight attributed to ||u||^2 in regularization functional","",user.beta,&user.beta,NULL);
137:   user.noise = 0.01;
138:   PetscOptionsReal("-noise","Amount of noise to add to data","",user.noise,&user.noise,NULL);

140:   user.use_ptap = PETSC_FALSE;
141:   PetscOptionsBool("-use_ptap","Use ptap matrix for DSG","",user.use_ptap,&user.use_ptap,NULL);
142:   user.use_lrc = PETSC_FALSE;
143:   PetscOptionsBool("-use_lrc","Use lrc matrix for Js","",user.use_lrc,&user.use_lrc,NULL);
144:   PetscOptionsInt("-ntests","Number of times to repeat TaoSolve","",ntests,&ntests,NULL);
145:   PetscOptionsEnd();

147:   user.m = user.ns*user.mx*user.mx*user.mx; /* number of constraints */
148:   user.nstate =  user.m;
149:   user.ndesign = user.mx*user.mx*user.mx;
150:   user.n = user.nstate + user.ndesign; /* number of variables */

152:   /* Create TAO solver and set desired solution method */
153:   TaoCreate(PETSC_COMM_WORLD,&tao);
154:   TaoSetType(tao,TAOLCL);

156:   /* Set up initial vectors and matrices */
157:   EllipticInitialize(&user);

159:   Gather(user.x,user.y,user.state_scatter,user.u,user.design_scatter);
160:   VecDuplicate(user.x,&x0);
161:   VecCopy(user.x,x0);

163:   /* Set solution vector with an initial guess */
164:   TaoSetInitialVector(tao,user.x);
165:   TaoSetObjectiveRoutine(tao, FormFunction, (void *)&user);
166:   TaoSetGradientRoutine(tao, FormGradient, (void *)&user);
167:   TaoSetConstraintsRoutine(tao, user.c, FormConstraints, (void *)&user);

169:   TaoSetJacobianStateRoutine(tao, user.Js, NULL, user.JsInv, FormJacobianState, (void *)&user);
170:   TaoSetJacobianDesignRoutine(tao, user.Jd, FormJacobianDesign, (void *)&user);

172:   TaoSetStateDesignIS(tao,user.s_is,user.d_is);
173:   TaoSetFromOptions(tao);

175:   /* SOLVE THE APPLICATION */
176:   PetscLogStageRegister("Trials",&user.stages[1]);
177:   PetscLogStagePush(user.stages[1]);
178:   for (i=0; i<ntests; i++) {
179:     TaoSolve(tao);
180:     PetscPrintf(PETSC_COMM_WORLD,"KSP Iterations = %D\n",user.ksp_its);
181:     VecCopy(x0,user.x);
182:   }
183:   PetscLogStagePop();
184:   PetscBarrier((PetscObject)user.x);
185:   PetscPrintf(PETSC_COMM_WORLD,"KSP iterations within initialization: ");
186:   PetscPrintf(PETSC_COMM_WORLD,"%D\n",user.ksp_its_initial);

188:   TaoDestroy(&tao);
189:   VecDestroy(&x0);
190:   EllipticDestroy(&user);
191:   PetscFinalize();
192:   return ierr;
193: }
194: /* ------------------------------------------------------------------- */
195: /*
196:    dwork = Qy - d
197:    lwork = L*(u-ur)
198:    f = 1/2 * (dwork.dwork + alpha*lwork.lwork)
199: */
200: PetscErrorCode FormFunction(Tao tao,Vec X,PetscReal *f,void *ptr)
201: {
203:   PetscReal      d1=0,d2=0;
204:   AppCtx         *user = (AppCtx*)ptr;

207:   Scatter(X,user->y,user->state_scatter,user->u,user->design_scatter);
208:   MatMult(user->MQ,user->y,user->dwork);
209:   VecAXPY(user->dwork,-1.0,user->d);
210:   VecDot(user->dwork,user->dwork,&d1);
211:   VecWAXPY(user->uwork,-1.0,user->ur,user->u);
212:   MatMult(user->L,user->uwork,user->lwork);
213:   VecDot(user->lwork,user->lwork,&d2);
214:   *f = 0.5 * (d1 + user->alpha*d2);
215:   return(0);
216: }

218: /* ------------------------------------------------------------------- */
219: /*
220:     state: g_s = Q' *(Qy - d)
221:     design: g_d = alpha*L'*L*(u-ur)
222: */
223: PetscErrorCode FormGradient(Tao tao,Vec X,Vec G,void *ptr)
224: {
226:   AppCtx         *user = (AppCtx*)ptr;

229:   Scatter(X,user->y,user->state_scatter,user->u,user->design_scatter);
230:   MatMult(user->MQ,user->y,user->dwork);
231:   VecAXPY(user->dwork,-1.0,user->d);
232:   MatMultTranspose(user->MQ,user->dwork,user->ywork);
233:   VecWAXPY(user->uwork,-1.0,user->ur,user->u);
234:   MatMult(user->L,user->uwork,user->lwork);
235:   MatMultTranspose(user->L,user->lwork,user->uwork);
236:   VecScale(user->uwork, user->alpha);
237:   Gather(G,user->ywork,user->state_scatter,user->uwork,user->design_scatter);
238:   return(0);
239: }

241: PetscErrorCode FormFunctionGradient(Tao tao, Vec X, PetscReal *f, Vec G, void *ptr)
242: {
244:   PetscReal      d1,d2;
245:   AppCtx         *user = (AppCtx*)ptr;

248:   Scatter(X,user->y,user->state_scatter,user->u,user->design_scatter);
249:   MatMult(user->MQ,user->y,user->dwork);
250:   VecAXPY(user->dwork,-1.0,user->d);
251:   VecDot(user->dwork,user->dwork,&d1);
252:   MatMultTranspose(user->MQ,user->dwork,user->ywork);

254:   VecWAXPY(user->uwork,-1.0,user->ur,user->u);
255:   MatMult(user->L,user->uwork,user->lwork);
256:   VecDot(user->lwork,user->lwork,&d2);
257:   MatMultTranspose(user->L,user->lwork,user->uwork);
258:   VecScale(user->uwork, user->alpha);
259:   *f = 0.5 * (d1 + user->alpha*d2);
260:   Gather(G,user->ywork,user->state_scatter,user->uwork,user->design_scatter);
261:   return(0);
262: }

264: /* ------------------------------------------------------------------- */
265: /* A
266: MatShell object
267: */
268: PetscErrorCode FormJacobianState(Tao tao, Vec X, Mat J, Mat JPre, Mat JInv, void *ptr)
269: {
271:   AppCtx         *user = (AppCtx*)ptr;

274:   Scatter(X,user->y,user->state_scatter,user->u,user->design_scatter);
275:   /* DSG = Div * (1/Av_u) * Grad */
276:   VecSet(user->uwork,0);
277:   VecAXPY(user->uwork,-1.0,user->u);
278:   VecExp(user->uwork);
279:   MatMult(user->Av,user->uwork,user->Av_u);
280:   VecCopy(user->Av_u,user->Swork);
281:   VecReciprocal(user->Swork);
282:   if (user->use_ptap) {
283:     MatDiagonalSet(user->Diag,user->Swork,INSERT_VALUES);
284:     MatPtAP(user->Diag,user->Grad,MAT_REUSE_MATRIX,1.0,&user->DSG);
285:   } else {
286:     MatCopy(user->Div,user->Divwork,SAME_NONZERO_PATTERN);
287:     MatDiagonalScale(user->Divwork,NULL,user->Swork);
288:     MatProductNumeric(user->DSG);
289:   }
290:   return(0);
291: }
292: /* ------------------------------------------------------------------- */
293: /* B */
294: PetscErrorCode FormJacobianDesign(Tao tao, Vec X, Mat J, void *ptr)
295: {
297:   AppCtx         *user = (AppCtx*)ptr;

300:   Scatter(X,user->y,user->state_scatter,user->u,user->design_scatter);
301:   return(0);
302: }

304: PetscErrorCode StateBlockMatMult(Mat J_shell, Vec X, Vec Y)
305: {
307:   PetscReal      sum;
308:   AppCtx         *user;

311:   MatShellGetContext(J_shell,(void**)&user);
312:   MatMult(user->DSG,X,Y);
313:   VecSum(X,&sum);
314:   sum /= user->ndesign;
315:   VecShift(Y,sum);
316:   return(0);
317: }

319: PetscErrorCode StateMatMult(Mat J_shell, Vec X, Vec Y)
320: {
322:   PetscInt       i;
323:   AppCtx         *user;

326:   MatShellGetContext(J_shell,(void**)&user);
327:   if (user->ns == 1) {
328:     MatMult(user->JsBlock,X,Y);
329:   } else {
330:     for (i=0;i<user->ns;i++) {
331:       Scatter(X,user->subq,user->yi_scatter[i],0,0);
332:       Scatter(Y,user->suby,user->yi_scatter[i],0,0);
333:       MatMult(user->JsBlock,user->subq,user->suby);
334:       Gather(Y,user->suby,user->yi_scatter[i],0,0);
335:     }
336:   }
337:   return(0);
338: }

340: PetscErrorCode StateInvMatMult(Mat J_shell, Vec X, Vec Y)
341: {
343:   PetscInt       its,i;
344:   AppCtx         *user;

347:   MatShellGetContext(J_shell,(void**)&user);
348:   KSPSetOperators(user->solver,user->JsBlock,user->DSG);
349:   if (Y == user->ytrue) {
350:     /* First solve is done using true solution to set up problem */
351:     KSPSetTolerances(user->solver,1e-8,PETSC_DEFAULT,PETSC_DEFAULT,PETSC_DEFAULT);
352:   } else {
353:     KSPSetTolerances(user->solver,PETSC_DEFAULT,PETSC_DEFAULT,PETSC_DEFAULT,PETSC_DEFAULT);
354:   }
355:   if (user->ns == 1) {
356:     KSPSolve(user->solver,X,Y);
357:     KSPGetIterationNumber(user->solver,&its);
358:     user->ksp_its+=its;
359:   } else {
360:     for (i=0;i<user->ns;i++) {
361:       Scatter(X,user->subq,user->yi_scatter[i],0,0);
362:       Scatter(Y,user->suby,user->yi_scatter[i],0,0);
363:       KSPSolve(user->solver,user->subq,user->suby);
364:       KSPGetIterationNumber(user->solver,&its);
365:       user->ksp_its+=its;
366:       Gather(Y,user->suby,user->yi_scatter[i],0,0);
367:     }
368:   }
369:   return(0);
370: }
371: PetscErrorCode QMatMult(Mat J_shell, Vec X, Vec Y)
372: {
374:   AppCtx         *user;
375:   PetscInt       i;

378:   MatShellGetContext(J_shell,(void**)&user);
379:   if (user->ns == 1) {
380:     MatMult(user->Q,X,Y);
381:   } else {
382:     for (i=0;i<user->ns;i++) {
383:       Scatter(X,user->subq,user->yi_scatter[i],0,0);
384:       Scatter(Y,user->subd,user->di_scatter[i],0,0);
385:       MatMult(user->Q,user->subq,user->subd);
386:       Gather(Y,user->subd,user->di_scatter[i],0,0);
387:     }
388:   }
389:   return(0);
390: }

392: PetscErrorCode QMatMultTranspose(Mat J_shell, Vec X, Vec Y)
393: {
395:   AppCtx         *user;
396:   PetscInt       i;

399:   MatShellGetContext(J_shell,(void**)&user);
400:   if (user->ns == 1) {
401:     MatMultTranspose(user->Q,X,Y);
402:   } else {
403:     for (i=0;i<user->ns;i++) {
404:       Scatter(X,user->subd,user->di_scatter[i],0,0);
405:       Scatter(Y,user->suby,user->yi_scatter[i],0,0);
406:       MatMultTranspose(user->Q,user->subd,user->suby);
407:       Gather(Y,user->suby,user->yi_scatter[i],0,0);
408:     }
409:   }
410:   return(0);
411: }

413: PetscErrorCode DesignMatMult(Mat J_shell, Vec X, Vec Y)
414: {
416:   PetscInt       i;
417:   AppCtx         *user;

420:   MatShellGetContext(J_shell,(void**)&user);

422:   /* sdiag(1./v) */
423:   VecSet(user->uwork,0);
424:   VecAXPY(user->uwork,-1.0,user->u);
425:   VecExp(user->uwork);

427:   /* sdiag(1./((Av*(1./v)).^2)) */
428:   MatMult(user->Av,user->uwork,user->Swork);
429:   VecPointwiseMult(user->Swork,user->Swork,user->Swork);
430:   VecReciprocal(user->Swork);

432:   /* (Av * (sdiag(1./v) * b)) */
433:   VecPointwiseMult(user->uwork,user->uwork,X);
434:   MatMult(user->Av,user->uwork,user->Twork);

436:   /* (sdiag(1./((Av*(1./v)).^2)) * (Av * (sdiag(1./v) * b))) */
437:   VecPointwiseMult(user->Swork,user->Twork,user->Swork);

439:   if (user->ns == 1) {
440:     /* (sdiag(Grad*y(:,i)) */
441:     MatMult(user->Grad,user->y,user->Twork);

443:     /* Div * (sdiag(Grad*y(:,i)) * (sdiag(1./((Av*(1./v)).^2)) * (Av * (sdiag(1./v) * b)))) */
444:     VecPointwiseMult(user->Swork,user->Twork,user->Swork);
445:     MatMultTranspose(user->Grad,user->Swork,Y);
446:   } else {
447:     for (i=0;i<user->ns;i++) {
448:       Scatter(user->y,user->suby,user->yi_scatter[i],0,0);
449:       Scatter(Y,user->subq,user->yi_scatter[i],0,0);

451:       MatMult(user->Grad,user->suby,user->Twork);
452:       VecPointwiseMult(user->Twork,user->Twork,user->Swork);
453:       MatMultTranspose(user->Grad,user->Twork,user->subq);
454:       Gather(user->y,user->suby,user->yi_scatter[i],0,0);
455:       Gather(Y,user->subq,user->yi_scatter[i],0,0);
456:     }
457:   }
458:   return(0);
459: }

461: PetscErrorCode DesignMatMultTranspose(Mat J_shell, Vec X, Vec Y)
462: {
464:   PetscInt       i;
465:   AppCtx         *user;

468:   MatShellGetContext(J_shell,(void**)&user);
469:   VecZeroEntries(Y);

471:   /* Sdiag = 1./((Av*(1./v)).^2) */
472:   VecSet(user->uwork,0);
473:   VecAXPY(user->uwork,-1.0,user->u);
474:   VecExp(user->uwork);
475:   MatMult(user->Av,user->uwork,user->Swork);
476:   VecPointwiseMult(user->Sdiag,user->Swork,user->Swork);
477:   VecReciprocal(user->Sdiag);

479:   for (i=0;i<user->ns;i++) {
480:     Scatter(X,user->subq,user->yi_scatter[i],0,0);
481:     Scatter(user->y,user->suby,user->yi_scatter[i],0,0);

483:     /* Swork = (Div' * b(:,i)) */
484:     MatMult(user->Grad,user->subq,user->Swork);

486:     /* Twork = Grad*y(:,i) */
487:     MatMult(user->Grad,user->suby,user->Twork);

489:     /* Twork = sdiag(Twork) * Swork */
490:     VecPointwiseMult(user->Twork,user->Swork,user->Twork);

492:     /* Swork = pointwisemult(Sdiag,Twork) */
493:     VecPointwiseMult(user->Swork,user->Twork,user->Sdiag);

495:     /* Ywork = Av' * Swork */
496:     MatMultTranspose(user->Av,user->Swork,user->Ywork);

498:     /* Ywork = pointwisemult(uwork,Ywork) */
499:     VecPointwiseMult(user->Ywork,user->uwork,user->Ywork);
500:     VecAXPY(Y,1.0,user->Ywork);
501:     Gather(user->y,user->suby,user->yi_scatter[i],0,0);
502:   }
503:   return(0);
504: }

506: PetscErrorCode FormConstraints(Tao tao, Vec X, Vec C, void *ptr)
507: {
508:    /* C=Ay - q      A = Div * Sigma * Grad + hx*hx*hx*ones(n,n) */
510:    PetscReal      sum;
511:    PetscInt       i;
512:    AppCtx         *user = (AppCtx*)ptr;

515:    Scatter(X,user->y,user->state_scatter,user->u,user->design_scatter);
516:    if (user->ns == 1) {
517:      MatMult(user->Grad,user->y,user->Swork);
518:      VecPointwiseDivide(user->Swork,user->Swork,user->Av_u);
519:      MatMultTranspose(user->Grad,user->Swork,C);
520:      VecSum(user->y,&sum);
521:      sum /= user->ndesign;
522:      VecShift(C,sum);
523:    } else {
524:      for (i=0;i<user->ns;i++) {
525:       Scatter(user->y,user->suby,user->yi_scatter[i],0,0);
526:       Scatter(C,user->subq,user->yi_scatter[i],0,0);
527:       MatMult(user->Grad,user->suby,user->Swork);
528:       VecPointwiseDivide(user->Swork,user->Swork,user->Av_u);
529:       MatMultTranspose(user->Grad,user->Swork,user->subq);

531:       VecSum(user->suby,&sum);
532:       sum /= user->ndesign;
533:       VecShift(user->subq,sum);

535:       Gather(user->y,user->suby,user->yi_scatter[i],0,0);
536:       Gather(C,user->subq,user->yi_scatter[i],0,0);
537:      }
538:    }
539:    VecAXPY(C,-1.0,user->q);
540:    return(0);
541: }

543: PetscErrorCode Scatter(Vec x, Vec sub1, VecScatter scat1, Vec sub2, VecScatter scat2)
544: {

548:   VecScatterBegin(scat1,x,sub1,INSERT_VALUES,SCATTER_FORWARD);
549:   VecScatterEnd(scat1,x,sub1,INSERT_VALUES,SCATTER_FORWARD);
550:   if (sub2) {
551:     VecScatterBegin(scat2,x,sub2,INSERT_VALUES,SCATTER_FORWARD);
552:     VecScatterEnd(scat2,x,sub2,INSERT_VALUES,SCATTER_FORWARD);
553:   }
554:   return(0);
555: }

557: PetscErrorCode Gather(Vec x, Vec sub1, VecScatter scat1, Vec sub2, VecScatter scat2)
558: {

562:   VecScatterBegin(scat1,sub1,x,INSERT_VALUES,SCATTER_REVERSE);
563:   VecScatterEnd(scat1,sub1,x,INSERT_VALUES,SCATTER_REVERSE);
564:   if (sub2) {
565:     VecScatterBegin(scat2,sub2,x,INSERT_VALUES,SCATTER_REVERSE);
566:     VecScatterEnd(scat2,sub2,x,INSERT_VALUES,SCATTER_REVERSE);
567:   }
568:   return(0);
569: }

571: PetscErrorCode EllipticInitialize(AppCtx *user)
572: {
574:   PetscInt       m,n,i,j,k,l,linear_index,is,js,ks,ls,istart,iend,iblock;
575:   Vec            XX,YY,ZZ,XXwork,YYwork,ZZwork,UTwork;
576:   PetscReal      *x,*y,*z;
577:   PetscReal      h,meanut;
578:   PetscScalar    hinv,neg_hinv,half = 0.5,sqrt_beta;
579:   PetscInt       im,indx1,indx2,indy1,indy2,indz1,indz2,nx,ny,nz;
580:   IS             is_alldesign,is_allstate;
581:   IS             is_from_d;
582:   IS             is_from_y;
583:   PetscInt       lo,hi,hi2,lo2,ysubnlocal,dsubnlocal;
584:   const PetscInt *ranges, *subranges;
585:   PetscMPIInt    size;
586:   PetscReal      xri,yri,zri,xim,yim,zim,dx1,dx2,dy1,dy2,dz1,dz2,Dx,Dy,Dz;
587:   PetscScalar    v,vx,vy,vz;
588:   PetscInt       offset,subindex,subvec,nrank,kk;

590:   PetscScalar xr[64] = {0.4970,     0.8498,     0.7814,     0.6268,     0.7782,     0.6402,     0.3617,     0.3160,
591:                         0.3610,     0.5298,     0.6987,     0.3331,     0.7962,     0.5596,     0.3866,     0.6774,
592:                         0.5407,     0.4518,     0.6702,     0.6061,     0.7580,     0.8997,     0.5198,     0.8326,
593:                         0.2138,     0.9198,     0.3000,     0.2833,     0.8288,     0.7076,     0.1820,     0.0728,
594:                         0.8447,     0.2367,     0.3239,     0.6413,     0.3114,     0.4731,     0.1192,     0.9273,
595:                         0.5724,     0.4331,     0.5136,     0.3547,     0.4413,     0.2602,     0.5698,     0.7278,
596:                         0.5261,     0.6230,     0.2454,     0.3948,     0.7479,     0.6582,     0.4660,     0.5594,
597:                         0.7574,     0.1143,     0.5900,     0.1065,     0.4260,     0.3294,     0.8276,     0.0756};

599:   PetscScalar yr[64] = {0.7345,     0.9120,     0.9288,     0.7528,     0.4463,     0.4985,     0.2497,     0.6256,
600:                         0.3425,     0.9026,     0.6983,     0.4230,     0.7140,     0.2970,     0.4474,     0.8792,
601:                         0.6604,     0.2485,     0.7968,     0.6127,     0.1796,     0.2437,     0.5938,     0.6137,
602:                         0.3867,     0.5658,     0.4575,     0.1009,     0.0863,     0.3361,     0.0738,     0.3985,
603:                         0.6602,     0.1437,     0.0934,     0.5983,     0.5950,     0.0763,     0.0768,     0.2288,
604:                         0.5761,     0.1129,     0.3841,     0.6150,     0.6904,     0.6686,     0.1361,     0.4601,
605:                         0.4491,     0.3716,     0.1969,     0.6537,     0.6743,     0.6991,     0.4811,     0.5480,
606:                         0.1684,     0.4569,     0.6889,     0.8437,     0.3015,     0.2854,     0.8199,     0.2658};

608:   PetscScalar zr[64] = {0.7668,     0.8573,     0.2654,     0.2719,     0.1060,     0.1311,     0.6232,     0.2295,
609:                         0.8009,     0.2147,     0.2119,     0.9325,     0.4473,     0.3600,     0.3374,     0.3819,
610:                         0.4066,     0.5801,     0.1673,     0.0959,     0.4638,     0.8236,     0.8800,     0.2939,
611:                         0.2028,     0.8262,     0.2706,     0.6276,     0.9085,     0.6443,     0.8241,     0.0712,
612:                         0.1824,     0.7789,     0.4389,     0.8415,     0.7055,     0.6639,     0.3653,     0.2078,
613:                         0.1987,     0.2297,     0.4321,     0.8115,     0.4915,     0.7764,     0.4657,     0.4627,
614:                         0.4569,     0.4232,     0.8514,     0.0674,     0.3227,     0.1055,     0.6690,     0.6313,
615:                         0.9226,     0.5461,     0.4126,     0.2364,     0.6096,     0.7042,     0.3914,     0.0711};

618:   MPI_Comm_size(PETSC_COMM_WORLD,&size);
619:   PetscLogStageRegister("Elliptic Setup",&user->stages[0]);
620:   PetscLogStagePush(user->stages[0]);

622:   /* Create u,y,c,x */
623:   VecCreate(PETSC_COMM_WORLD,&user->u);
624:   VecCreate(PETSC_COMM_WORLD,&user->y);
625:   VecCreate(PETSC_COMM_WORLD,&user->c);
626:   VecSetSizes(user->u,PETSC_DECIDE,user->ndesign);
627:   VecSetFromOptions(user->u);
628:   VecGetLocalSize(user->u,&ysubnlocal);
629:   VecSetSizes(user->y,ysubnlocal*user->ns,user->nstate);
630:   VecSetSizes(user->c,ysubnlocal*user->ns,user->m);
631:   VecSetFromOptions(user->y);
632:   VecSetFromOptions(user->c);

634:   /*
635:      *******************************
636:      Create scatters for x <-> y,u
637:      *******************************

639:      If the state vector y and design vector u are partitioned as
640:      [y_1; y_2; ...; y_np] and [u_1; u_2; ...; u_np] (with np = # of processors),
641:      then the solution vector x is organized as
642:      [y_1; u_1; y_2; u_2; ...; y_np; u_np].
643:      The index sets user->s_is and user->d_is correspond to the indices of the
644:      state and design variables owned by the current processor.
645:   */
646:   VecCreate(PETSC_COMM_WORLD,&user->x);

648:   VecGetOwnershipRange(user->y,&lo,&hi);
649:   VecGetOwnershipRange(user->u,&lo2,&hi2);

651:   ISCreateStride(PETSC_COMM_SELF,hi-lo,lo,1,&is_allstate);
652:   ISCreateStride(PETSC_COMM_SELF,hi-lo,lo+lo2,1,&user->s_is);
653:   ISCreateStride(PETSC_COMM_SELF,hi2-lo2,lo2,1,&is_alldesign);
654:   ISCreateStride(PETSC_COMM_SELF,hi2-lo2,hi+lo2,1,&user->d_is);

656:   VecSetSizes(user->x,hi-lo+hi2-lo2,user->n);
657:   VecSetFromOptions(user->x);

659:   VecScatterCreate(user->x,user->s_is,user->y,is_allstate,&user->state_scatter);
660:   VecScatterCreate(user->x,user->d_is,user->u,is_alldesign,&user->design_scatter);
661:   ISDestroy(&is_alldesign);
662:   ISDestroy(&is_allstate);
663:   /*
664:      *******************************
665:      Create scatter from y to y_1,y_2,...,y_ns
666:      *******************************
667:   */
668:   PetscMalloc1(user->ns,&user->yi_scatter);
669:   VecDuplicate(user->u,&user->suby);
670:   VecDuplicate(user->u,&user->subq);

672:   VecGetOwnershipRange(user->y,&lo2,&hi2);
673:   istart = 0;
674:   for (i=0; i<user->ns; i++) {
675:     VecGetOwnershipRange(user->suby,&lo,&hi);
676:     ISCreateStride(PETSC_COMM_SELF,hi-lo,lo2+istart,1,&is_from_y);
677:     VecScatterCreate(user->y,is_from_y,user->suby,NULL,&user->yi_scatter[i]);
678:     istart = istart + hi-lo;
679:     ISDestroy(&is_from_y);
680:   }
681:   /*
682:      *******************************
683:      Create scatter from d to d_1,d_2,...,d_ns
684:      *******************************
685:   */
686:   VecCreate(PETSC_COMM_WORLD,&user->subd);
687:   VecSetSizes(user->subd,PETSC_DECIDE,user->ndata);
688:   VecSetFromOptions(user->subd);
689:   VecCreate(PETSC_COMM_WORLD,&user->d);
690:   VecGetLocalSize(user->subd,&dsubnlocal);
691:   VecSetSizes(user->d,dsubnlocal*user->ns,user->ndata*user->ns);
692:   VecSetFromOptions(user->d);
693:   PetscMalloc1(user->ns,&user->di_scatter);

695:   VecGetOwnershipRange(user->d,&lo2,&hi2);
696:   istart = 0;
697:   for (i=0; i<user->ns; i++) {
698:     VecGetOwnershipRange(user->subd,&lo,&hi);
699:     ISCreateStride(PETSC_COMM_SELF,hi-lo,lo2+istart,1,&is_from_d);
700:     VecScatterCreate(user->d,is_from_d,user->subd,NULL,&user->di_scatter[i]);
701:     istart = istart + hi-lo;
702:     ISDestroy(&is_from_d);
703:   }

705:   PetscMalloc1(user->mx,&x);
706:   PetscMalloc1(user->mx,&y);
707:   PetscMalloc1(user->mx,&z);

709:   user->ksp_its = 0;
710:   user->ksp_its_initial = 0;

712:   n = user->mx * user->mx * user->mx;
713:   m = 3 * user->mx * user->mx * (user->mx-1);
714:   sqrt_beta = PetscSqrtScalar(user->beta);

716:   VecCreate(PETSC_COMM_WORLD,&XX);
717:   VecCreate(PETSC_COMM_WORLD,&user->q);
718:   VecSetSizes(XX,ysubnlocal,n);
719:   VecSetSizes(user->q,ysubnlocal*user->ns,user->m);
720:   VecSetFromOptions(XX);
721:   VecSetFromOptions(user->q);

723:   VecDuplicate(XX,&YY);
724:   VecDuplicate(XX,&ZZ);
725:   VecDuplicate(XX,&XXwork);
726:   VecDuplicate(XX,&YYwork);
727:   VecDuplicate(XX,&ZZwork);
728:   VecDuplicate(XX,&UTwork);
729:   VecDuplicate(XX,&user->utrue);

731:   /* map for striding q */
732:   VecGetOwnershipRanges(user->q,&ranges);
733:   VecGetOwnershipRanges(user->u,&subranges);

735:   VecGetOwnershipRange(user->q,&lo2,&hi2);
736:   VecGetOwnershipRange(user->u,&lo,&hi);
737:   /* Generate 3D grid, and collect ns (1<=ns<=8) right-hand-side vectors into user->q */
738:   h = 1.0/user->mx;
739:   hinv = user->mx;
740:   neg_hinv = -hinv;

742:   VecGetOwnershipRange(XX,&istart,&iend);
743:   for (linear_index=istart; linear_index<iend; linear_index++) {
744:     i = linear_index % user->mx;
745:     j = ((linear_index-i)/user->mx) % user->mx;
746:     k = ((linear_index-i)/user->mx-j) / user->mx;
747:     vx = h*(i+0.5);
748:     vy = h*(j+0.5);
749:     vz = h*(k+0.5);
750:     VecSetValues(XX,1,&linear_index,&vx,INSERT_VALUES);
751:     VecSetValues(YY,1,&linear_index,&vy,INSERT_VALUES);
752:     VecSetValues(ZZ,1,&linear_index,&vz,INSERT_VALUES);
753:     for (is=0; is<2; is++) {
754:       for (js=0; js<2; js++) {
755:         for (ks=0; ks<2; ks++) {
756:           ls = is*4 + js*2 + ks;
757:           if (ls<user->ns) {
758:             l =ls*n + linear_index;
759:             /* remap */
760:             subindex = l%n;
761:             subvec = l/n;
762:             nrank=0;
763:             while (subindex >= subranges[nrank+1]) nrank++;
764:             offset = subindex - subranges[nrank];
765:             istart=0;
766:             for (kk=0;kk<nrank;kk++) istart+=user->ns*(subranges[kk+1]-subranges[kk]);
767:             istart += (subranges[nrank+1]-subranges[nrank])*subvec;
768:             l = istart+offset;
769:             v = 100*PetscSinScalar(2*PETSC_PI*(vx+0.25*is))*PetscSinScalar(2*PETSC_PI*(vy+0.25*js))*PetscSinScalar(2*PETSC_PI*(vz+0.25*ks));
770:             VecSetValues(user->q,1,&l,&v,INSERT_VALUES);
771:           }
772:         }
773:       }
774:     }
775:   }

777:   VecAssemblyBegin(XX);
778:   VecAssemblyEnd(XX);
779:   VecAssemblyBegin(YY);
780:   VecAssemblyEnd(YY);
781:   VecAssemblyBegin(ZZ);
782:   VecAssemblyEnd(ZZ);
783:   VecAssemblyBegin(user->q);
784:   VecAssemblyEnd(user->q);

786:   /* Compute true parameter function
787:      ut = exp(-((x-0.25)^2+(y-0.25)^2+(z-0.25)^2)/0.05) - exp((x-0.75)^2-(y-0.75)^2-(z-0.75))^2/0.05) */
788:   VecCopy(XX,XXwork);
789:   VecCopy(YY,YYwork);
790:   VecCopy(ZZ,ZZwork);

792:   VecShift(XXwork,-0.25);
793:   VecShift(YYwork,-0.25);
794:   VecShift(ZZwork,-0.25);

796:   VecPointwiseMult(XXwork,XXwork,XXwork);
797:   VecPointwiseMult(YYwork,YYwork,YYwork);
798:   VecPointwiseMult(ZZwork,ZZwork,ZZwork);

800:   VecCopy(XXwork,UTwork);
801:   VecAXPY(UTwork,1.0,YYwork);
802:   VecAXPY(UTwork,1.0,ZZwork);
803:   VecScale(UTwork,-20.0);
804:   VecExp(UTwork);
805:   VecCopy(UTwork,user->utrue);

807:   VecCopy(XX,XXwork);
808:   VecCopy(YY,YYwork);
809:   VecCopy(ZZ,ZZwork);

811:   VecShift(XXwork,-0.75);
812:   VecShift(YYwork,-0.75);
813:   VecShift(ZZwork,-0.75);

815:   VecPointwiseMult(XXwork,XXwork,XXwork);
816:   VecPointwiseMult(YYwork,YYwork,YYwork);
817:   VecPointwiseMult(ZZwork,ZZwork,ZZwork);

819:   VecCopy(XXwork,UTwork);
820:   VecAXPY(UTwork,1.0,YYwork);
821:   VecAXPY(UTwork,1.0,ZZwork);
822:   VecScale(UTwork,-20.0);
823:   VecExp(UTwork);

825:   VecAXPY(user->utrue,-1.0,UTwork);

827:   VecDestroy(&XX);
828:   VecDestroy(&YY);
829:   VecDestroy(&ZZ);
830:   VecDestroy(&XXwork);
831:   VecDestroy(&YYwork);
832:   VecDestroy(&ZZwork);
833:   VecDestroy(&UTwork);

835:   /* Initial guess and reference model */
836:   VecDuplicate(user->utrue,&user->ur);
837:   VecSum(user->utrue,&meanut);
838:   meanut = meanut / n;
839:   VecSet(user->ur,meanut);
840:   VecCopy(user->ur,user->u);

842:   /* Generate Grad matrix */
843:   MatCreate(PETSC_COMM_WORLD,&user->Grad);
844:   MatSetSizes(user->Grad,PETSC_DECIDE,ysubnlocal,m,n);
845:   MatSetFromOptions(user->Grad);
846:   MatMPIAIJSetPreallocation(user->Grad,2,NULL,2,NULL);
847:   MatSeqAIJSetPreallocation(user->Grad,2,NULL);
848:   MatGetOwnershipRange(user->Grad,&istart,&iend);

850:   for (i=istart; i<iend; i++) {
851:     if (i<m/3) {
852:       iblock = i / (user->mx-1);
853:       j = iblock*user->mx + (i % (user->mx-1));
854:       MatSetValues(user->Grad,1,&i,1,&j,&neg_hinv,INSERT_VALUES);
855:       j = j+1;
856:       MatSetValues(user->Grad,1,&i,1,&j,&hinv,INSERT_VALUES);
857:     }
858:     if (i>=m/3 && i<2*m/3) {
859:       iblock = (i-m/3) / (user->mx*(user->mx-1));
860:       j = iblock*user->mx*user->mx + ((i-m/3) % (user->mx*(user->mx-1)));
861:       MatSetValues(user->Grad,1,&i,1,&j,&neg_hinv,INSERT_VALUES);
862:       j = j + user->mx;
863:       MatSetValues(user->Grad,1,&i,1,&j,&hinv,INSERT_VALUES);
864:     }
865:     if (i>=2*m/3) {
866:       j = i-2*m/3;
867:       MatSetValues(user->Grad,1,&i,1,&j,&neg_hinv,INSERT_VALUES);
868:       j = j + user->mx*user->mx;
869:       MatSetValues(user->Grad,1,&i,1,&j,&hinv,INSERT_VALUES);
870:     }
871:   }

873:   MatAssemblyBegin(user->Grad,MAT_FINAL_ASSEMBLY);
874:   MatAssemblyEnd(user->Grad,MAT_FINAL_ASSEMBLY);

876:   /* Generate arithmetic averaging matrix Av */
877:   MatCreate(PETSC_COMM_WORLD,&user->Av);
878:   MatSetSizes(user->Av,PETSC_DECIDE,ysubnlocal,m,n);
879:   MatSetFromOptions(user->Av);
880:   MatMPIAIJSetPreallocation(user->Av,2,NULL,2,NULL);
881:   MatSeqAIJSetPreallocation(user->Av,2,NULL);
882:   MatGetOwnershipRange(user->Av,&istart,&iend);

884:   for (i=istart; i<iend; i++) {
885:     if (i<m/3) {
886:       iblock = i / (user->mx-1);
887:       j = iblock*user->mx + (i % (user->mx-1));
888:       MatSetValues(user->Av,1,&i,1,&j,&half,INSERT_VALUES);
889:       j = j+1;
890:       MatSetValues(user->Av,1,&i,1,&j,&half,INSERT_VALUES);
891:     }
892:     if (i>=m/3 && i<2*m/3) {
893:       iblock = (i-m/3) / (user->mx*(user->mx-1));
894:       j = iblock*user->mx*user->mx + ((i-m/3) % (user->mx*(user->mx-1)));
895:       MatSetValues(user->Av,1,&i,1,&j,&half,INSERT_VALUES);
896:       j = j + user->mx;
897:       MatSetValues(user->Av,1,&i,1,&j,&half,INSERT_VALUES);
898:     }
899:     if (i>=2*m/3) {
900:       j = i-2*m/3;
901:       MatSetValues(user->Av,1,&i,1,&j,&half,INSERT_VALUES);
902:       j = j + user->mx*user->mx;
903:       MatSetValues(user->Av,1,&i,1,&j,&half,INSERT_VALUES);
904:     }
905:   }

907:   MatAssemblyBegin(user->Av,MAT_FINAL_ASSEMBLY);
908:   MatAssemblyEnd(user->Av,MAT_FINAL_ASSEMBLY);

910:   MatCreate(PETSC_COMM_WORLD,&user->L);
911:   MatSetSizes(user->L,PETSC_DECIDE,ysubnlocal,m+n,n);
912:   MatSetFromOptions(user->L);
913:   MatMPIAIJSetPreallocation(user->L,2,NULL,2,NULL);
914:   MatSeqAIJSetPreallocation(user->L,2,NULL);
915:   MatGetOwnershipRange(user->L,&istart,&iend);

917:   for (i=istart; i<iend; i++) {
918:     if (i<m/3) {
919:       iblock = i / (user->mx-1);
920:       j = iblock*user->mx + (i % (user->mx-1));
921:       MatSetValues(user->L,1,&i,1,&j,&neg_hinv,INSERT_VALUES);
922:       j = j+1;
923:       MatSetValues(user->L,1,&i,1,&j,&hinv,INSERT_VALUES);
924:     }
925:     if (i>=m/3 && i<2*m/3) {
926:       iblock = (i-m/3) / (user->mx*(user->mx-1));
927:       j = iblock*user->mx*user->mx + ((i-m/3) % (user->mx*(user->mx-1)));
928:       MatSetValues(user->L,1,&i,1,&j,&neg_hinv,INSERT_VALUES);
929:       j = j + user->mx;
930:       MatSetValues(user->L,1,&i,1,&j,&hinv,INSERT_VALUES);
931:     }
932:     if (i>=2*m/3 && i<m) {
933:       j = i-2*m/3;
934:       MatSetValues(user->L,1,&i,1,&j,&neg_hinv,INSERT_VALUES);
935:       j = j + user->mx*user->mx;
936:       MatSetValues(user->L,1,&i,1,&j,&hinv,INSERT_VALUES);
937:     }
938:     if (i>=m) {
939:       j = i - m;
940:       MatSetValues(user->L,1,&i,1,&j,&sqrt_beta,INSERT_VALUES);
941:     }
942:   }
943:   MatAssemblyBegin(user->L,MAT_FINAL_ASSEMBLY);
944:   MatAssemblyEnd(user->L,MAT_FINAL_ASSEMBLY);
945:   MatScale(user->L,PetscPowScalar(h,1.5));

947:   /* Generate Div matrix */
948:   if (!user->use_ptap) {
949:     /* Generate Div matrix */
950:     MatCreate(PETSC_COMM_WORLD,&user->Div);
951:     MatSetSizes(user->Div,ysubnlocal,PETSC_DECIDE,n,m);
952:     MatSetFromOptions(user->Div);
953:     MatMPIAIJSetPreallocation(user->Div,4,NULL,4,NULL);
954:     MatSeqAIJSetPreallocation(user->Div,6,NULL);
955:     MatGetOwnershipRange(user->Grad,&istart,&iend);

957:     for (i=istart; i<iend; i++) {
958:       if (i<m/3) {
959:         iblock = i / (user->mx-1);
960:         j = iblock*user->mx + (i % (user->mx-1));
961:         MatSetValues(user->Div,1,&j,1,&i,&neg_hinv,INSERT_VALUES);
962:         j = j+1;
963:         MatSetValues(user->Div,1,&j,1,&i,&hinv,INSERT_VALUES);
964:       }
965:       if (i>=m/3 && i<2*m/3) {
966:         iblock = (i-m/3) / (user->mx*(user->mx-1));
967:         j = iblock*user->mx*user->mx + ((i-m/3) % (user->mx*(user->mx-1)));
968:         MatSetValues(user->Div,1,&j,1,&i,&neg_hinv,INSERT_VALUES);
969:         j = j + user->mx;
970:         MatSetValues(user->Div,1,&j,1,&i,&hinv,INSERT_VALUES);
971:       }
972:       if (i>=2*m/3) {
973:         j = i-2*m/3;
974:         MatSetValues(user->Div,1,&j,1,&i,&neg_hinv,INSERT_VALUES);
975:         j = j + user->mx*user->mx;
976:         MatSetValues(user->Div,1,&j,1,&i,&hinv,INSERT_VALUES);
977:       }
978:     }

980:     MatAssemblyBegin(user->Div,MAT_FINAL_ASSEMBLY);
981:     MatAssemblyEnd(user->Div,MAT_FINAL_ASSEMBLY);
982:     MatDuplicate(user->Div,MAT_SHARE_NONZERO_PATTERN,&user->Divwork);
983:   } else {
984:     MatCreate(PETSC_COMM_WORLD,&user->Diag);
985:     MatSetSizes(user->Diag,PETSC_DECIDE,PETSC_DECIDE,m,m);
986:     MatSetFromOptions(user->Diag);
987:     MatMPIAIJSetPreallocation(user->Diag,1,NULL,0,NULL);
988:     MatSeqAIJSetPreallocation(user->Diag,1,NULL);
989:   }

991:   /* Build work vectors and matrices */
992:   VecCreate(PETSC_COMM_WORLD,&user->S);
993:   VecSetSizes(user->S, PETSC_DECIDE, m);
994:   VecSetFromOptions(user->S);

996:   VecCreate(PETSC_COMM_WORLD,&user->lwork);
997:   VecSetSizes(user->lwork,PETSC_DECIDE,m+user->mx*user->mx*user->mx);
998:   VecSetFromOptions(user->lwork);

1000:   MatDuplicate(user->Av,MAT_SHARE_NONZERO_PATTERN,&user->Avwork);

1002:   VecDuplicate(user->S,&user->Swork);
1003:   VecDuplicate(user->S,&user->Sdiag);
1004:   VecDuplicate(user->S,&user->Av_u);
1005:   VecDuplicate(user->S,&user->Twork);
1006:   VecDuplicate(user->y,&user->ywork);
1007:   VecDuplicate(user->u,&user->Ywork);
1008:   VecDuplicate(user->u,&user->uwork);
1009:   VecDuplicate(user->u,&user->js_diag);
1010:   VecDuplicate(user->c,&user->cwork);
1011:   VecDuplicate(user->d,&user->dwork);

1013:   /* Create a matrix-free shell user->Jd for computing B*x */
1014:   MatCreateShell(PETSC_COMM_WORLD,ysubnlocal*user->ns,ysubnlocal,user->nstate,user->ndesign,user,&user->Jd);
1015:   MatShellSetOperation(user->Jd,MATOP_MULT,(void(*)(void))DesignMatMult);
1016:   MatShellSetOperation(user->Jd,MATOP_MULT_TRANSPOSE,(void(*)(void))DesignMatMultTranspose);

1018:   /* Compute true state function ytrue given utrue */
1019:   VecDuplicate(user->y,&user->ytrue);

1021:   /* First compute Av_u = Av*exp(-u) */
1022:   VecSet(user->uwork, 0);
1023:   VecAXPY(user->uwork,-1.0,user->utrue); /* Note: user->utrue */
1024:   VecExp(user->uwork);
1025:   MatMult(user->Av,user->uwork,user->Av_u);

1027:   /* Next form DSG = Div*S*Grad */
1028:   VecCopy(user->Av_u,user->Swork);
1029:   VecReciprocal(user->Swork);
1030:   if (user->use_ptap) {
1031:     MatDiagonalSet(user->Diag,user->Swork,INSERT_VALUES);
1032:     MatPtAP(user->Diag,user->Grad,MAT_INITIAL_MATRIX,1.0,&user->DSG);
1033:   } else {
1034:     MatCopy(user->Div,user->Divwork,SAME_NONZERO_PATTERN);
1035:     MatDiagonalScale(user->Divwork,NULL,user->Swork);

1037:     MatMatMult(user->Divwork,user->Grad,MAT_INITIAL_MATRIX,1.0,&user->DSG);
1038:   }

1040:   MatSetOption(user->DSG,MAT_SYMMETRIC,PETSC_TRUE);
1041:   MatSetOption(user->DSG,MAT_SYMMETRY_ETERNAL,PETSC_TRUE);

1043:   if (user->use_lrc == PETSC_TRUE) {
1044:     v=PetscSqrtReal(1.0 /user->ndesign);
1045:     PetscMalloc1(user->ndesign,&user->ones);

1047:     for (i=0;i<user->ndesign;i++) {
1048:       user->ones[i]=v;
1049:     }
1050:     MatCreateDense(PETSC_COMM_WORLD,ysubnlocal,PETSC_DECIDE,user->ndesign,1,user->ones,&user->Ones);
1051:     MatAssemblyBegin(user->Ones, MAT_FINAL_ASSEMBLY);
1052:     MatAssemblyEnd(user->Ones, MAT_FINAL_ASSEMBLY);
1053:     MatCreateLRC(user->DSG,user->Ones,NULL,user->Ones,&user->JsBlock);
1054:     MatSetUp(user->JsBlock);
1055:   } else {
1056:     /* Create matrix-free shell user->Js for computing (A + h^3*e*e^T)*x */
1057:     MatCreateShell(PETSC_COMM_WORLD,ysubnlocal,ysubnlocal,user->ndesign,user->ndesign,user,&user->JsBlock);
1058:     MatShellSetOperation(user->JsBlock,MATOP_MULT,(void(*)(void))StateBlockMatMult);
1059:     MatShellSetOperation(user->JsBlock,MATOP_MULT_TRANSPOSE,(void(*)(void))StateBlockMatMult);
1060:   }
1061:   MatSetOption(user->JsBlock,MAT_SYMMETRIC,PETSC_TRUE);
1062:   MatSetOption(user->JsBlock,MAT_SYMMETRY_ETERNAL,PETSC_TRUE);
1063:   MatCreateShell(PETSC_COMM_WORLD,ysubnlocal*user->ns,ysubnlocal*user->ns,user->nstate,user->nstate,user,&user->Js);
1064:   MatShellSetOperation(user->Js,MATOP_MULT,(void(*)(void))StateMatMult);
1065:   MatShellSetOperation(user->Js,MATOP_MULT_TRANSPOSE,(void(*)(void))StateMatMult);
1066:   MatSetOption(user->Js,MAT_SYMMETRIC,PETSC_TRUE);
1067:   MatSetOption(user->Js,MAT_SYMMETRY_ETERNAL,PETSC_TRUE);

1069:   MatCreateShell(PETSC_COMM_WORLD,ysubnlocal*user->ns,ysubnlocal*user->ns,user->nstate,user->nstate,user,&user->JsInv);
1070:   MatShellSetOperation(user->JsInv,MATOP_MULT,(void(*)(void))StateInvMatMult);
1071:   MatShellSetOperation(user->JsInv,MATOP_MULT_TRANSPOSE,(void(*)(void))StateInvMatMult);
1072:   MatSetOption(user->JsInv,MAT_SYMMETRIC,PETSC_TRUE);
1073:   MatSetOption(user->JsInv,MAT_SYMMETRY_ETERNAL,PETSC_TRUE);

1075:   MatSetOption(user->DSG,MAT_SYMMETRIC,PETSC_TRUE);
1076:   MatSetOption(user->DSG,MAT_SYMMETRY_ETERNAL,PETSC_TRUE);
1077:   /* Now solve for ytrue */
1078:   KSPCreate(PETSC_COMM_WORLD,&user->solver);
1079:   KSPSetFromOptions(user->solver);

1081:   KSPSetOperators(user->solver,user->JsBlock,user->DSG);

1083:   MatMult(user->JsInv,user->q,user->ytrue);
1084:   /* First compute Av_u = Av*exp(-u) */
1085:   VecSet(user->uwork,0);
1086:   VecAXPY(user->uwork,-1.0,user->u); /* Note: user->u */
1087:   VecExp(user->uwork);
1088:   MatMult(user->Av,user->uwork,user->Av_u);

1090:   /* Next update DSG = Div*S*Grad  with user->u */
1091:   VecCopy(user->Av_u,user->Swork);
1092:   VecReciprocal(user->Swork);
1093:   if (user->use_ptap) {
1094:     MatDiagonalSet(user->Diag,user->Swork,INSERT_VALUES);
1095:     MatPtAP(user->Diag,user->Grad,MAT_REUSE_MATRIX,1.0,&user->DSG);
1096:   } else {
1097:     MatCopy(user->Div,user->Divwork,SAME_NONZERO_PATTERN);
1098:     MatDiagonalScale(user->Divwork,NULL,user->Av_u);
1099:     MatProductNumeric(user->DSG);
1100:   }

1102:   /* Now solve for y */

1104:   MatMult(user->JsInv,user->q,user->y);

1106:   user->ksp_its_initial = user->ksp_its;
1107:   user->ksp_its = 0;
1108:   /* Construct projection matrix Q (blocks) */
1109:   MatCreate(PETSC_COMM_WORLD,&user->Q);
1110:   MatSetSizes(user->Q,dsubnlocal,ysubnlocal,user->ndata,user->ndesign);
1111:   MatSetFromOptions(user->Q);
1112:   MatMPIAIJSetPreallocation(user->Q,8,NULL,8,NULL);
1113:   MatSeqAIJSetPreallocation(user->Q,8,NULL);

1115:   for (i=0; i<user->mx; i++) {
1116:     x[i] = h*(i+0.5);
1117:     y[i] = h*(i+0.5);
1118:     z[i] = h*(i+0.5);
1119:   }
1120:   MatGetOwnershipRange(user->Q,&istart,&iend);

1122:   nx = user->mx; ny = user->mx; nz = user->mx;
1123:   for (i=istart; i<iend; i++) {

1125:     xri = xr[i];
1126:     im = 0;
1127:     xim = x[im];
1128:     while (xri>xim && im<nx) {
1129:       im = im+1;
1130:       xim = x[im];
1131:     }
1132:     indx1 = im-1;
1133:     indx2 = im;
1134:     dx1 = xri - x[indx1];
1135:     dx2 = x[indx2] - xri;

1137:     yri = yr[i];
1138:     im = 0;
1139:     yim = y[im];
1140:     while (yri>yim && im<ny) {
1141:       im = im+1;
1142:       yim = y[im];
1143:     }
1144:     indy1 = im-1;
1145:     indy2 = im;
1146:     dy1 = yri - y[indy1];
1147:     dy2 = y[indy2] - yri;

1149:     zri = zr[i];
1150:     im = 0;
1151:     zim = z[im];
1152:     while (zri>zim && im<nz) {
1153:       im = im+1;
1154:       zim = z[im];
1155:     }
1156:     indz1 = im-1;
1157:     indz2 = im;
1158:     dz1 = zri - z[indz1];
1159:     dz2 = z[indz2] - zri;

1161:     Dx = x[indx2] - x[indx1];
1162:     Dy = y[indy2] - y[indy1];
1163:     Dz = z[indz2] - z[indz1];

1165:     j = indx1 + indy1*nx + indz1*nx*ny;
1166:     v = (1-dx1/Dx)*(1-dy1/Dy)*(1-dz1/Dz);
1167:     MatSetValues(user->Q,1,&i,1,&j,&v,INSERT_VALUES);

1169:     j = indx1 + indy1*nx + indz2*nx*ny;
1170:     v = (1-dx1/Dx)*(1-dy1/Dy)*(1-dz2/Dz);
1171:     MatSetValues(user->Q,1,&i,1,&j,&v,INSERT_VALUES);

1173:     j = indx1 + indy2*nx + indz1*nx*ny;
1174:     v = (1-dx1/Dx)*(1-dy2/Dy)*(1-dz1/Dz);
1175:     MatSetValues(user->Q,1,&i,1,&j,&v,INSERT_VALUES);

1177:     j = indx1 + indy2*nx + indz2*nx*ny;
1178:     v = (1-dx1/Dx)*(1-dy2/Dy)*(1-dz2/Dz);
1179:     MatSetValues(user->Q,1,&i,1,&j,&v,INSERT_VALUES);

1181:     j = indx2 + indy1*nx + indz1*nx*ny;
1182:     v = (1-dx2/Dx)*(1-dy1/Dy)*(1-dz1/Dz);
1183:     MatSetValues(user->Q,1,&i,1,&j,&v,INSERT_VALUES);

1185:     j = indx2 + indy1*nx + indz2*nx*ny;
1186:     v = (1-dx2/Dx)*(1-dy1/Dy)*(1-dz2/Dz);
1187:     MatSetValues(user->Q,1,&i,1,&j,&v,INSERT_VALUES);

1189:     j = indx2 + indy2*nx + indz1*nx*ny;
1190:     v = (1-dx2/Dx)*(1-dy2/Dy)*(1-dz1/Dz);
1191:     MatSetValues(user->Q,1,&i,1,&j,&v,INSERT_VALUES);

1193:     j = indx2 + indy2*nx + indz2*nx*ny;
1194:     v = (1-dx2/Dx)*(1-dy2/Dy)*(1-dz2/Dz);
1195:     MatSetValues(user->Q,1,&i,1,&j,&v,INSERT_VALUES);
1196:   }

1198:   MatAssemblyBegin(user->Q,MAT_FINAL_ASSEMBLY);
1199:   MatAssemblyEnd(user->Q,MAT_FINAL_ASSEMBLY);
1200:   /* Create MQ (composed of blocks of Q */
1201:   MatCreateShell(PETSC_COMM_WORLD,dsubnlocal*user->ns,PETSC_DECIDE,user->ndata*user->ns,user->nstate,user,&user->MQ);
1202:   MatShellSetOperation(user->MQ,MATOP_MULT,(void(*)(void))QMatMult);
1203:   MatShellSetOperation(user->MQ,MATOP_MULT_TRANSPOSE,(void(*)(void))QMatMultTranspose);

1205:   /* Add noise to the measurement data */
1206:   VecSet(user->ywork,1.0);
1207:   VecAYPX(user->ywork,user->noise,user->ytrue);
1208:   MatMult(user->MQ,user->ywork,user->d);

1210:   /* Now that initial conditions have been set, let the user pass tolerance options to the KSP solver */
1211:   PetscFree(x);
1212:   PetscFree(y);
1213:   PetscFree(z);
1214:   PetscLogStagePop();
1215:   return(0);
1216: }

1218: PetscErrorCode EllipticDestroy(AppCtx *user)
1219: {
1221:   PetscInt       i;

1224:   MatDestroy(&user->DSG);
1225:   KSPDestroy(&user->solver);
1226:   MatDestroy(&user->Q);
1227:   MatDestroy(&user->MQ);
1228:   if (!user->use_ptap) {
1229:     MatDestroy(&user->Div);
1230:     MatDestroy(&user->Divwork);
1231:   } else {
1232:     MatDestroy(&user->Diag);
1233:   }
1234:   if (user->use_lrc) {
1235:     MatDestroy(&user->Ones);
1236:   }

1238:   MatDestroy(&user->Grad);
1239:   MatDestroy(&user->Av);
1240:   MatDestroy(&user->Avwork);
1241:   MatDestroy(&user->L);
1242:   MatDestroy(&user->Js);
1243:   MatDestroy(&user->Jd);
1244:   MatDestroy(&user->JsBlock);
1245:   MatDestroy(&user->JsInv);

1247:   VecDestroy(&user->x);
1248:   VecDestroy(&user->u);
1249:   VecDestroy(&user->uwork);
1250:   VecDestroy(&user->utrue);
1251:   VecDestroy(&user->y);
1252:   VecDestroy(&user->ywork);
1253:   VecDestroy(&user->ytrue);
1254:   VecDestroy(&user->c);
1255:   VecDestroy(&user->cwork);
1256:   VecDestroy(&user->ur);
1257:   VecDestroy(&user->q);
1258:   VecDestroy(&user->d);
1259:   VecDestroy(&user->dwork);
1260:   VecDestroy(&user->lwork);
1261:   VecDestroy(&user->S);
1262:   VecDestroy(&user->Swork);
1263:   VecDestroy(&user->Sdiag);
1264:   VecDestroy(&user->Ywork);
1265:   VecDestroy(&user->Twork);
1266:   VecDestroy(&user->Av_u);
1267:   VecDestroy(&user->js_diag);
1268:   ISDestroy(&user->s_is);
1269:   ISDestroy(&user->d_is);
1270:   VecDestroy(&user->suby);
1271:   VecDestroy(&user->subd);
1272:   VecDestroy(&user->subq);
1273:   VecScatterDestroy(&user->state_scatter);
1274:   VecScatterDestroy(&user->design_scatter);
1275:   for (i=0;i<user->ns;i++) {
1276:     VecScatterDestroy(&user->yi_scatter[i]);
1277:     VecScatterDestroy(&user->di_scatter[i]);
1278:   }
1279:   PetscFree(user->yi_scatter);
1280:   PetscFree(user->di_scatter);
1281:   if (user->use_lrc) {
1282:     PetscFree(user->ones);
1283:     MatDestroy(&user->Ones);
1284:   }
1285:   return(0);
1286: }

1288: PetscErrorCode EllipticMonitor(Tao tao, void *ptr)
1289: {
1291:   Vec            X;
1292:   PetscReal      unorm,ynorm;
1293:   AppCtx         *user = (AppCtx*)ptr;

1296:   TaoGetSolutionVector(tao,&X);
1297:   Scatter(X,user->ywork,user->state_scatter,user->uwork,user->design_scatter);
1298:   VecAXPY(user->ywork,-1.0,user->ytrue);
1299:   VecAXPY(user->uwork,-1.0,user->utrue);
1300:   VecNorm(user->uwork,NORM_2,&unorm);
1301:   VecNorm(user->ywork,NORM_2,&ynorm);
1302:   PetscPrintf(MPI_COMM_WORLD, "||u-ut||=%g ||y-yt||=%g\n",(double)unorm,(double)ynorm);
1303:   return(0);
1304: }

1306: /*TEST

1308:    build:
1309:       requires: !complex

1311:    test:
1312:       args: -tao_cmonitor -ns 1 -tao_type lcl -tao_gatol 1.e-3 -tao_max_it 11
1313:       requires: !single

1315:    test:
1316:       suffix: 2
1317:       args: -tao_cmonitor -tao_type lcl -tao_max_it 11 -use_ptap -use_lrc -ns 1 -tao_gatol 1.e-3
1318:       requires: !single

1320: TEST*/