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:   Vec           x, x0;
 99:   Tao           tao;
100:   AppCtx        user;
101:   IS            is_allstate, is_alldesign;
102:   PetscInt      lo, hi, hi2, lo2, ksp_old;
103:   PetscInt      ntests = 1;
104:   PetscInt      i;
105:   PetscLogStage stages[1];

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

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

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

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

151:   PetscCall(VecGetOwnershipRange(user.y, &lo, &hi));
152:   PetscCall(VecGetOwnershipRange(user.u, &lo2, &hi2));

154:   PetscCall(ISCreateStride(PETSC_COMM_SELF, hi - lo, lo, 1, &is_allstate));
155:   PetscCall(ISCreateStride(PETSC_COMM_SELF, hi - lo, lo + lo2, 1, &user.s_is));

157:   PetscCall(ISCreateStride(PETSC_COMM_SELF, hi2 - lo2, lo2, 1, &is_alldesign));
158:   PetscCall(ISCreateStride(PETSC_COMM_SELF, hi2 - lo2, hi + lo2, 1, &user.d_is));

160:   PetscCall(VecSetSizes(x, hi - lo + hi2 - lo2, user.n));
161:   PetscCall(VecSetFromOptions(x));

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

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

172:   /* Set up initial vectors and matrices */
173:   PetscCall(ParabolicInitialize(&user));

175:   PetscCall(Gather(x, user.y, user.state_scatter, user.u, user.design_scatter));
176:   PetscCall(VecDuplicate(x, &x0));
177:   PetscCall(VecCopy(x, x0));

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

185:   PetscCall(TaoSetJacobianStateRoutine(tao, user.Js, user.JsBlockPrec, user.JsInv, FormJacobianState, &user));
186:   PetscCall(TaoSetJacobianDesignRoutine(tao, user.Jd, FormJacobianDesign, &user));

188:   PetscCall(TaoSetFromOptions(tao));
189:   PetscCall(TaoSetStateDesignIS(tao, user.s_is, user.d_is));

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

211:   PetscCall(TaoDestroy(&tao));
212:   PetscCall(VecDestroy(&x));
213:   PetscCall(VecDestroy(&x0));
214:   PetscCall(ParabolicDestroy(&user));

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

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

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

246:   *f = 0.5 * (d1 + user->alpha * d2);
247:   PetscFunctionReturn(PETSC_SUCCESS);
248: }

250: /* ------------------------------------------------------------------- */
251: /*
252:     state: g_s = Q' *(Qy - d)
253:     design: g_d = alpha*L'*L*(u-ur)
254: */
255: PetscErrorCode FormGradient(Tao tao, Vec X, Vec G, void *ptr)
256: {
257:   PetscInt i, j;
258:   AppCtx  *user = (AppCtx *)ptr;

260:   PetscFunctionBegin;
261:   PetscCall(Scatter(X, user->y, user->state_scatter, user->u, user->design_scatter));
262:   PetscCall(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:     PetscCall(MatMult(user->Qblock, user->yi[i], user->di[j]));
266:   }
267:   PetscCall(Gather_i(user->dwork, user->di, user->di_scatter, user->ns));
268:   PetscCall(VecAXPY(user->dwork, -1.0, user->d));
269:   PetscCall(Scatter_i(user->dwork, user->di, user->di_scatter, user->ns));
270:   PetscCall(VecSet(user->ywork, 0.0));
271:   PetscCall(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:     PetscCall(MatMult(user->QblockT, user->di[j], user->yiwork[i]));
275:   }
276:   PetscCall(Gather_i(user->ywork, user->yiwork, user->yi_scatter, user->nt));

278:   PetscCall(VecWAXPY(user->uwork, -1.0, user->ur, user->u));
279:   PetscCall(MatMult(user->L, user->uwork, user->lwork));
280:   PetscCall(MatMult(user->LT, user->lwork, user->uwork));
281:   PetscCall(VecScale(user->uwork, user->alpha));
282:   PetscCall(Gather(G, user->ywork, user->state_scatter, user->uwork, user->design_scatter));
283:   PetscFunctionReturn(PETSC_SUCCESS);
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:   PetscFunctionBegin;
293:   PetscCall(Scatter(X, user->y, user->state_scatter, user->u, user->design_scatter));
294:   PetscCall(Scatter_i(user->y, user->yi, user->yi_scatter, user->nt));
295:   for (j = 0; j < user->ns; j++) {
296:     i = user->sample_times[j];
297:     PetscCall(MatMult(user->Qblock, user->yi[i], user->di[j]));
298:   }
299:   PetscCall(Gather_i(user->dwork, user->di, user->di_scatter, user->ns));
300:   PetscCall(VecAXPY(user->dwork, -1.0, user->d));
301:   PetscCall(VecDot(user->dwork, user->dwork, &d1));
302:   PetscCall(Scatter_i(user->dwork, user->di, user->di_scatter, user->ns));
303:   PetscCall(VecSet(user->ywork, 0.0));
304:   PetscCall(Scatter_i(user->ywork, user->yiwork, user->yi_scatter, user->nt));
305:   for (j = 0; j < user->ns; j++) {
306:     i = user->sample_times[j];
307:     PetscCall(MatMult(user->QblockT, user->di[j], user->yiwork[i]));
308:   }
309:   PetscCall(Gather_i(user->ywork, user->yiwork, user->yi_scatter, user->nt));

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

318:   PetscCall(Gather(G, user->ywork, user->state_scatter, user->uwork, user->design_scatter));
319:   PetscFunctionReturn(PETSC_SUCCESS);
320: }

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

330:   PetscFunctionBegin;
331:   PetscCall(Scatter(X, user->y, user->state_scatter, user->u, user->design_scatter));
332:   PetscCall(VecSet(user->uwork, 0));
333:   PetscCall(VecAXPY(user->uwork, -1.0, user->u));
334:   PetscCall(VecExp(user->uwork));
335:   PetscCall(MatMult(user->Av, user->uwork, user->Av_u));
336:   PetscCall(VecCopy(user->Av_u, user->Swork));
337:   PetscCall(VecReciprocal(user->Swork));
338:   PetscCall(MatCopy(user->Div, user->Divwork, SAME_NONZERO_PATTERN));
339:   PetscCall(MatDiagonalScale(user->Divwork, NULL, user->Swork));
340:   if (user->dsg_formed) {
341:     PetscCall(MatProductNumeric(user->DSG));
342:   } else {
343:     PetscCall(MatMatMult(user->Divwork, user->Grad, MAT_INITIAL_MATRIX, PETSC_DETERMINE, &user->DSG));
344:     user->dsg_formed = PETSC_TRUE;
345:   }

347:   /* B = speye(nx^3) + ht*DSG; */
348:   PetscCall(MatScale(user->DSG, user->ht));
349:   PetscCall(MatShift(user->DSG, 1.0));
350:   PetscFunctionReturn(PETSC_SUCCESS);
351: }

353: /* ------------------------------------------------------------------- */
354: /* B */
355: PetscErrorCode FormJacobianDesign(Tao tao, Vec X, Mat J, void *ptr)
356: {
357:   AppCtx *user = (AppCtx *)ptr;

359:   PetscFunctionBegin;
360:   PetscCall(Scatter(X, user->y, user->state_scatter, user->u, user->design_scatter));
361:   PetscFunctionReturn(PETSC_SUCCESS);
362: }

364: PetscErrorCode StateMatMult(Mat J_shell, Vec X, Vec Y)
365: {
366:   PetscInt i;
367:   AppCtx  *user;

369:   PetscFunctionBegin;
370:   PetscCall(MatShellGetContext(J_shell, &user));
371:   PetscCall(Scatter_i(X, user->yi, user->yi_scatter, user->nt));
372:   PetscCall(MatMult(user->JsBlock, user->yi[0], user->yiwork[0]));
373:   for (i = 1; i < user->nt; i++) {
374:     PetscCall(MatMult(user->JsBlock, user->yi[i], user->yiwork[i]));
375:     PetscCall(VecAXPY(user->yiwork[i], -1.0, user->yi[i - 1]));
376:   }
377:   PetscCall(Gather_i(Y, user->yiwork, user->yi_scatter, user->nt));
378:   PetscFunctionReturn(PETSC_SUCCESS);
379: }

381: PetscErrorCode StateMatMultTranspose(Mat J_shell, Vec X, Vec Y)
382: {
383:   PetscInt i;
384:   AppCtx  *user;

386:   PetscFunctionBegin;
387:   PetscCall(MatShellGetContext(J_shell, &user));
388:   PetscCall(Scatter_i(X, user->yi, user->yi_scatter, user->nt));
389:   for (i = 0; i < user->nt - 1; i++) {
390:     PetscCall(MatMult(user->JsBlock, user->yi[i], user->yiwork[i]));
391:     PetscCall(VecAXPY(user->yiwork[i], -1.0, user->yi[i + 1]));
392:   }
393:   i = user->nt - 1;
394:   PetscCall(MatMult(user->JsBlock, user->yi[i], user->yiwork[i]));
395:   PetscCall(Gather_i(Y, user->yiwork, user->yi_scatter, user->nt));
396:   PetscFunctionReturn(PETSC_SUCCESS);
397: }

399: PetscErrorCode StateMatBlockMult(Mat J_shell, Vec X, Vec Y)
400: {
401:   AppCtx *user;

403:   PetscFunctionBegin;
404:   PetscCall(MatShellGetContext(J_shell, &user));
405:   PetscCall(MatMult(user->Grad, X, user->Swork));
406:   PetscCall(VecPointwiseDivide(user->Swork, user->Swork, user->Av_u));
407:   PetscCall(MatMult(user->Div, user->Swork, Y));
408:   PetscCall(VecAYPX(Y, user->ht, X));
409:   PetscFunctionReturn(PETSC_SUCCESS);
410: }

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

417:   PetscFunctionBegin;
418:   PetscCall(MatShellGetContext(J_shell, &user));

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

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

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

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

437:   PetscCall(Scatter_i(user->y, user->yi, user->yi_scatter, user->nt));
438:   for (i = 0; i < user->nt; i++) {
439:     /* (sdiag(Grad*y(:,i)) */
440:     PetscCall(MatMult(user->Grad, user->yi[i], user->Twork));

442:     /* ht * Div * (sdiag(Grad*y(:,i)) * (sdiag(1./((Av*(1./v)).^2)) * (Av * (sdiag(1./v) * b)))) */
443:     PetscCall(VecPointwiseMult(user->Twork, user->Twork, user->Swork));
444:     PetscCall(MatMult(user->Div, user->Twork, user->yiwork[i]));
445:     PetscCall(VecScale(user->yiwork[i], user->ht));
446:   }
447:   PetscCall(Gather_i(Y, user->yiwork, user->yi_scatter, user->nt));
448:   PetscFunctionReturn(PETSC_SUCCESS);
449: }

451: PetscErrorCode DesignMatMultTranspose(Mat J_shell, Vec X, Vec Y)
452: {
453:   PetscInt i;
454:   AppCtx  *user;

456:   PetscFunctionBegin;
457:   PetscCall(MatShellGetContext(J_shell, &user));

459:   /* sdiag(1./((Av*(1./v)).^2)) */
460:   PetscCall(VecSet(user->uwork, 0));
461:   PetscCall(VecAXPY(user->uwork, -1.0, user->u));
462:   PetscCall(VecExp(user->uwork));
463:   PetscCall(MatMult(user->Av, user->uwork, user->Rwork));
464:   PetscCall(VecPointwiseMult(user->Rwork, user->Rwork, user->Rwork));
465:   PetscCall(VecReciprocal(user->Rwork));

467:   PetscCall(VecSet(Y, 0.0));
468:   PetscCall(Scatter_i(user->y, user->yi, user->yi_scatter, user->nt));
469:   PetscCall(Scatter_i(X, user->yiwork, user->yi_scatter, user->nt));
470:   for (i = 0; i < user->nt; i++) {
471:     /* (Div' * b(:,i)) */
472:     PetscCall(MatMult(user->Grad, user->yiwork[i], user->Swork));

474:     /* sdiag(Grad*y(:,i)) */
475:     PetscCall(MatMult(user->Grad, user->yi[i], user->Twork));

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

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

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

486:     /* sdiag(1./v) * (Av' * (sdiag(1./((Av*(1./v)).^2)) * (sdiag(Grad*y(:,i)) * (Div' * b(:,i))))) */
487:     PetscCall(VecPointwiseMult(user->yiwork[i], user->uwork, user->yiwork[i]));
488:     PetscCall(VecAXPY(Y, user->ht, user->yiwork[i]));
489:   }
490:   PetscFunctionReturn(PETSC_SUCCESS);
491: }

493: PetscErrorCode StateMatBlockPrecMult(PC PC_shell, Vec X, Vec Y)
494: {
495:   AppCtx *user;

497:   PetscFunctionBegin;
498:   PetscCall(PCShellGetContext(PC_shell, &user));

500:   if (user->dsg_formed) {
501:     PetscCall(MatSOR(user->DSG, X, 1.0, (MatSORType)(SOR_ZERO_INITIAL_GUESS | SOR_LOCAL_SYMMETRIC_SWEEP), 0.0, 1, 1, Y));
502:   } else SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "DSG not formed");
503:   PetscFunctionReturn(PETSC_SUCCESS);
504: }

506: PetscErrorCode StateMatInvMult(Mat J_shell, Vec X, Vec Y)
507: {
508:   AppCtx  *user;
509:   PetscInt its, i;

511:   PetscFunctionBegin;
512:   PetscCall(MatShellGetContext(J_shell, &user));

514:   if (Y == user->ytrue) {
515:     /* First solve is done with true solution to set up problem */
516:     PetscCall(KSPSetTolerances(user->solver, 1e-8, PETSC_CURRENT, PETSC_CURRENT, PETSC_CURRENT));
517:   } else {
518:     PetscCall(KSPSetTolerances(user->solver, PETSC_CURRENT, PETSC_CURRENT, PETSC_CURRENT, PETSC_CURRENT));
519:   }

521:   PetscCall(Scatter_i(X, user->yi, user->yi_scatter, user->nt));
522:   PetscCall(KSPSolve(user->solver, user->yi[0], user->yiwork[0]));
523:   PetscCall(KSPGetIterationNumber(user->solver, &its));
524:   user->ksp_its = user->ksp_its + its;

526:   for (i = 1; i < user->nt; i++) {
527:     PetscCall(VecAXPY(user->yi[i], 1.0, user->yiwork[i - 1]));
528:     PetscCall(KSPSolve(user->solver, user->yi[i], user->yiwork[i]));
529:     PetscCall(KSPGetIterationNumber(user->solver, &its));
530:     user->ksp_its = user->ksp_its + its;
531:   }
532:   PetscCall(Gather_i(Y, user->yiwork, user->yi_scatter, user->nt));
533:   PetscFunctionReturn(PETSC_SUCCESS);
534: }

536: PetscErrorCode StateMatInvTransposeMult(Mat J_shell, Vec X, Vec Y)
537: {
538:   AppCtx  *user;
539:   PetscInt its, i;

541:   PetscFunctionBegin;
542:   PetscCall(MatShellGetContext(J_shell, &user));

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

546:   i = user->nt - 1;
547:   PetscCall(KSPSolve(user->solver, user->yi[i], user->yiwork[i]));

549:   PetscCall(KSPGetIterationNumber(user->solver, &its));
550:   user->ksp_its = user->ksp_its + its;

552:   for (i = user->nt - 2; i >= 0; i--) {
553:     PetscCall(VecAXPY(user->yi[i], 1.0, user->yiwork[i + 1]));
554:     PetscCall(KSPSolve(user->solver, user->yi[i], user->yiwork[i]));

556:     PetscCall(KSPGetIterationNumber(user->solver, &its));
557:     user->ksp_its = user->ksp_its + its;
558:   }

560:   PetscCall(Gather_i(Y, user->yiwork, user->yi_scatter, user->nt));
561:   PetscFunctionReturn(PETSC_SUCCESS);
562: }

564: PetscErrorCode StateMatDuplicate(Mat J_shell, MatDuplicateOption opt, Mat *new_shell)
565: {
566:   AppCtx *user;

568:   PetscFunctionBegin;
569:   PetscCall(MatShellGetContext(J_shell, &user));

571:   PetscCall(MatCreateShell(PETSC_COMM_WORLD, PETSC_DETERMINE, PETSC_DETERMINE, user->m, user->m, user, new_shell));
572:   PetscCall(MatShellSetOperation(*new_shell, MATOP_MULT, (void (*)(void))StateMatMult));
573:   PetscCall(MatShellSetOperation(*new_shell, MATOP_DUPLICATE, (void (*)(void))StateMatDuplicate));
574:   PetscCall(MatShellSetOperation(*new_shell, MATOP_MULT_TRANSPOSE, (void (*)(void))StateMatMultTranspose));
575:   PetscCall(MatShellSetOperation(*new_shell, MATOP_GET_DIAGONAL, (void (*)(void))StateMatGetDiagonal));
576:   PetscFunctionReturn(PETSC_SUCCESS);
577: }

579: PetscErrorCode StateMatGetDiagonal(Mat J_shell, Vec X)
580: {
581:   AppCtx *user;

583:   PetscFunctionBegin;
584:   PetscCall(MatShellGetContext(J_shell, &user));
585:   PetscCall(VecCopy(user->js_diag, X));
586:   PetscFunctionReturn(PETSC_SUCCESS);
587: }

589: PetscErrorCode FormConstraints(Tao tao, Vec X, Vec C, void *ptr)
590: {
591:   /* con = Ay - q, A = [B  0  0 ... 0;
592:                        -I  B  0 ... 0;
593:                         0 -I  B ... 0;
594:                              ...     ;
595:                         0    ... -I B]
596:      B = ht * Div * Sigma * Grad + eye */
597:   PetscInt i;
598:   AppCtx  *user = (AppCtx *)ptr;

600:   PetscFunctionBegin;
601:   PetscCall(Scatter(X, user->y, user->state_scatter, user->u, user->design_scatter));
602:   PetscCall(Scatter_i(user->y, user->yi, user->yi_scatter, user->nt));
603:   PetscCall(MatMult(user->JsBlock, user->yi[0], user->yiwork[0]));
604:   for (i = 1; i < user->nt; i++) {
605:     PetscCall(MatMult(user->JsBlock, user->yi[i], user->yiwork[i]));
606:     PetscCall(VecAXPY(user->yiwork[i], -1.0, user->yi[i - 1]));
607:   }
608:   PetscCall(Gather_i(C, user->yiwork, user->yi_scatter, user->nt));
609:   PetscCall(VecAXPY(C, -1.0, user->q));
610:   PetscFunctionReturn(PETSC_SUCCESS);
611: }

613: PetscErrorCode Scatter(Vec x, Vec state, VecScatter s_scat, Vec design, VecScatter d_scat)
614: {
615:   PetscFunctionBegin;
616:   PetscCall(VecScatterBegin(s_scat, x, state, INSERT_VALUES, SCATTER_FORWARD));
617:   PetscCall(VecScatterEnd(s_scat, x, state, INSERT_VALUES, SCATTER_FORWARD));
618:   PetscCall(VecScatterBegin(d_scat, x, design, INSERT_VALUES, SCATTER_FORWARD));
619:   PetscCall(VecScatterEnd(d_scat, x, design, INSERT_VALUES, SCATTER_FORWARD));
620:   PetscFunctionReturn(PETSC_SUCCESS);
621: }

623: PetscErrorCode Scatter_i(Vec y, Vec *yi, VecScatter *scat, PetscInt nt)
624: {
625:   PetscInt i;

627:   PetscFunctionBegin;
628:   for (i = 0; i < nt; i++) {
629:     PetscCall(VecScatterBegin(scat[i], y, yi[i], INSERT_VALUES, SCATTER_FORWARD));
630:     PetscCall(VecScatterEnd(scat[i], y, yi[i], INSERT_VALUES, SCATTER_FORWARD));
631:   }
632:   PetscFunctionReturn(PETSC_SUCCESS);
633: }

635: PetscErrorCode Gather(Vec x, Vec state, VecScatter s_scat, Vec design, VecScatter d_scat)
636: {
637:   PetscFunctionBegin;
638:   PetscCall(VecScatterBegin(s_scat, state, x, INSERT_VALUES, SCATTER_REVERSE));
639:   PetscCall(VecScatterEnd(s_scat, state, x, INSERT_VALUES, SCATTER_REVERSE));
640:   PetscCall(VecScatterBegin(d_scat, design, x, INSERT_VALUES, SCATTER_REVERSE));
641:   PetscCall(VecScatterEnd(d_scat, design, x, INSERT_VALUES, SCATTER_REVERSE));
642:   PetscFunctionReturn(PETSC_SUCCESS);
643: }

645: PetscErrorCode Gather_i(Vec y, Vec *yi, VecScatter *scat, PetscInt nt)
646: {
647:   PetscInt i;

649:   PetscFunctionBegin;
650:   for (i = 0; i < nt; i++) {
651:     PetscCall(VecScatterBegin(scat[i], yi[i], y, INSERT_VALUES, SCATTER_REVERSE));
652:     PetscCall(VecScatterEnd(scat[i], yi[i], y, INSERT_VALUES, SCATTER_REVERSE));
653:   }
654:   PetscFunctionReturn(PETSC_SUCCESS);
655: }

657: PetscErrorCode ParabolicInitialize(AppCtx *user)
658: {
659:   PetscInt    m, n, i, j, k, linear_index, istart, iend, iblock, lo, hi, lo2, hi2;
660:   Vec         XX, YY, ZZ, XXwork, YYwork, ZZwork, UTwork, yi, di, bc;
661:   PetscReal  *x, *y, *z;
662:   PetscReal   h, stime;
663:   PetscScalar hinv, neg_hinv, half = 0.5, sqrt_beta;
664:   PetscInt    im, indx1, indx2, indy1, indy2, indz1, indz2, nx, ny, nz;
665:   PetscReal   xri, yri, zri, xim, yim, zim, dx1, dx2, dy1, dy2, dz1, dz2, Dx, Dy, Dz;
666:   PetscScalar v, vx, vy, vz;
667:   IS          is_from_y, is_to_yi, is_from_d, is_to_di;
668:   /* Data locations */
669:   PetscScalar xr[64] = {0.4970, 0.8498, 0.7814, 0.6268, 0.7782, 0.6402, 0.3617, 0.3160, 0.3610, 0.5298, 0.6987, 0.3331, 0.7962, 0.5596, 0.3866, 0.6774, 0.5407, 0.4518, 0.6702, 0.6061, 0.7580, 0.8997,
670:                         0.5198, 0.8326, 0.2138, 0.9198, 0.3000, 0.2833, 0.8288, 0.7076, 0.1820, 0.0728, 0.8447, 0.2367, 0.3239, 0.6413, 0.3114, 0.4731, 0.1192, 0.9273, 0.5724, 0.4331, 0.5136, 0.3547,
671:                         0.4413, 0.2602, 0.5698, 0.7278, 0.5261, 0.6230, 0.2454, 0.3948, 0.7479, 0.6582, 0.4660, 0.5594, 0.7574, 0.1143, 0.5900, 0.1065, 0.4260, 0.3294, 0.8276, 0.0756};

673:   PetscScalar yr[64] = {0.7345, 0.9120, 0.9288, 0.7528, 0.4463, 0.4985, 0.2497, 0.6256, 0.3425, 0.9026, 0.6983, 0.4230, 0.7140, 0.2970, 0.4474, 0.8792, 0.6604, 0.2485, 0.7968, 0.6127, 0.1796, 0.2437,
674:                         0.5938, 0.6137, 0.3867, 0.5658, 0.4575, 0.1009, 0.0863, 0.3361, 0.0738, 0.3985, 0.6602, 0.1437, 0.0934, 0.5983, 0.5950, 0.0763, 0.0768, 0.2288, 0.5761, 0.1129, 0.3841, 0.6150,
675:                         0.6904, 0.6686, 0.1361, 0.4601, 0.4491, 0.3716, 0.1969, 0.6537, 0.6743, 0.6991, 0.4811, 0.5480, 0.1684, 0.4569, 0.6889, 0.8437, 0.3015, 0.2854, 0.8199, 0.2658};

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

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

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

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

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

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

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

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

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

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

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

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

756:   PetscCall(VecPointwiseMult(XXwork, XXwork, XXwork));
757:   PetscCall(VecPointwiseMult(YYwork, YYwork, YYwork));
758:   PetscCall(VecPointwiseMult(ZZwork, ZZwork, ZZwork));

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

1011:   /* Compute true state function yt given ut */
1012:   PetscCall(VecCreate(PETSC_COMM_WORLD, &user->ytrue));
1013:   PetscCall(VecSetSizes(user->ytrue, PETSC_DECIDE, n * user->nt));
1014:   PetscCall(VecSetFromOptions(user->ytrue));

1016:   /* First compute Av_u = Av*exp(-u) */
1017:   PetscCall(VecSet(user->uwork, 0));
1018:   PetscCall(VecAXPY(user->uwork, -1.0, user->utrue)); /*  Note: user->utrue */
1019:   PetscCall(VecExp(user->uwork));
1020:   PetscCall(MatMult(user->Av, user->uwork, user->Av_u));

1022:   /* Symbolic DSG = Div * Grad */
1023:   PetscCall(MatProductCreate(user->Div, user->Grad, NULL, &user->DSG));
1024:   PetscCall(MatProductSetType(user->DSG, MATPRODUCT_AB));
1025:   PetscCall(MatProductSetAlgorithm(user->DSG, "default"));
1026:   PetscCall(MatProductSetFill(user->DSG, PETSC_DETERMINE));
1027:   PetscCall(MatProductSetFromOptions(user->DSG));
1028:   PetscCall(MatProductSymbolic(user->DSG));

1030:   user->dsg_formed = PETSC_TRUE;

1032:   /* Next form DSG = Div*Grad */
1033:   PetscCall(MatCopy(user->Div, user->Divwork, SAME_NONZERO_PATTERN));
1034:   PetscCall(MatDiagonalScale(user->Divwork, NULL, user->Av_u));
1035:   if (user->dsg_formed) {
1036:     PetscCall(MatProductNumeric(user->DSG));
1037:   } else {
1038:     PetscCall(MatMatMult(user->Div, user->Grad, MAT_INITIAL_MATRIX, PETSC_DETERMINE, &user->DSG));
1039:     user->dsg_formed = PETSC_TRUE;
1040:   }
1041:   /* B = speye(nx^3) + ht*DSG; */
1042:   PetscCall(MatScale(user->DSG, user->ht));
1043:   PetscCall(MatShift(user->DSG, 1.0));

1045:   /* Now solve for ytrue */
1046:   PetscCall(StateMatInvMult(user->Js, user->q, user->ytrue));

1048:   /* Initial guess y0 for state given u0 */

1050:   /* First compute Av_u = Av*exp(-u) */
1051:   PetscCall(VecSet(user->uwork, 0));
1052:   PetscCall(VecAXPY(user->uwork, -1.0, user->u)); /*  Note: user->u */
1053:   PetscCall(VecExp(user->uwork));
1054:   PetscCall(MatMult(user->Av, user->uwork, user->Av_u));

1056:   /* Next form DSG = Div*S*Grad */
1057:   PetscCall(MatCopy(user->Div, user->Divwork, SAME_NONZERO_PATTERN));
1058:   PetscCall(MatDiagonalScale(user->Divwork, NULL, user->Av_u));
1059:   if (user->dsg_formed) {
1060:     PetscCall(MatProductNumeric(user->DSG));
1061:   } else {
1062:     PetscCall(MatMatMult(user->Div, user->Grad, MAT_INITIAL_MATRIX, PETSC_DETERMINE, &user->DSG));

1064:     user->dsg_formed = PETSC_TRUE;
1065:   }
1066:   /* B = speye(nx^3) + ht*DSG; */
1067:   PetscCall(MatScale(user->DSG, user->ht));
1068:   PetscCall(MatShift(user->DSG, 1.0));

1070:   /* Now solve for y */
1071:   PetscCall(StateMatInvMult(user->Js, user->q, user->y));

1073:   /* Construct projection matrix Q, a block diagonal matrix consisting of nt copies of Qblock along the diagonal */
1074:   PetscCall(MatCreate(PETSC_COMM_WORLD, &user->Qblock));
1075:   PetscCall(MatSetSizes(user->Qblock, PETSC_DECIDE, PETSC_DECIDE, user->ndata, n));
1076:   PetscCall(MatSetFromOptions(user->Qblock));
1077:   PetscCall(MatMPIAIJSetPreallocation(user->Qblock, 8, NULL, 8, NULL));
1078:   PetscCall(MatSeqAIJSetPreallocation(user->Qblock, 8, NULL));

1080:   for (i = 0; i < user->mx; i++) {
1081:     x[i] = h * (i + 0.5);
1082:     y[i] = h * (i + 0.5);
1083:     z[i] = h * (i + 0.5);
1084:   }

1086:   PetscCall(MatGetOwnershipRange(user->Qblock, &istart, &iend));
1087:   nx = user->mx;
1088:   ny = user->mx;
1089:   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:     PetscCall(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:     PetscCall(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:     PetscCall(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:     PetscCall(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:     PetscCall(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:     PetscCall(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:     PetscCall(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:     PetscCall(MatSetValues(user->Qblock, 1, &i, 1, &j, &v, INSERT_VALUES));
1162:   }
1163:   PetscCall(MatAssemblyBegin(user->Qblock, MAT_FINAL_ASSEMBLY));
1164:   PetscCall(MatAssemblyEnd(user->Qblock, MAT_FINAL_ASSEMBLY));

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

1169:   /* Add noise to the measurement data */
1170:   PetscCall(VecSet(user->ywork, 1.0));
1171:   PetscCall(VecAYPX(user->ywork, user->noise, user->ytrue));
1172:   PetscCall(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:     PetscCall(MatMult(user->Qblock, user->yiwork[i], user->di[j]));
1176:   }
1177:   PetscCall(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:   PetscCall(KSPSetFromOptions(user->solver));
1181:   PetscCall(PetscFree(x));
1182:   PetscCall(PetscFree(y));
1183:   PetscCall(PetscFree(z));
1184:   PetscFunctionReturn(PETSC_SUCCESS);
1185: }

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

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

1246: PetscErrorCode ParabolicMonitor(Tao tao, void *ptr)
1247: {
1248:   Vec       X;
1249:   PetscReal unorm, ynorm;
1250:   AppCtx   *user = (AppCtx *)ptr;

1252:   PetscFunctionBegin;
1253:   PetscCall(TaoGetSolution(tao, &X));
1254:   PetscCall(Scatter(X, user->ywork, user->state_scatter, user->uwork, user->design_scatter));
1255:   PetscCall(VecAXPY(user->ywork, -1.0, user->ytrue));
1256:   PetscCall(VecAXPY(user->uwork, -1.0, user->utrue));
1257:   PetscCall(VecNorm(user->uwork, NORM_2, &unorm));
1258:   PetscCall(VecNorm(user->ywork, NORM_2, &ynorm));
1259:   PetscCall(PetscPrintf(MPI_COMM_WORLD, "||u-ut||=%g ||y-yt||=%g\n", (double)unorm, (double)ynorm));
1260:   PetscFunctionReturn(PETSC_SUCCESS);
1261: }

1263: /*TEST

1265:    build:
1266:       requires: !complex

1268:    test:
1269:       args: -tao_monitor_constraint_norm -tao_type lcl -ns 1 -tao_gatol 1.e-4 -ksp_max_it 30
1270:       requires: !single

1272: TEST*/