Actual source code: parabolic.c

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

  3: typedef struct {
  4:   PetscInt n; /*  Number of variables */
  5:   PetscInt m; /*  Number of constraints per time step */
  6:   PetscInt mx; /*  grid points in each direction */
  7:   PetscInt nt; /*  Number of time steps; as of now, must be divisible by 8 */
  8:   PetscInt ndata; /*  Number of data points per sample */
  9:   PetscInt ns; /*  Number of samples */
 10:   PetscInt *sample_times; /*  Times of samples */
 11:   IS       s_is;
 12:   IS       d_is;

 14:   VecScatter state_scatter;
 15:   VecScatter design_scatter;
 16:   VecScatter *yi_scatter;
 17:   VecScatter *di_scatter;

 19:   Mat       Js,Jd,JsBlockPrec,JsInv,JsBlock;
 20:   PetscBool jformed,dsg_formed;

 22:   PetscReal alpha; /*  Regularization parameter */
 23:   PetscReal beta; /*  Weight attributed to ||u||^2 in regularization functional */
 24:   PetscReal noise; /*  Amount of noise to add to data */
 25:   PetscReal ht; /*  Time step */

 27:   Mat Qblock,QblockT;
 28:   Mat L,LT;
 29:   Mat Div,Divwork;
 30:   Mat Grad;
 31:   Mat Av,Avwork,AvT;
 32:   Mat DSG;
 33:   Vec q;
 34:   Vec ur; /*  reference */

 36:   Vec d;
 37:   Vec dwork;
 38:   Vec *di;

 40:   Vec y; /*  state variables */
 41:   Vec ywork;

 43:   Vec ytrue;
 44:   Vec *yi,*yiwork;

 46:   Vec u; /*  design variables */
 47:   Vec uwork;

 49:   Vec utrue;
 50:   Vec js_diag;
 51:   Vec c; /*  constraint vector */
 52:   Vec cwork;

 54:   Vec lwork;
 55:   Vec S;
 56:   Vec Rwork,Swork,Twork;
 57:   Vec Av_u;

 59:   KSP solver;
 60:   PC prec;

 62:   PetscInt ksp_its;
 63:   PetscInt ksp_its_initial;
 64: } AppCtx;

 66: PetscErrorCode FormFunction(Tao, Vec, PetscReal*, void*);
 67: PetscErrorCode FormGradient(Tao, Vec, Vec, void*);
 68: PetscErrorCode FormFunctionGradient(Tao, Vec, PetscReal*, Vec, void*);
 69: PetscErrorCode FormJacobianState(Tao, Vec, Mat, Mat, Mat, void*);
 70: PetscErrorCode FormJacobianDesign(Tao, Vec, Mat, void*);
 71: PetscErrorCode FormConstraints(Tao, Vec, Vec, void*);
 72: PetscErrorCode FormHessian(Tao, Vec, Mat, Mat, void*);
 73: PetscErrorCode Gather(Vec x, Vec state, VecScatter s_scat, Vec design, VecScatter d_scat);
 74: PetscErrorCode Scatter(Vec x, Vec state, VecScatter s_scat, Vec design, VecScatter d_scat);
 75: PetscErrorCode ParabolicInitialize(AppCtx *user);
 76: PetscErrorCode ParabolicDestroy(AppCtx *user);
 77: PetscErrorCode ParabolicMonitor(Tao, void*);

 79: PetscErrorCode StateMatMult(Mat,Vec,Vec);
 80: PetscErrorCode StateMatBlockMult(Mat,Vec,Vec);
 81: PetscErrorCode StateMatMultTranspose(Mat,Vec,Vec);
 82: PetscErrorCode StateMatGetDiagonal(Mat,Vec);
 83: PetscErrorCode StateMatDuplicate(Mat,MatDuplicateOption,Mat*);
 84: PetscErrorCode StateMatInvMult(Mat,Vec,Vec);
 85: PetscErrorCode StateMatInvTransposeMult(Mat,Vec,Vec);
 86: PetscErrorCode StateMatBlockPrecMult(PC,Vec,Vec);

 88: PetscErrorCode DesignMatMult(Mat,Vec,Vec);
 89: PetscErrorCode DesignMatMultTranspose(Mat,Vec,Vec);

 91: PetscErrorCode Gather_i(Vec,Vec*,VecScatter*,PetscInt);
 92: PetscErrorCode Scatter_i(Vec,Vec*,VecScatter*,PetscInt);

 94: static  char help[]="";

 96: int main(int argc, char **argv)
 97: {
 98:   PetscErrorCode     ierr;
 99:   Vec                x,x0;
100:   Tao                tao;
101:   AppCtx             user;
102:   IS                 is_allstate,is_alldesign;
103:   PetscInt           lo,hi,hi2,lo2,ksp_old;
104:   PetscInt           ntests = 1;
105:   PetscInt           i;
106: #if defined(PETSC_USE_LOG)
107:   PetscLogStage      stages[1];
108: #endif

110:   PetscInitialize(&argc, &argv, (char*)0,help);
111:   user.mx = 8;
112:   PetscOptionsBegin(PETSC_COMM_WORLD,NULL,"parabolic example",NULL);
113:   PetscOptionsInt("-mx","Number of grid points in each direction","",user.mx,&user.mx,NULL);
114:   user.nt = 8;
115:   PetscOptionsInt("-nt","Number of time steps","",user.nt,&user.nt,NULL);
116:   user.ndata = 64;
117:   PetscOptionsInt("-ndata","Numbers of data points per sample","",user.ndata,&user.ndata,NULL);
118:   user.ns = 8;
119:   PetscOptionsInt("-ns","Number of samples","",user.ns,&user.ns,NULL);
120:   user.alpha = 1.0;
121:   PetscOptionsReal("-alpha","Regularization parameter","",user.alpha,&user.alpha,NULL);
122:   user.beta = 0.01;
123:   PetscOptionsReal("-beta","Weight attributed to ||u||^2 in regularization functional","",user.beta,&user.beta,NULL);
124:   user.noise = 0.01;
125:   PetscOptionsReal("-noise","Amount of noise to add to data","",user.noise,&user.noise,NULL);
126:   PetscOptionsInt("-ntests","Number of times to repeat TaoSolve","",ntests,&ntests,NULL);
127:   PetscOptionsEnd();

129:   user.m = user.mx*user.mx*user.mx; /*  number of constraints per time step */
130:   user.n = user.m*(user.nt+1); /*  number of variables */
131:   user.ht = (PetscReal)1/user.nt;

133:   VecCreate(PETSC_COMM_WORLD,&user.u);
134:   VecCreate(PETSC_COMM_WORLD,&user.y);
135:   VecCreate(PETSC_COMM_WORLD,&user.c);
136:   VecSetSizes(user.u,PETSC_DECIDE,user.n-user.m*user.nt);
137:   VecSetSizes(user.y,PETSC_DECIDE,user.m*user.nt);
138:   VecSetSizes(user.c,PETSC_DECIDE,user.m*user.nt);
139:   VecSetFromOptions(user.u);
140:   VecSetFromOptions(user.y);
141:   VecSetFromOptions(user.c);

143:   /* Create scatters for reduced spaces.
144:      If the state vector y and design vector u are partitioned as
145:      [y_1; y_2; ...; y_np] and [u_1; u_2; ...; u_np] (with np = # of processors),
146:      then the solution vector x is organized as
147:      [y_1; u_1; y_2; u_2; ...; y_np; u_np].
148:      The index sets user.s_is and user.d_is correspond to the indices of the
149:      state and design variables owned by the current processor.
150:   */
151:   VecCreate(PETSC_COMM_WORLD,&x);

153:   VecGetOwnershipRange(user.y,&lo,&hi);
154:   VecGetOwnershipRange(user.u,&lo2,&hi2);

156:   ISCreateStride(PETSC_COMM_SELF,hi-lo,lo,1,&is_allstate);
157:   ISCreateStride(PETSC_COMM_SELF,hi-lo,lo+lo2,1,&user.s_is);

159:   ISCreateStride(PETSC_COMM_SELF,hi2-lo2,lo2,1,&is_alldesign);
160:   ISCreateStride(PETSC_COMM_SELF,hi2-lo2,hi+lo2,1,&user.d_is);

162:   VecSetSizes(x,hi-lo+hi2-lo2,user.n);
163:   VecSetFromOptions(x);

165:   VecScatterCreate(x,user.s_is,user.y,is_allstate,&user.state_scatter);
166:   VecScatterCreate(x,user.d_is,user.u,is_alldesign,&user.design_scatter);
167:   ISDestroy(&is_alldesign);
168:   ISDestroy(&is_allstate);

170:   /* Create TAO solver and set desired solution method */
171:   TaoCreate(PETSC_COMM_WORLD,&tao);
172:   TaoSetType(tao,TAOLCL);

174:   /* Set up initial vectors and matrices */
175:   ParabolicInitialize(&user);

177:   Gather(x,user.y,user.state_scatter,user.u,user.design_scatter);
178:   VecDuplicate(x,&x0);
179:   VecCopy(x,x0);

181:   /* Set solution vector with an initial guess */
182:   TaoSetSolution(tao,x);
183:   TaoSetObjective(tao, FormFunction, &user);
184:   TaoSetGradient(tao, NULL, FormGradient, &user);
185:   TaoSetConstraintsRoutine(tao, user.c, FormConstraints, &user);

187:   TaoSetJacobianStateRoutine(tao, user.Js, user.JsBlockPrec, user.JsInv, FormJacobianState, &user);
188:   TaoSetJacobianDesignRoutine(tao, user.Jd, FormJacobianDesign, &user);

190:   TaoSetFromOptions(tao);
191:   TaoSetStateDesignIS(tao,user.s_is,user.d_is);

193:  /* SOLVE THE APPLICATION */
194:   PetscLogStageRegister("Trials",&stages[0]);
195:   PetscLogStagePush(stages[0]);
196:   user.ksp_its_initial = user.ksp_its;
197:   ksp_old = user.ksp_its;
198:   for (i=0; i<ntests; i++) {
199:     TaoSolve(tao);
200:     PetscPrintf(PETSC_COMM_WORLD,"KSP Iterations = %D\n",user.ksp_its-ksp_old);
201:     VecCopy(x0,x);
202:     TaoSetSolution(tao,x);
203:   }
204:   PetscLogStagePop();
205:   PetscBarrier((PetscObject)x);
206:   PetscPrintf(PETSC_COMM_WORLD,"KSP iterations within initialization: ");
207:   PetscPrintf(PETSC_COMM_WORLD,"%D\n",user.ksp_its_initial);
208:   PetscPrintf(PETSC_COMM_WORLD,"Total KSP iterations over %D trial(s): ",ntests);
209:   PetscPrintf(PETSC_COMM_WORLD,"%D\n",user.ksp_its);
210:   PetscPrintf(PETSC_COMM_WORLD,"KSP iterations per trial: ");
211:   PetscPrintf(PETSC_COMM_WORLD,"%D\n",(user.ksp_its-user.ksp_its_initial)/ntests);

213:   TaoDestroy(&tao);
214:   VecDestroy(&x);
215:   VecDestroy(&x0);
216:   ParabolicDestroy(&user);

218:   PetscFinalize();
219:   return 0;
220: }
221: /* ------------------------------------------------------------------- */
222: /*
223:    dwork = Qy - d
224:    lwork = L*(u-ur)
225:    f = 1/2 * (dwork.dork + alpha*lwork.lwork)
226: */
227: PetscErrorCode FormFunction(Tao tao,Vec X,PetscReal *f,void *ptr)
228: {
229:   PetscReal      d1=0,d2=0;
230:   PetscInt       i,j;
231:   AppCtx         *user = (AppCtx*)ptr;

233:   Scatter(X,user->y,user->state_scatter,user->u,user->design_scatter);
234:   Scatter_i(user->y,user->yi,user->yi_scatter,user->nt);
235:   for (j=0; j<user->ns; j++) {
236:     i = user->sample_times[j];
237:     MatMult(user->Qblock,user->yi[i],user->di[j]);
238:   }
239:   Gather_i(user->dwork,user->di,user->di_scatter,user->ns);
240:   VecAXPY(user->dwork,-1.0,user->d);
241:   VecDot(user->dwork,user->dwork,&d1);

243:   VecWAXPY(user->uwork,-1.0,user->ur,user->u);
244:   MatMult(user->L,user->uwork,user->lwork);
245:   VecDot(user->lwork,user->lwork,&d2);

247:   *f = 0.5 * (d1 + user->alpha*d2);
248:   return 0;
249: }

251: /* ------------------------------------------------------------------- */
252: /*
253:     state: g_s = Q' *(Qy - d)
254:     design: g_d = alpha*L'*L*(u-ur)
255: */
256: PetscErrorCode FormGradient(Tao tao,Vec X,Vec G,void *ptr)
257: {
258:   PetscInt       i,j;
259:   AppCtx         *user = (AppCtx*)ptr;

261:   Scatter(X,user->y,user->state_scatter,user->u,user->design_scatter);
262:   Scatter_i(user->y,user->yi,user->yi_scatter,user->nt);
263:   for (j=0; j<user->ns; j++) {
264:     i = user->sample_times[j];
265:     MatMult(user->Qblock,user->yi[i],user->di[j]);
266:   }
267:   Gather_i(user->dwork,user->di,user->di_scatter,user->ns);
268:   VecAXPY(user->dwork,-1.0,user->d);
269:   Scatter_i(user->dwork,user->di,user->di_scatter,user->ns);
270:   VecSet(user->ywork,0.0);
271:   Scatter_i(user->ywork,user->yiwork,user->yi_scatter,user->nt);
272:   for (j=0; j<user->ns; j++) {
273:     i = user->sample_times[j];
274:     MatMult(user->QblockT,user->di[j],user->yiwork[i]);
275:   }
276:   Gather_i(user->ywork,user->yiwork,user->yi_scatter,user->nt);

278:   VecWAXPY(user->uwork,-1.0,user->ur,user->u);
279:   MatMult(user->L,user->uwork,user->lwork);
280:   MatMult(user->LT,user->lwork,user->uwork);
281:   VecScale(user->uwork, user->alpha);
282:   Gather(G,user->ywork,user->state_scatter,user->uwork,user->design_scatter);
283:   return 0;
284: }

286: PetscErrorCode FormFunctionGradient(Tao tao, Vec X, PetscReal *f, Vec G, void *ptr)
287: {
288:   PetscReal      d1,d2;
289:   PetscInt       i,j;
290:   AppCtx         *user = (AppCtx*)ptr;

292:   Scatter(X,user->y,user->state_scatter,user->u,user->design_scatter);
293:   Scatter_i(user->y,user->yi,user->yi_scatter,user->nt);
294:   for (j=0; j<user->ns; j++) {
295:     i = user->sample_times[j];
296:     MatMult(user->Qblock,user->yi[i],user->di[j]);
297:   }
298:   Gather_i(user->dwork,user->di,user->di_scatter,user->ns);
299:   VecAXPY(user->dwork,-1.0,user->d);
300:   VecDot(user->dwork,user->dwork,&d1);
301:   Scatter_i(user->dwork,user->di,user->di_scatter,user->ns);
302:   VecSet(user->ywork,0.0);
303:   Scatter_i(user->ywork,user->yiwork,user->yi_scatter,user->nt);
304:   for (j=0; j<user->ns; j++) {
305:     i = user->sample_times[j];
306:     MatMult(user->QblockT,user->di[j],user->yiwork[i]);
307:   }
308:   Gather_i(user->ywork,user->yiwork,user->yi_scatter,user->nt);

310:   VecWAXPY(user->uwork,-1.0,user->ur,user->u);
311:   MatMult(user->L,user->uwork,user->lwork);
312:   VecDot(user->lwork,user->lwork,&d2);
313:   MatMult(user->LT,user->lwork,user->uwork);
314:   VecScale(user->uwork, user->alpha);
315:   *f = 0.5 * (d1 + user->alpha*d2);

317:   Gather(G,user->ywork,user->state_scatter,user->uwork,user->design_scatter);
318:   return 0;
319: }

321: /* ------------------------------------------------------------------- */
322: /* A
323: MatShell object
324: */
325: PetscErrorCode FormJacobianState(Tao tao, Vec X, Mat J, Mat JPre, Mat JInv, void *ptr)
326: {
327:   AppCtx         *user = (AppCtx*)ptr;

329:   Scatter(X,user->y,user->state_scatter,user->u,user->design_scatter);
330:   VecSet(user->uwork,0);
331:   VecAXPY(user->uwork,-1.0,user->u);
332:   VecExp(user->uwork);
333:   MatMult(user->Av,user->uwork,user->Av_u);
334:   VecCopy(user->Av_u,user->Swork);
335:   VecReciprocal(user->Swork);
336:   MatCopy(user->Div,user->Divwork,SAME_NONZERO_PATTERN);
337:   MatDiagonalScale(user->Divwork,NULL,user->Swork);
338:   if (user->dsg_formed) {
339:     MatProductNumeric(user->DSG);
340:   } else {
341:     MatMatMult(user->Divwork,user->Grad,MAT_INITIAL_MATRIX,PETSC_DEFAULT,&user->DSG);
342:     user->dsg_formed = PETSC_TRUE;
343:   }

345:   /* B = speye(nx^3) + ht*DSG; */
346:   MatScale(user->DSG,user->ht);
347:   MatShift(user->DSG,1.0);
348:   return 0;
349: }

351: /* ------------------------------------------------------------------- */
352: /* B */
353: PetscErrorCode FormJacobianDesign(Tao tao, Vec X, Mat J, void *ptr)
354: {
355:   AppCtx         *user = (AppCtx*)ptr;

357:   Scatter(X,user->y,user->state_scatter,user->u,user->design_scatter);
358:   return 0;
359: }

361: PetscErrorCode StateMatMult(Mat J_shell, Vec X, Vec Y)
362: {
363:   PetscInt       i;
364:   AppCtx         *user;

366:   MatShellGetContext(J_shell,&user);
367:   Scatter_i(X,user->yi,user->yi_scatter,user->nt);
368:   MatMult(user->JsBlock,user->yi[0],user->yiwork[0]);
369:   for (i=1; i<user->nt; i++) {
370:     MatMult(user->JsBlock,user->yi[i],user->yiwork[i]);
371:     VecAXPY(user->yiwork[i],-1.0,user->yi[i-1]);
372:   }
373:   Gather_i(Y,user->yiwork,user->yi_scatter,user->nt);
374:   return 0;
375: }

377: PetscErrorCode StateMatMultTranspose(Mat J_shell, Vec X, Vec Y)
378: {
379:   PetscInt       i;
380:   AppCtx         *user;

382:   MatShellGetContext(J_shell,&user);
383:   Scatter_i(X,user->yi,user->yi_scatter,user->nt);
384:   for (i=0; i<user->nt-1; i++) {
385:     MatMult(user->JsBlock,user->yi[i],user->yiwork[i]);
386:     VecAXPY(user->yiwork[i],-1.0,user->yi[i+1]);
387:   }
388:   i = user->nt-1;
389:   MatMult(user->JsBlock,user->yi[i],user->yiwork[i]);
390:   Gather_i(Y,user->yiwork,user->yi_scatter,user->nt);
391:   return 0;
392: }

394: PetscErrorCode StateMatBlockMult(Mat J_shell, Vec X, Vec Y)
395: {
396:   AppCtx         *user;

398:   MatShellGetContext(J_shell,&user);
399:   MatMult(user->Grad,X,user->Swork);
400:   VecPointwiseDivide(user->Swork,user->Swork,user->Av_u);
401:   MatMult(user->Div,user->Swork,Y);
402:   VecAYPX(Y,user->ht,X);
403:   return 0;
404: }

406: PetscErrorCode DesignMatMult(Mat J_shell, Vec X, Vec Y)
407: {
408:   PetscInt       i;
409:   AppCtx         *user;

411:   MatShellGetContext(J_shell,&user);

413:   /* sdiag(1./v) */
414:   VecSet(user->uwork,0);
415:   VecAXPY(user->uwork,-1.0,user->u);
416:   VecExp(user->uwork);

418:   /* sdiag(1./((Av*(1./v)).^2)) */
419:   MatMult(user->Av,user->uwork,user->Swork);
420:   VecPointwiseMult(user->Swork,user->Swork,user->Swork);
421:   VecReciprocal(user->Swork);

423:   /* (Av * (sdiag(1./v) * b)) */
424:   VecPointwiseMult(user->uwork,user->uwork,X);
425:   MatMult(user->Av,user->uwork,user->Twork);

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

430:   Scatter_i(user->y,user->yi,user->yi_scatter,user->nt);
431:   for (i=0; i<user->nt; i++) {
432:     /* (sdiag(Grad*y(:,i)) */
433:     MatMult(user->Grad,user->yi[i],user->Twork);

435:     /* ht * Div * (sdiag(Grad*y(:,i)) * (sdiag(1./((Av*(1./v)).^2)) * (Av * (sdiag(1./v) * b)))) */
436:     VecPointwiseMult(user->Twork,user->Twork,user->Swork);
437:     MatMult(user->Div,user->Twork,user->yiwork[i]);
438:     VecScale(user->yiwork[i],user->ht);
439:   }
440:   Gather_i(Y,user->yiwork,user->yi_scatter,user->nt);

442:   return 0;
443: }

445: PetscErrorCode DesignMatMultTranspose(Mat J_shell, Vec X, Vec Y)
446: {
447:   PetscInt       i;
448:   AppCtx         *user;

450:   MatShellGetContext(J_shell,&user);

452:   /* sdiag(1./((Av*(1./v)).^2)) */
453:   VecSet(user->uwork,0);
454:   VecAXPY(user->uwork,-1.0,user->u);
455:   VecExp(user->uwork);
456:   MatMult(user->Av,user->uwork,user->Rwork);
457:   VecPointwiseMult(user->Rwork,user->Rwork,user->Rwork);
458:   VecReciprocal(user->Rwork);

460:   VecSet(Y,0.0);
461:   Scatter_i(user->y,user->yi,user->yi_scatter,user->nt);
462:   Scatter_i(X,user->yiwork,user->yi_scatter,user->nt);
463:   for (i=0; i<user->nt; i++) {
464:     /* (Div' * b(:,i)) */
465:     MatMult(user->Grad,user->yiwork[i],user->Swork);

467:     /* sdiag(Grad*y(:,i)) */
468:     MatMult(user->Grad,user->yi[i],user->Twork);

470:     /* (sdiag(Grad*y(:,i)) * (Div' * b(:,i))) */
471:     VecPointwiseMult(user->Twork,user->Swork,user->Twork);

473:     /* (sdiag(1./((Av*(1./v)).^2)) * (sdiag(Grad*y(:,i)) * (Div' * b(:,i)))) */
474:     VecPointwiseMult(user->Twork,user->Rwork,user->Twork);

476:     /* (Av' * (sdiag(1./((Av*(1./v)).^2)) * (sdiag(Grad*y(:,i)) * (Div' * b(:,i))))) */
477:     MatMult(user->AvT,user->Twork,user->yiwork[i]);

479:     /* sdiag(1./v) * (Av' * (sdiag(1./((Av*(1./v)).^2)) * (sdiag(Grad*y(:,i)) * (Div' * b(:,i))))) */
480:     VecPointwiseMult(user->yiwork[i],user->uwork,user->yiwork[i]);
481:     VecAXPY(Y,user->ht,user->yiwork[i]);
482:   }
483:   return 0;
484: }

486: PetscErrorCode StateMatBlockPrecMult(PC PC_shell, Vec X, Vec Y)
487: {
488:   AppCtx         *user;

490:   PCShellGetContext(PC_shell,&user);

492:   if (user->dsg_formed) {
493:     MatSOR(user->DSG,X,1.0,(MatSORType)(SOR_ZERO_INITIAL_GUESS | SOR_LOCAL_SYMMETRIC_SWEEP),0.0,1,1,Y);
494:   } else SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"DSG not formed");
495:   return 0;
496: }

498: PetscErrorCode StateMatInvMult(Mat J_shell, Vec X, Vec Y)
499: {
500:   AppCtx         *user;
501:   PetscInt       its,i;

503:   MatShellGetContext(J_shell,&user);

505:   if (Y == user->ytrue) {
506:     /* First solve is done with true solution to set up problem */
507:     KSPSetTolerances(user->solver,1e-8,PETSC_DEFAULT,PETSC_DEFAULT,PETSC_DEFAULT);
508:   } else {
509:     KSPSetTolerances(user->solver,PETSC_DEFAULT,PETSC_DEFAULT,PETSC_DEFAULT,PETSC_DEFAULT);
510:   }

512:   Scatter_i(X,user->yi,user->yi_scatter,user->nt);
513:   KSPSolve(user->solver,user->yi[0],user->yiwork[0]);
514:   KSPGetIterationNumber(user->solver,&its);
515:   user->ksp_its = user->ksp_its + its;

517:   for (i=1; i<user->nt; i++) {
518:     VecAXPY(user->yi[i],1.0,user->yiwork[i-1]);
519:     KSPSolve(user->solver,user->yi[i],user->yiwork[i]);
520:     KSPGetIterationNumber(user->solver,&its);
521:     user->ksp_its = user->ksp_its + its;
522:   }
523:   Gather_i(Y,user->yiwork,user->yi_scatter,user->nt);
524:   return 0;
525: }

527: PetscErrorCode StateMatInvTransposeMult(Mat J_shell, Vec X, Vec Y)
528: {
529:   AppCtx         *user;
530:   PetscInt       its,i;

532:   MatShellGetContext(J_shell,&user);

534:   Scatter_i(X,user->yi,user->yi_scatter,user->nt);

536:   i = user->nt - 1;
537:   KSPSolve(user->solver,user->yi[i],user->yiwork[i]);

539:   KSPGetIterationNumber(user->solver,&its);
540:   user->ksp_its = user->ksp_its + its;

542:   for (i=user->nt-2; i>=0; i--) {
543:     VecAXPY(user->yi[i],1.0,user->yiwork[i+1]);
544:     KSPSolve(user->solver,user->yi[i],user->yiwork[i]);

546:     KSPGetIterationNumber(user->solver,&its);
547:     user->ksp_its = user->ksp_its + its;

549:   }

551:   Gather_i(Y,user->yiwork,user->yi_scatter,user->nt);
552:   return 0;
553: }

555: PetscErrorCode StateMatDuplicate(Mat J_shell, MatDuplicateOption opt, Mat *new_shell)
556: {
557:   AppCtx         *user;

559:   MatShellGetContext(J_shell,&user);

561:   MatCreateShell(PETSC_COMM_WORLD,PETSC_DETERMINE,PETSC_DETERMINE,user->m,user->m,user,new_shell);
562:   MatShellSetOperation(*new_shell,MATOP_MULT,(void(*)(void))StateMatMult);
563:   MatShellSetOperation(*new_shell,MATOP_DUPLICATE,(void(*)(void))StateMatDuplicate);
564:   MatShellSetOperation(*new_shell,MATOP_MULT_TRANSPOSE,(void(*)(void))StateMatMultTranspose);
565:   MatShellSetOperation(*new_shell,MATOP_GET_DIAGONAL,(void(*)(void))StateMatGetDiagonal);
566:   return 0;
567: }

569: PetscErrorCode StateMatGetDiagonal(Mat J_shell, Vec X)
570: {
571:   AppCtx         *user;

573:   MatShellGetContext(J_shell,&user);
574:   VecCopy(user->js_diag,X);
575:   return 0;

577: }

579: PetscErrorCode FormConstraints(Tao tao, Vec X, Vec C, void *ptr)
580: {
581:   /* con = Ay - q, A = [B  0  0 ... 0;
582:                        -I  B  0 ... 0;
583:                         0 -I  B ... 0;
584:                              ...     ;
585:                         0    ... -I B]
586:      B = ht * Div * Sigma * Grad + eye */
587:   PetscInt       i;
588:   AppCtx         *user = (AppCtx*)ptr;

590:   Scatter(X,user->y,user->state_scatter,user->u,user->design_scatter);
591:   Scatter_i(user->y,user->yi,user->yi_scatter,user->nt);
592:   MatMult(user->JsBlock,user->yi[0],user->yiwork[0]);
593:   for (i=1; i<user->nt; i++) {
594:     MatMult(user->JsBlock,user->yi[i],user->yiwork[i]);
595:     VecAXPY(user->yiwork[i],-1.0,user->yi[i-1]);
596:   }
597:   Gather_i(C,user->yiwork,user->yi_scatter,user->nt);
598:   VecAXPY(C,-1.0,user->q);
599:   return 0;
600: }

602: PetscErrorCode Scatter(Vec x, Vec state, VecScatter s_scat, Vec design, VecScatter d_scat)
603: {
604:   VecScatterBegin(s_scat,x,state,INSERT_VALUES,SCATTER_FORWARD);
605:   VecScatterEnd(s_scat,x,state,INSERT_VALUES,SCATTER_FORWARD);
606:   VecScatterBegin(d_scat,x,design,INSERT_VALUES,SCATTER_FORWARD);
607:   VecScatterEnd(d_scat,x,design,INSERT_VALUES,SCATTER_FORWARD);
608:   return 0;
609: }

611: PetscErrorCode Scatter_i(Vec y, Vec *yi, VecScatter *scat, PetscInt nt)
612: {
613:   PetscInt       i;

615:   for (i=0; i<nt; i++) {
616:     VecScatterBegin(scat[i],y,yi[i],INSERT_VALUES,SCATTER_FORWARD);
617:     VecScatterEnd(scat[i],y,yi[i],INSERT_VALUES,SCATTER_FORWARD);
618:   }
619:   return 0;
620: }

622: PetscErrorCode Gather(Vec x, Vec state, VecScatter s_scat, Vec design, VecScatter d_scat)
623: {
624:   VecScatterBegin(s_scat,state,x,INSERT_VALUES,SCATTER_REVERSE);
625:   VecScatterEnd(s_scat,state,x,INSERT_VALUES,SCATTER_REVERSE);
626:   VecScatterBegin(d_scat,design,x,INSERT_VALUES,SCATTER_REVERSE);
627:   VecScatterEnd(d_scat,design,x,INSERT_VALUES,SCATTER_REVERSE);
628:   return 0;
629: }

631: PetscErrorCode Gather_i(Vec y, Vec *yi, VecScatter *scat, PetscInt nt)
632: {
633:   PetscInt       i;

635:   for (i=0; i<nt; i++) {
636:     VecScatterBegin(scat[i],yi[i],y,INSERT_VALUES,SCATTER_REVERSE);
637:     VecScatterEnd(scat[i],yi[i],y,INSERT_VALUES,SCATTER_REVERSE);
638:   }
639:   return 0;
640: }

642: PetscErrorCode ParabolicInitialize(AppCtx *user)
643: {
644:   PetscInt       m,n,i,j,k,linear_index,istart,iend,iblock,lo,hi,lo2,hi2;
645:   Vec            XX,YY,ZZ,XXwork,YYwork,ZZwork,UTwork,yi,di,bc;
646:   PetscReal      *x, *y, *z;
647:   PetscReal      h,stime;
648:   PetscScalar    hinv,neg_hinv,half = 0.5,sqrt_beta;
649:   PetscInt       im,indx1,indx2,indy1,indy2,indz1,indz2,nx,ny,nz;
650:   PetscReal      xri,yri,zri,xim,yim,zim,dx1,dx2,dy1,dy2,dz1,dz2,Dx,Dy,Dz;
651:   PetscScalar    v,vx,vy,vz;
652:   IS             is_from_y,is_to_yi,is_from_d,is_to_di;
653:   /* Data locations */
654:   PetscScalar xr[64] = {0.4970,     0.8498,     0.7814,     0.6268,     0.7782,     0.6402,     0.3617,     0.3160,
655:                         0.3610,     0.5298,     0.6987,     0.3331,     0.7962,     0.5596,     0.3866,     0.6774,
656:                         0.5407,     0.4518,     0.6702,     0.6061,     0.7580,     0.8997,     0.5198,     0.8326,
657:                         0.2138,     0.9198,     0.3000,     0.2833,     0.8288,     0.7076,     0.1820,     0.0728,
658:                         0.8447,     0.2367,     0.3239,     0.6413,     0.3114,     0.4731,     0.1192,     0.9273,
659:                         0.5724,     0.4331,     0.5136,     0.3547,     0.4413,     0.2602,     0.5698,     0.7278,
660:                         0.5261,     0.6230,     0.2454,     0.3948,     0.7479,     0.6582,     0.4660,     0.5594,
661:                         0.7574,     0.1143,     0.5900,     0.1065,     0.4260,     0.3294,     0.8276,     0.0756};

663:   PetscScalar yr[64] = {0.7345,     0.9120,     0.9288,     0.7528,     0.4463,     0.4985,     0.2497,     0.6256,
664:                         0.3425,     0.9026,     0.6983,     0.4230,     0.7140,     0.2970,     0.4474,     0.8792,
665:                         0.6604,     0.2485,     0.7968,     0.6127,     0.1796,     0.2437,     0.5938,     0.6137,
666:                         0.3867,     0.5658,     0.4575,     0.1009,     0.0863,     0.3361,     0.0738,     0.3985,
667:                         0.6602,     0.1437,     0.0934,     0.5983,     0.5950,     0.0763,     0.0768,     0.2288,
668:                         0.5761,     0.1129,     0.3841,     0.6150,     0.6904,     0.6686,     0.1361,     0.4601,
669:                         0.4491,     0.3716,     0.1969,     0.6537,     0.6743,     0.6991,     0.4811,     0.5480,
670:                         0.1684,     0.4569,     0.6889,     0.8437,     0.3015,     0.2854,     0.8199,     0.2658};

672:   PetscScalar zr[64] = {0.7668,     0.8573,     0.2654,     0.2719,     0.1060,     0.1311,     0.6232,     0.2295,
673:                         0.8009,     0.2147,     0.2119,     0.9325,     0.4473,     0.3600,     0.3374,     0.3819,
674:                         0.4066,     0.5801,     0.1673,     0.0959,     0.4638,     0.8236,     0.8800,     0.2939,
675:                         0.2028,     0.8262,     0.2706,     0.6276,     0.9085,     0.6443,     0.8241,     0.0712,
676:                         0.1824,     0.7789,     0.4389,     0.8415,     0.7055,     0.6639,     0.3653,     0.2078,
677:                         0.1987,     0.2297,     0.4321,     0.8115,     0.4915,     0.7764,     0.4657,     0.4627,
678:                         0.4569,     0.4232,     0.8514,     0.0674,     0.3227,     0.1055,     0.6690,     0.6313,
679:                         0.9226,     0.5461,     0.4126,     0.2364,     0.6096,     0.7042,     0.3914,     0.0711};

681:   PetscMalloc1(user->mx,&x);
682:   PetscMalloc1(user->mx,&y);
683:   PetscMalloc1(user->mx,&z);
684:   user->jformed = PETSC_FALSE;
685:   user->dsg_formed = PETSC_FALSE;

687:   n = user->mx * user->mx * user->mx;
688:   m = 3 * user->mx * user->mx * (user->mx-1);
689:   sqrt_beta = PetscSqrtScalar(user->beta);

691:   user->ksp_its = 0;
692:   user->ksp_its_initial = 0;

694:   stime = (PetscReal)user->nt/user->ns;
695:   PetscMalloc1(user->ns,&user->sample_times);
696:   for (i=0; i<user->ns; i++) {
697:     user->sample_times[i] = (PetscInt)(stime*((PetscReal)i+1.0)-0.5);
698:   }

700:   VecCreate(PETSC_COMM_WORLD,&XX);
701:   VecCreate(PETSC_COMM_WORLD,&user->q);
702:   VecSetSizes(XX,PETSC_DECIDE,n);
703:   VecSetSizes(user->q,PETSC_DECIDE,n*user->nt);
704:   VecSetFromOptions(XX);
705:   VecSetFromOptions(user->q);

707:   VecDuplicate(XX,&YY);
708:   VecDuplicate(XX,&ZZ);
709:   VecDuplicate(XX,&XXwork);
710:   VecDuplicate(XX,&YYwork);
711:   VecDuplicate(XX,&ZZwork);
712:   VecDuplicate(XX,&UTwork);
713:   VecDuplicate(XX,&user->utrue);
714:   VecDuplicate(XX,&bc);

716:   /* Generate 3D grid, and collect ns (1<=ns<=8) right-hand-side vectors into user->q */
717:   h = 1.0/user->mx;
718:   hinv = user->mx;
719:   neg_hinv = -hinv;

721:   VecGetOwnershipRange(XX,&istart,&iend);
722:   for (linear_index=istart; linear_index<iend; linear_index++) {
723:     i = linear_index % user->mx;
724:     j = ((linear_index-i)/user->mx) % user->mx;
725:     k = ((linear_index-i)/user->mx-j) / user->mx;
726:     vx = h*(i+0.5);
727:     vy = h*(j+0.5);
728:     vz = h*(k+0.5);
729:     VecSetValues(XX,1,&linear_index,&vx,INSERT_VALUES);
730:     VecSetValues(YY,1,&linear_index,&vy,INSERT_VALUES);
731:     VecSetValues(ZZ,1,&linear_index,&vz,INSERT_VALUES);
732:     if ((vx<0.6) && (vx>0.4) && (vy<0.6) && (vy>0.4) && (vy<0.6) && (vz<0.6) && (vz>0.4)) {
733:       v = 1000.0;
734:       VecSetValues(bc,1,&linear_index,&v,INSERT_VALUES);
735:     }
736:   }

738:   VecAssemblyBegin(XX);
739:   VecAssemblyEnd(XX);
740:   VecAssemblyBegin(YY);
741:   VecAssemblyEnd(YY);
742:   VecAssemblyBegin(ZZ);
743:   VecAssemblyEnd(ZZ);
744:   VecAssemblyBegin(bc);
745:   VecAssemblyEnd(bc);

747:   /* Compute true parameter function
748:      ut = 0.5 + exp(-10*((x-0.5)^2+(y-0.5)^2+(z-0.5)^2)) */
749:   VecCopy(XX,XXwork);
750:   VecCopy(YY,YYwork);
751:   VecCopy(ZZ,ZZwork);

753:   VecShift(XXwork,-0.5);
754:   VecShift(YYwork,-0.5);
755:   VecShift(ZZwork,-0.5);

757:   VecPointwiseMult(XXwork,XXwork,XXwork);
758:   VecPointwiseMult(YYwork,YYwork,YYwork);
759:   VecPointwiseMult(ZZwork,ZZwork,ZZwork);

761:   VecCopy(XXwork,user->utrue);
762:   VecAXPY(user->utrue,1.0,YYwork);
763:   VecAXPY(user->utrue,1.0,ZZwork);
764:   VecScale(user->utrue,-10.0);
765:   VecExp(user->utrue);
766:   VecShift(user->utrue,0.5);

768:   VecDestroy(&XX);
769:   VecDestroy(&YY);
770:   VecDestroy(&ZZ);
771:   VecDestroy(&XXwork);
772:   VecDestroy(&YYwork);
773:   VecDestroy(&ZZwork);
774:   VecDestroy(&UTwork);

776:   /* Initial guess and reference model */
777:   VecDuplicate(user->utrue,&user->ur);
778:   VecSet(user->ur,0.0);

780:   /* Generate Grad matrix */
781:   MatCreate(PETSC_COMM_WORLD,&user->Grad);
782:   MatSetSizes(user->Grad,PETSC_DECIDE,PETSC_DECIDE,m,n);
783:   MatSetFromOptions(user->Grad);
784:   MatMPIAIJSetPreallocation(user->Grad,2,NULL,2,NULL);
785:   MatSeqAIJSetPreallocation(user->Grad,2,NULL);
786:   MatGetOwnershipRange(user->Grad,&istart,&iend);

788:   for (i=istart; i<iend; i++) {
789:     if (i<m/3) {
790:       iblock = i / (user->mx-1);
791:       j = iblock*user->mx + (i % (user->mx-1));
792:       MatSetValues(user->Grad,1,&i,1,&j,&neg_hinv,INSERT_VALUES);
793:       j = j+1;
794:       MatSetValues(user->Grad,1,&i,1,&j,&hinv,INSERT_VALUES);
795:     }
796:     if (i>=m/3 && i<2*m/3) {
797:       iblock = (i-m/3) / (user->mx*(user->mx-1));
798:       j = iblock*user->mx*user->mx + ((i-m/3) % (user->mx*(user->mx-1)));
799:       MatSetValues(user->Grad,1,&i,1,&j,&neg_hinv,INSERT_VALUES);
800:       j = j + user->mx;
801:       MatSetValues(user->Grad,1,&i,1,&j,&hinv,INSERT_VALUES);
802:     }
803:     if (i>=2*m/3) {
804:       j = i-2*m/3;
805:       MatSetValues(user->Grad,1,&i,1,&j,&neg_hinv,INSERT_VALUES);
806:       j = j + user->mx*user->mx;
807:       MatSetValues(user->Grad,1,&i,1,&j,&hinv,INSERT_VALUES);
808:     }
809:   }

811:   MatAssemblyBegin(user->Grad,MAT_FINAL_ASSEMBLY);
812:   MatAssemblyEnd(user->Grad,MAT_FINAL_ASSEMBLY);

814:   /* Generate arithmetic averaging matrix Av */
815:   MatCreate(PETSC_COMM_WORLD,&user->Av);
816:   MatSetSizes(user->Av,PETSC_DECIDE,PETSC_DECIDE,m,n);
817:   MatSetFromOptions(user->Av);
818:   MatMPIAIJSetPreallocation(user->Av,2,NULL,2,NULL);
819:   MatSeqAIJSetPreallocation(user->Av,2,NULL);
820:   MatGetOwnershipRange(user->Av,&istart,&iend);

822:   for (i=istart; i<iend; i++) {
823:     if (i<m/3) {
824:       iblock = i / (user->mx-1);
825:       j = iblock*user->mx + (i % (user->mx-1));
826:       MatSetValues(user->Av,1,&i,1,&j,&half,INSERT_VALUES);
827:       j = j+1;
828:       MatSetValues(user->Av,1,&i,1,&j,&half,INSERT_VALUES);
829:     }
830:     if (i>=m/3 && i<2*m/3) {
831:       iblock = (i-m/3) / (user->mx*(user->mx-1));
832:       j = iblock*user->mx*user->mx + ((i-m/3) % (user->mx*(user->mx-1)));
833:       MatSetValues(user->Av,1,&i,1,&j,&half,INSERT_VALUES);
834:       j = j + user->mx;
835:       MatSetValues(user->Av,1,&i,1,&j,&half,INSERT_VALUES);
836:     }
837:     if (i>=2*m/3) {
838:       j = i-2*m/3;
839:       MatSetValues(user->Av,1,&i,1,&j,&half,INSERT_VALUES);
840:       j = j + user->mx*user->mx;
841:       MatSetValues(user->Av,1,&i,1,&j,&half,INSERT_VALUES);
842:     }
843:   }

845:   MatAssemblyBegin(user->Av,MAT_FINAL_ASSEMBLY);
846:   MatAssemblyEnd(user->Av,MAT_FINAL_ASSEMBLY);

848:   /* Generate transpose of averaging matrix Av */
849:   MatTranspose(user->Av,MAT_INITIAL_MATRIX,&user->AvT);

851:   MatCreate(PETSC_COMM_WORLD,&user->L);
852:   MatSetSizes(user->L,PETSC_DECIDE,PETSC_DECIDE,m+n,n);
853:   MatSetFromOptions(user->L);
854:   MatMPIAIJSetPreallocation(user->L,2,NULL,2,NULL);
855:   MatSeqAIJSetPreallocation(user->L,2,NULL);
856:   MatGetOwnershipRange(user->L,&istart,&iend);

858:   for (i=istart; i<iend; i++) {
859:     if (i<m/3) {
860:       iblock = i / (user->mx-1);
861:       j = iblock*user->mx + (i % (user->mx-1));
862:       MatSetValues(user->L,1,&i,1,&j,&neg_hinv,INSERT_VALUES);
863:       j = j+1;
864:       MatSetValues(user->L,1,&i,1,&j,&hinv,INSERT_VALUES);
865:     }
866:     if (i>=m/3 && i<2*m/3) {
867:       iblock = (i-m/3) / (user->mx*(user->mx-1));
868:       j = iblock*user->mx*user->mx + ((i-m/3) % (user->mx*(user->mx-1)));
869:       MatSetValues(user->L,1,&i,1,&j,&neg_hinv,INSERT_VALUES);
870:       j = j + user->mx;
871:       MatSetValues(user->L,1,&i,1,&j,&hinv,INSERT_VALUES);
872:     }
873:     if (i>=2*m/3 && i<m) {
874:       j = i-2*m/3;
875:       MatSetValues(user->L,1,&i,1,&j,&neg_hinv,INSERT_VALUES);
876:       j = j + user->mx*user->mx;
877:       MatSetValues(user->L,1,&i,1,&j,&hinv,INSERT_VALUES);
878:     }
879:     if (i>=m) {
880:       j = i - m;
881:       MatSetValues(user->L,1,&i,1,&j,&sqrt_beta,INSERT_VALUES);
882:     }
883:   }

885:   MatAssemblyBegin(user->L,MAT_FINAL_ASSEMBLY);
886:   MatAssemblyEnd(user->L,MAT_FINAL_ASSEMBLY);

888:   MatScale(user->L,PetscPowScalar(h,1.5));

890:   /* Generate Div matrix */
891:   MatTranspose(user->Grad,MAT_INITIAL_MATRIX,&user->Div);

893:   /* Build work vectors and matrices */
894:   VecCreate(PETSC_COMM_WORLD,&user->S);
895:   VecSetSizes(user->S, PETSC_DECIDE, user->mx*user->mx*(user->mx-1)*3);
896:   VecSetFromOptions(user->S);

898:   VecCreate(PETSC_COMM_WORLD,&user->lwork);
899:   VecSetSizes(user->lwork,PETSC_DECIDE,m+user->mx*user->mx*user->mx);
900:   VecSetFromOptions(user->lwork);

902:   MatDuplicate(user->Div,MAT_SHARE_NONZERO_PATTERN,&user->Divwork);
903:   MatDuplicate(user->Av,MAT_SHARE_NONZERO_PATTERN,&user->Avwork);

905:   VecCreate(PETSC_COMM_WORLD,&user->d);
906:   VecSetSizes(user->d,PETSC_DECIDE,user->ndata*user->nt);
907:   VecSetFromOptions(user->d);

909:   VecDuplicate(user->S,&user->Swork);
910:   VecDuplicate(user->S,&user->Av_u);
911:   VecDuplicate(user->S,&user->Twork);
912:   VecDuplicate(user->S,&user->Rwork);
913:   VecDuplicate(user->y,&user->ywork);
914:   VecDuplicate(user->u,&user->uwork);
915:   VecDuplicate(user->u,&user->js_diag);
916:   VecDuplicate(user->c,&user->cwork);
917:   VecDuplicate(user->d,&user->dwork);

919:   /* Create matrix-free shell user->Js for computing A*x */
920:   MatCreateShell(PETSC_COMM_WORLD,PETSC_DETERMINE,PETSC_DETERMINE,user->m*user->nt,user->m*user->nt,user,&user->Js);
921:   MatShellSetOperation(user->Js,MATOP_MULT,(void(*)(void))StateMatMult);
922:   MatShellSetOperation(user->Js,MATOP_DUPLICATE,(void(*)(void))StateMatDuplicate);
923:   MatShellSetOperation(user->Js,MATOP_MULT_TRANSPOSE,(void(*)(void))StateMatMultTranspose);
924:   MatShellSetOperation(user->Js,MATOP_GET_DIAGONAL,(void(*)(void))StateMatGetDiagonal);

926:   /* Diagonal blocks of user->Js */
927:   MatCreateShell(PETSC_COMM_WORLD,PETSC_DETERMINE,PETSC_DETERMINE,user->m,user->m,user,&user->JsBlock);
928:   MatShellSetOperation(user->JsBlock,MATOP_MULT,(void(*)(void))StateMatBlockMult);
929:   /* Blocks are symmetric */
930:   MatShellSetOperation(user->JsBlock,MATOP_MULT_TRANSPOSE,(void(*)(void))StateMatBlockMult);

932:   /* Create a matrix-free shell user->JsBlockPrec for computing (U+D)\D*(L+D)\x, where JsBlock = L+D+U,
933:      D is diagonal, L is strictly lower triangular, and U is strictly upper triangular.
934:      This is an SSOR preconditioner for user->JsBlock. */
935:   MatCreateShell(PETSC_COMM_WORLD,PETSC_DETERMINE,PETSC_DETERMINE,user->m,user->m,user,&user->JsBlockPrec);
936:   MatShellSetOperation(user->JsBlockPrec,MATOP_MULT,(void(*)(void))StateMatBlockPrecMult);
937:   /* JsBlockPrec is symmetric */
938:   MatShellSetOperation(user->JsBlockPrec,MATOP_MULT_TRANSPOSE,(void(*)(void))StateMatBlockPrecMult);
939:   MatSetOption(user->JsBlockPrec,MAT_SYMMETRY_ETERNAL,PETSC_TRUE);

941:   /* Create a matrix-free shell user->Jd for computing B*x */
942:   MatCreateShell(PETSC_COMM_WORLD,PETSC_DETERMINE,PETSC_DETERMINE,user->m*user->nt,user->m,user,&user->Jd);
943:   MatShellSetOperation(user->Jd,MATOP_MULT,(void(*)(void))DesignMatMult);
944:   MatShellSetOperation(user->Jd,MATOP_MULT_TRANSPOSE,(void(*)(void))DesignMatMultTranspose);

946:   /* User-defined routines for computing user->Js\x and user->Js^T\x*/
947:   MatCreateShell(PETSC_COMM_WORLD,PETSC_DETERMINE,PETSC_DETERMINE,user->m*user->nt,user->m*user->nt,user,&user->JsInv);
948:   MatShellSetOperation(user->JsInv,MATOP_MULT,(void(*)(void))StateMatInvMult);
949:   MatShellSetOperation(user->JsInv,MATOP_MULT_TRANSPOSE,(void(*)(void))StateMatInvTransposeMult);

951:   /* Solver options and tolerances */
952:   KSPCreate(PETSC_COMM_WORLD,&user->solver);
953:   KSPSetType(user->solver,KSPCG);
954:   KSPSetOperators(user->solver,user->JsBlock,user->JsBlockPrec);
955:   KSPSetInitialGuessNonzero(user->solver,PETSC_FALSE);
956:   KSPSetTolerances(user->solver,1e-4,1e-20,1e3,500);
957:   KSPSetFromOptions(user->solver);
958:   KSPGetPC(user->solver,&user->prec);
959:   PCSetType(user->prec,PCSHELL);

961:   PCShellSetApply(user->prec,StateMatBlockPrecMult);
962:   PCShellSetApplyTranspose(user->prec,StateMatBlockPrecMult);
963:   PCShellSetContext(user->prec,user);

965:   /* Create scatter from y to y_1,y_2,...,y_nt */
966:   PetscMalloc1(user->nt*user->m,&user->yi_scatter);
967:   VecCreate(PETSC_COMM_WORLD,&yi);
968:   VecSetSizes(yi,PETSC_DECIDE,user->mx*user->mx*user->mx);
969:   VecSetFromOptions(yi);
970:   VecDuplicateVecs(yi,user->nt,&user->yi);
971:   VecDuplicateVecs(yi,user->nt,&user->yiwork);

973:   VecGetOwnershipRange(user->y,&lo2,&hi2);
974:   istart = 0;
975:   for (i=0; i<user->nt; i++) {
976:     VecGetOwnershipRange(user->yi[i],&lo,&hi);
977:     ISCreateStride(PETSC_COMM_SELF,hi-lo,lo,1,&is_to_yi);
978:     ISCreateStride(PETSC_COMM_SELF,hi-lo,lo2+istart,1,&is_from_y);
979:     VecScatterCreate(user->y,is_from_y,user->yi[i],is_to_yi,&user->yi_scatter[i]);
980:     istart = istart + hi-lo;
981:     ISDestroy(&is_to_yi);
982:     ISDestroy(&is_from_y);
983:   }
984:   VecDestroy(&yi);

986:   /* Create scatter from d to d_1,d_2,...,d_ns */
987:   PetscMalloc1(user->ns*user->ndata,&user->di_scatter);
988:   VecCreate(PETSC_COMM_WORLD,&di);
989:   VecSetSizes(di,PETSC_DECIDE,user->ndata);
990:   VecSetFromOptions(di);
991:   VecDuplicateVecs(di,user->ns,&user->di);
992:   VecGetOwnershipRange(user->d,&lo2,&hi2);
993:   istart = 0;
994:   for (i=0; i<user->ns; i++) {
995:     VecGetOwnershipRange(user->di[i],&lo,&hi);
996:     ISCreateStride(PETSC_COMM_SELF,hi-lo,lo,1,&is_to_di);
997:     ISCreateStride(PETSC_COMM_SELF,hi-lo,lo2+istart,1,&is_from_d);
998:     VecScatterCreate(user->d,is_from_d,user->di[i],is_to_di,&user->di_scatter[i]);
999:     istart = istart + hi-lo;
1000:     ISDestroy(&is_to_di);
1001:     ISDestroy(&is_from_d);
1002:   }
1003:   VecDestroy(&di);

1005:   /* Assemble RHS of forward problem */
1006:   VecCopy(bc,user->yiwork[0]);
1007:   for (i=1; i<user->nt; i++) {
1008:     VecSet(user->yiwork[i],0.0);
1009:   }
1010:   Gather_i(user->q,user->yiwork,user->yi_scatter,user->nt);
1011:   VecDestroy(&bc);

1013:   /* Compute true state function yt given ut */
1014:   VecCreate(PETSC_COMM_WORLD,&user->ytrue);
1015:   VecSetSizes(user->ytrue,PETSC_DECIDE,n*user->nt);
1016:   VecSetFromOptions(user->ytrue);

1018:   /* First compute Av_u = Av*exp(-u) */
1019:   VecSet(user->uwork,0);
1020:   VecAXPY(user->uwork,-1.0,user->utrue); /*  Note: user->utrue */
1021:   VecExp(user->uwork);
1022:   MatMult(user->Av,user->uwork,user->Av_u);

1024:   /* Symbolic DSG = Div * Grad */
1025:   MatProductCreate(user->Div,user->Grad,NULL,&user->DSG);
1026:   MatProductSetType(user->DSG,MATPRODUCT_AB);
1027:   MatProductSetAlgorithm(user->DSG,"default");
1028:   MatProductSetFill(user->DSG,PETSC_DEFAULT);
1029:   MatProductSetFromOptions(user->DSG);
1030:   MatProductSymbolic(user->DSG);

1032:   user->dsg_formed = PETSC_TRUE;

1034:   /* Next form DSG = Div*Grad */
1035:   MatCopy(user->Div,user->Divwork,SAME_NONZERO_PATTERN);
1036:   MatDiagonalScale(user->Divwork,NULL,user->Av_u);
1037:   if (user->dsg_formed) {
1038:     MatProductNumeric(user->DSG);
1039:   } else {
1040:     MatMatMult(user->Div,user->Grad,MAT_INITIAL_MATRIX,PETSC_DEFAULT,&user->DSG);
1041:     user->dsg_formed = PETSC_TRUE;
1042:   }
1043:   /* B = speye(nx^3) + ht*DSG; */
1044:   MatScale(user->DSG,user->ht);
1045:   MatShift(user->DSG,1.0);

1047:   /* Now solve for ytrue */
1048:   StateMatInvMult(user->Js,user->q,user->ytrue);

1050:   /* Initial guess y0 for state given u0 */

1052:   /* First compute Av_u = Av*exp(-u) */
1053:   VecSet(user->uwork,0);
1054:   VecAXPY(user->uwork,-1.0,user->u); /*  Note: user->u */
1055:   VecExp(user->uwork);
1056:   MatMult(user->Av,user->uwork,user->Av_u);

1058:   /* Next form DSG = Div*S*Grad */
1059:   MatCopy(user->Div,user->Divwork,SAME_NONZERO_PATTERN);
1060:   MatDiagonalScale(user->Divwork,NULL,user->Av_u);
1061:   if (user->dsg_formed) {
1062:     MatProductNumeric(user->DSG);
1063:   } else {
1064:     MatMatMult(user->Div,user->Grad,MAT_INITIAL_MATRIX,PETSC_DEFAULT,&user->DSG);

1066:     user->dsg_formed = PETSC_TRUE;
1067:   }
1068:   /* B = speye(nx^3) + ht*DSG; */
1069:   MatScale(user->DSG,user->ht);
1070:   MatShift(user->DSG,1.0);

1072:   /* Now solve for y */
1073:   StateMatInvMult(user->Js,user->q,user->y);

1075:   /* Construct projection matrix Q, a block diagonal matrix consisting of nt copies of Qblock along the diagonal */
1076:   MatCreate(PETSC_COMM_WORLD,&user->Qblock);
1077:   MatSetSizes(user->Qblock,PETSC_DECIDE,PETSC_DECIDE,user->ndata,n);
1078:   MatSetFromOptions(user->Qblock);
1079:   MatMPIAIJSetPreallocation(user->Qblock,8,NULL,8,NULL);
1080:   MatSeqAIJSetPreallocation(user->Qblock,8,NULL);

1082:   for (i=0; i<user->mx; i++) {
1083:     x[i] = h*(i+0.5);
1084:     y[i] = h*(i+0.5);
1085:     z[i] = h*(i+0.5);
1086:   }

1088:   MatGetOwnershipRange(user->Qblock,&istart,&iend);
1089:   nx = user->mx; ny = user->mx; nz = user->mx;
1090:   for (i=istart; i<iend; i++) {
1091:     xri = xr[i];
1092:     im = 0;
1093:     xim = x[im];
1094:     while (xri>xim && im<nx) {
1095:       im = im+1;
1096:       xim = x[im];
1097:     }
1098:     indx1 = im-1;
1099:     indx2 = im;
1100:     dx1 = xri - x[indx1];
1101:     dx2 = x[indx2] - xri;

1103:     yri = yr[i];
1104:     im = 0;
1105:     yim = y[im];
1106:     while (yri>yim && im<ny) {
1107:       im = im+1;
1108:       yim = y[im];
1109:     }
1110:     indy1 = im-1;
1111:     indy2 = im;
1112:     dy1 = yri - y[indy1];
1113:     dy2 = y[indy2] - yri;

1115:     zri = zr[i];
1116:     im = 0;
1117:     zim = z[im];
1118:     while (zri>zim && im<nz) {
1119:       im = im+1;
1120:       zim = z[im];
1121:     }
1122:     indz1 = im-1;
1123:     indz2 = im;
1124:     dz1 = zri - z[indz1];
1125:     dz2 = z[indz2] - zri;

1127:     Dx = x[indx2] - x[indx1];
1128:     Dy = y[indy2] - y[indy1];
1129:     Dz = z[indz2] - z[indz1];

1131:     j = indx1 + indy1*nx + indz1*nx*ny;
1132:     v = (1-dx1/Dx)*(1-dy1/Dy)*(1-dz1/Dz);
1133:     MatSetValues(user->Qblock,1,&i,1,&j,&v,INSERT_VALUES);

1135:     j = indx1 + indy1*nx + indz2*nx*ny;
1136:     v = (1-dx1/Dx)*(1-dy1/Dy)*(1-dz2/Dz);
1137:     MatSetValues(user->Qblock,1,&i,1,&j,&v,INSERT_VALUES);

1139:     j = indx1 + indy2*nx + indz1*nx*ny;
1140:     v = (1-dx1/Dx)*(1-dy2/Dy)*(1-dz1/Dz);
1141:     MatSetValues(user->Qblock,1,&i,1,&j,&v,INSERT_VALUES);

1143:     j = indx1 + indy2*nx + indz2*nx*ny;
1144:     v = (1-dx1/Dx)*(1-dy2/Dy)*(1-dz2/Dz);
1145:     MatSetValues(user->Qblock,1,&i,1,&j,&v,INSERT_VALUES);

1147:     j = indx2 + indy1*nx + indz1*nx*ny;
1148:     v = (1-dx2/Dx)*(1-dy1/Dy)*(1-dz1/Dz);
1149:     MatSetValues(user->Qblock,1,&i,1,&j,&v,INSERT_VALUES);

1151:     j = indx2 + indy1*nx + indz2*nx*ny;
1152:     v = (1-dx2/Dx)*(1-dy1/Dy)*(1-dz2/Dz);
1153:     MatSetValues(user->Qblock,1,&i,1,&j,&v,INSERT_VALUES);

1155:     j = indx2 + indy2*nx + indz1*nx*ny;
1156:     v = (1-dx2/Dx)*(1-dy2/Dy)*(1-dz1/Dz);
1157:     MatSetValues(user->Qblock,1,&i,1,&j,&v,INSERT_VALUES);

1159:     j = indx2 + indy2*nx + indz2*nx*ny;
1160:     v = (1-dx2/Dx)*(1-dy2/Dy)*(1-dz2/Dz);
1161:     MatSetValues(user->Qblock,1,&i,1,&j,&v,INSERT_VALUES);
1162:   }
1163:   MatAssemblyBegin(user->Qblock,MAT_FINAL_ASSEMBLY);
1164:   MatAssemblyEnd(user->Qblock,MAT_FINAL_ASSEMBLY);

1166:   MatTranspose(user->Qblock,MAT_INITIAL_MATRIX,&user->QblockT);
1167:   MatTranspose(user->L,MAT_INITIAL_MATRIX,&user->LT);

1169:   /* Add noise to the measurement data */
1170:   VecSet(user->ywork,1.0);
1171:   VecAYPX(user->ywork,user->noise,user->ytrue);
1172:   Scatter_i(user->ywork,user->yiwork,user->yi_scatter,user->nt);
1173:   for (j=0; j<user->ns; j++) {
1174:     i = user->sample_times[j];
1175:     MatMult(user->Qblock,user->yiwork[i],user->di[j]);
1176:   }
1177:   Gather_i(user->d,user->di,user->di_scatter,user->ns);

1179:   /* Now that initial conditions have been set, let the user pass tolerance options to the KSP solver */
1180:   KSPSetFromOptions(user->solver);
1181:   PetscFree(x);
1182:   PetscFree(y);
1183:   PetscFree(z);
1184:   return 0;
1185: }

1187: PetscErrorCode ParabolicDestroy(AppCtx *user)
1188: {
1189:   PetscInt       i;

1191:   MatDestroy(&user->Qblock);
1192:   MatDestroy(&user->QblockT);
1193:   MatDestroy(&user->Div);
1194:   MatDestroy(&user->Divwork);
1195:   MatDestroy(&user->Grad);
1196:   MatDestroy(&user->Av);
1197:   MatDestroy(&user->Avwork);
1198:   MatDestroy(&user->AvT);
1199:   MatDestroy(&user->DSG);
1200:   MatDestroy(&user->L);
1201:   MatDestroy(&user->LT);
1202:   KSPDestroy(&user->solver);
1203:   MatDestroy(&user->Js);
1204:   MatDestroy(&user->Jd);
1205:   MatDestroy(&user->JsInv);
1206:   MatDestroy(&user->JsBlock);
1207:   MatDestroy(&user->JsBlockPrec);
1208:   VecDestroy(&user->u);
1209:   VecDestroy(&user->uwork);
1210:   VecDestroy(&user->utrue);
1211:   VecDestroy(&user->y);
1212:   VecDestroy(&user->ywork);
1213:   VecDestroy(&user->ytrue);
1214:   VecDestroyVecs(user->nt,&user->yi);
1215:   VecDestroyVecs(user->nt,&user->yiwork);
1216:   VecDestroyVecs(user->ns,&user->di);
1217:   PetscFree(user->yi);
1218:   PetscFree(user->yiwork);
1219:   PetscFree(user->di);
1220:   VecDestroy(&user->c);
1221:   VecDestroy(&user->cwork);
1222:   VecDestroy(&user->ur);
1223:   VecDestroy(&user->q);
1224:   VecDestroy(&user->d);
1225:   VecDestroy(&user->dwork);
1226:   VecDestroy(&user->lwork);
1227:   VecDestroy(&user->S);
1228:   VecDestroy(&user->Swork);
1229:   VecDestroy(&user->Av_u);
1230:   VecDestroy(&user->Twork);
1231:   VecDestroy(&user->Rwork);
1232:   VecDestroy(&user->js_diag);
1233:   ISDestroy(&user->s_is);
1234:   ISDestroy(&user->d_is);
1235:   VecScatterDestroy(&user->state_scatter);
1236:   VecScatterDestroy(&user->design_scatter);
1237:   for (i=0; i<user->nt; i++) {
1238:     VecScatterDestroy(&user->yi_scatter[i]);
1239:   }
1240:   for (i=0; i<user->ns; i++) {
1241:     VecScatterDestroy(&user->di_scatter[i]);
1242:   }
1243:   PetscFree(user->yi_scatter);
1244:   PetscFree(user->di_scatter);
1245:   PetscFree(user->sample_times);
1246:   return 0;
1247: }

1249: PetscErrorCode ParabolicMonitor(Tao tao, void *ptr)
1250: {
1251:   Vec            X;
1252:   PetscReal      unorm,ynorm;
1253:   AppCtx         *user = (AppCtx*)ptr;

1255:   TaoGetSolution(tao,&X);
1256:   Scatter(X,user->ywork,user->state_scatter,user->uwork,user->design_scatter);
1257:   VecAXPY(user->ywork,-1.0,user->ytrue);
1258:   VecAXPY(user->uwork,-1.0,user->utrue);
1259:   VecNorm(user->uwork,NORM_2,&unorm);
1260:   VecNorm(user->ywork,NORM_2,&ynorm);
1261:   PetscPrintf(MPI_COMM_WORLD, "||u-ut||=%g ||y-yt||=%g\n",(double)unorm,(double)ynorm);
1262:   return 0;
1263: }

1265: /*TEST

1267:    build:
1268:       requires: !complex

1270:    test:
1271:       args: -tao_cmonitor -tao_type lcl -ns 1 -tao_gatol 1.e-4 -ksp_max_it 30
1272:       requires: !single

1274: TEST*/