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: #if defined(PETSC_USE_LOG)
106:   PetscLogStage stages[1];
107: #endif

109:   PetscFunctionBeginUser;
110:   PetscCall(PetscInitialize(&argc, &argv, (char *)0, help));
111:   user.mx = 8;
112:   PetscOptionsBegin(PETSC_COMM_WORLD, NULL, "parabolic example", NULL);
113:   PetscCall(PetscOptionsInt("-mx", "Number of grid points in each direction", "", user.mx, &user.mx, NULL));
114:   user.nt = 8;
115:   PetscCall(PetscOptionsInt("-nt", "Number of time steps", "", user.nt, &user.nt, NULL));
116:   user.ndata = 64;
117:   PetscCall(PetscOptionsInt("-ndata", "Numbers of data points per sample", "", user.ndata, &user.ndata, NULL));
118:   user.ns = 8;
119:   PetscCall(PetscOptionsInt("-ns", "Number of samples", "", user.ns, &user.ns, NULL));
120:   user.alpha = 1.0;
121:   PetscCall(PetscOptionsReal("-alpha", "Regularization parameter", "", user.alpha, &user.alpha, NULL));
122:   user.beta = 0.01;
123:   PetscCall(PetscOptionsReal("-beta", "Weight attributed to ||u||^2 in regularization functional", "", user.beta, &user.beta, NULL));
124:   user.noise = 0.01;
125:   PetscCall(PetscOptionsReal("-noise", "Amount of noise to add to data", "", user.noise, &user.noise, NULL));
126:   PetscCall(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:   PetscCall(VecCreate(PETSC_COMM_WORLD, &user.u));
134:   PetscCall(VecCreate(PETSC_COMM_WORLD, &user.y));
135:   PetscCall(VecCreate(PETSC_COMM_WORLD, &user.c));
136:   PetscCall(VecSetSizes(user.u, PETSC_DECIDE, user.n - user.m * user.nt));
137:   PetscCall(VecSetSizes(user.y, PETSC_DECIDE, user.m * user.nt));
138:   PetscCall(VecSetSizes(user.c, PETSC_DECIDE, user.m * user.nt));
139:   PetscCall(VecSetFromOptions(user.u));
140:   PetscCall(VecSetFromOptions(user.y));
141:   PetscCall(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:   PetscCall(VecCreate(PETSC_COMM_WORLD, &x));

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

248:   *f = 0.5 * (d1 + user->alpha * d2);
249:   PetscFunctionReturn(PETSC_SUCCESS);
250: }

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

262:   PetscFunctionBegin;
263:   PetscCall(Scatter(X, user->y, user->state_scatter, user->u, user->design_scatter));
264:   PetscCall(Scatter_i(user->y, user->yi, user->yi_scatter, user->nt));
265:   for (j = 0; j < user->ns; j++) {
266:     i = user->sample_times[j];
267:     PetscCall(MatMult(user->Qblock, user->yi[i], user->di[j]));
268:   }
269:   PetscCall(Gather_i(user->dwork, user->di, user->di_scatter, user->ns));
270:   PetscCall(VecAXPY(user->dwork, -1.0, user->d));
271:   PetscCall(Scatter_i(user->dwork, user->di, user->di_scatter, user->ns));
272:   PetscCall(VecSet(user->ywork, 0.0));
273:   PetscCall(Scatter_i(user->ywork, user->yiwork, user->yi_scatter, user->nt));
274:   for (j = 0; j < user->ns; j++) {
275:     i = user->sample_times[j];
276:     PetscCall(MatMult(user->QblockT, user->di[j], user->yiwork[i]));
277:   }
278:   PetscCall(Gather_i(user->ywork, user->yiwork, user->yi_scatter, user->nt));

280:   PetscCall(VecWAXPY(user->uwork, -1.0, user->ur, user->u));
281:   PetscCall(MatMult(user->L, user->uwork, user->lwork));
282:   PetscCall(MatMult(user->LT, user->lwork, user->uwork));
283:   PetscCall(VecScale(user->uwork, user->alpha));
284:   PetscCall(Gather(G, user->ywork, user->state_scatter, user->uwork, user->design_scatter));
285:   PetscFunctionReturn(PETSC_SUCCESS);
286: }

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

294:   PetscFunctionBegin;
295:   PetscCall(Scatter(X, user->y, user->state_scatter, user->u, user->design_scatter));
296:   PetscCall(Scatter_i(user->y, user->yi, user->yi_scatter, user->nt));
297:   for (j = 0; j < user->ns; j++) {
298:     i = user->sample_times[j];
299:     PetscCall(MatMult(user->Qblock, user->yi[i], user->di[j]));
300:   }
301:   PetscCall(Gather_i(user->dwork, user->di, user->di_scatter, user->ns));
302:   PetscCall(VecAXPY(user->dwork, -1.0, user->d));
303:   PetscCall(VecDot(user->dwork, user->dwork, &d1));
304:   PetscCall(Scatter_i(user->dwork, user->di, user->di_scatter, user->ns));
305:   PetscCall(VecSet(user->ywork, 0.0));
306:   PetscCall(Scatter_i(user->ywork, user->yiwork, user->yi_scatter, user->nt));
307:   for (j = 0; j < user->ns; j++) {
308:     i = user->sample_times[j];
309:     PetscCall(MatMult(user->QblockT, user->di[j], user->yiwork[i]));
310:   }
311:   PetscCall(Gather_i(user->ywork, user->yiwork, user->yi_scatter, user->nt));

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

320:   PetscCall(Gather(G, user->ywork, user->state_scatter, user->uwork, user->design_scatter));
321:   PetscFunctionReturn(PETSC_SUCCESS);
322: }

324: /* ------------------------------------------------------------------- */
325: /* A
326: MatShell object
327: */
328: PetscErrorCode FormJacobianState(Tao tao, Vec X, Mat J, Mat JPre, Mat JInv, void *ptr)
329: {
330:   AppCtx *user = (AppCtx *)ptr;

332:   PetscFunctionBegin;
333:   PetscCall(Scatter(X, user->y, user->state_scatter, user->u, user->design_scatter));
334:   PetscCall(VecSet(user->uwork, 0));
335:   PetscCall(VecAXPY(user->uwork, -1.0, user->u));
336:   PetscCall(VecExp(user->uwork));
337:   PetscCall(MatMult(user->Av, user->uwork, user->Av_u));
338:   PetscCall(VecCopy(user->Av_u, user->Swork));
339:   PetscCall(VecReciprocal(user->Swork));
340:   PetscCall(MatCopy(user->Div, user->Divwork, SAME_NONZERO_PATTERN));
341:   PetscCall(MatDiagonalScale(user->Divwork, NULL, user->Swork));
342:   if (user->dsg_formed) {
343:     PetscCall(MatProductNumeric(user->DSG));
344:   } else {
345:     PetscCall(MatMatMult(user->Divwork, user->Grad, MAT_INITIAL_MATRIX, PETSC_DEFAULT, &user->DSG));
346:     user->dsg_formed = PETSC_TRUE;
347:   }

349:   /* B = speye(nx^3) + ht*DSG; */
350:   PetscCall(MatScale(user->DSG, user->ht));
351:   PetscCall(MatShift(user->DSG, 1.0));
352:   PetscFunctionReturn(PETSC_SUCCESS);
353: }

355: /* ------------------------------------------------------------------- */
356: /* B */
357: PetscErrorCode FormJacobianDesign(Tao tao, Vec X, Mat J, void *ptr)
358: {
359:   AppCtx *user = (AppCtx *)ptr;

361:   PetscFunctionBegin;
362:   PetscCall(Scatter(X, user->y, user->state_scatter, user->u, user->design_scatter));
363:   PetscFunctionReturn(PETSC_SUCCESS);
364: }

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

371:   PetscFunctionBegin;
372:   PetscCall(MatShellGetContext(J_shell, &user));
373:   PetscCall(Scatter_i(X, user->yi, user->yi_scatter, user->nt));
374:   PetscCall(MatMult(user->JsBlock, user->yi[0], user->yiwork[0]));
375:   for (i = 1; i < user->nt; i++) {
376:     PetscCall(MatMult(user->JsBlock, user->yi[i], user->yiwork[i]));
377:     PetscCall(VecAXPY(user->yiwork[i], -1.0, user->yi[i - 1]));
378:   }
379:   PetscCall(Gather_i(Y, user->yiwork, user->yi_scatter, user->nt));
380:   PetscFunctionReturn(PETSC_SUCCESS);
381: }

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

388:   PetscFunctionBegin;
389:   PetscCall(MatShellGetContext(J_shell, &user));
390:   PetscCall(Scatter_i(X, user->yi, user->yi_scatter, user->nt));
391:   for (i = 0; i < user->nt - 1; i++) {
392:     PetscCall(MatMult(user->JsBlock, user->yi[i], user->yiwork[i]));
393:     PetscCall(VecAXPY(user->yiwork[i], -1.0, user->yi[i + 1]));
394:   }
395:   i = user->nt - 1;
396:   PetscCall(MatMult(user->JsBlock, user->yi[i], user->yiwork[i]));
397:   PetscCall(Gather_i(Y, user->yiwork, user->yi_scatter, user->nt));
398:   PetscFunctionReturn(PETSC_SUCCESS);
399: }

401: PetscErrorCode StateMatBlockMult(Mat J_shell, Vec X, Vec Y)
402: {
403:   AppCtx *user;

405:   PetscFunctionBegin;
406:   PetscCall(MatShellGetContext(J_shell, &user));
407:   PetscCall(MatMult(user->Grad, X, user->Swork));
408:   PetscCall(VecPointwiseDivide(user->Swork, user->Swork, user->Av_u));
409:   PetscCall(MatMult(user->Div, user->Swork, Y));
410:   PetscCall(VecAYPX(Y, user->ht, X));
411:   PetscFunctionReturn(PETSC_SUCCESS);
412: }

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

419:   PetscFunctionBegin;
420:   PetscCall(MatShellGetContext(J_shell, &user));

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

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

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

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

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

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

451:   PetscFunctionReturn(PETSC_SUCCESS);
452: }

454: PetscErrorCode DesignMatMultTranspose(Mat J_shell, Vec X, Vec Y)
455: {
456:   PetscInt i;
457:   AppCtx  *user;

459:   PetscFunctionBegin;
460:   PetscCall(MatShellGetContext(J_shell, &user));

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

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

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

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

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

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

489:     /* sdiag(1./v) * (Av' * (sdiag(1./((Av*(1./v)).^2)) * (sdiag(Grad*y(:,i)) * (Div' * b(:,i))))) */
490:     PetscCall(VecPointwiseMult(user->yiwork[i], user->uwork, user->yiwork[i]));
491:     PetscCall(VecAXPY(Y, user->ht, user->yiwork[i]));
492:   }
493:   PetscFunctionReturn(PETSC_SUCCESS);
494: }

496: PetscErrorCode StateMatBlockPrecMult(PC PC_shell, Vec X, Vec Y)
497: {
498:   AppCtx *user;

500:   PetscFunctionBegin;
501:   PetscCall(PCShellGetContext(PC_shell, &user));

503:   if (user->dsg_formed) {
504:     PetscCall(MatSOR(user->DSG, X, 1.0, (MatSORType)(SOR_ZERO_INITIAL_GUESS | SOR_LOCAL_SYMMETRIC_SWEEP), 0.0, 1, 1, Y));
505:   } else SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "DSG not formed");
506:   PetscFunctionReturn(PETSC_SUCCESS);
507: }

509: PetscErrorCode StateMatInvMult(Mat J_shell, Vec X, Vec Y)
510: {
511:   AppCtx  *user;
512:   PetscInt its, i;

514:   PetscFunctionBegin;
515:   PetscCall(MatShellGetContext(J_shell, &user));

517:   if (Y == user->ytrue) {
518:     /* First solve is done with true solution to set up problem */
519:     PetscCall(KSPSetTolerances(user->solver, 1e-8, PETSC_DEFAULT, PETSC_DEFAULT, PETSC_DEFAULT));
520:   } else {
521:     PetscCall(KSPSetTolerances(user->solver, PETSC_DEFAULT, PETSC_DEFAULT, PETSC_DEFAULT, PETSC_DEFAULT));
522:   }

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

529:   for (i = 1; i < user->nt; i++) {
530:     PetscCall(VecAXPY(user->yi[i], 1.0, user->yiwork[i - 1]));
531:     PetscCall(KSPSolve(user->solver, user->yi[i], user->yiwork[i]));
532:     PetscCall(KSPGetIterationNumber(user->solver, &its));
533:     user->ksp_its = user->ksp_its + its;
534:   }
535:   PetscCall(Gather_i(Y, user->yiwork, user->yi_scatter, user->nt));
536:   PetscFunctionReturn(PETSC_SUCCESS);
537: }

539: PetscErrorCode StateMatInvTransposeMult(Mat J_shell, Vec X, Vec Y)
540: {
541:   AppCtx  *user;
542:   PetscInt its, i;

544:   PetscFunctionBegin;
545:   PetscCall(MatShellGetContext(J_shell, &user));

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

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

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

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

559:     PetscCall(KSPGetIterationNumber(user->solver, &its));
560:     user->ksp_its = user->ksp_its + its;
561:   }

563:   PetscCall(Gather_i(Y, user->yiwork, user->yi_scatter, user->nt));
564:   PetscFunctionReturn(PETSC_SUCCESS);
565: }

567: PetscErrorCode StateMatDuplicate(Mat J_shell, MatDuplicateOption opt, Mat *new_shell)
568: {
569:   AppCtx *user;

571:   PetscFunctionBegin;
572:   PetscCall(MatShellGetContext(J_shell, &user));

574:   PetscCall(MatCreateShell(PETSC_COMM_WORLD, PETSC_DETERMINE, PETSC_DETERMINE, user->m, user->m, user, new_shell));
575:   PetscCall(MatShellSetOperation(*new_shell, MATOP_MULT, (void (*)(void))StateMatMult));
576:   PetscCall(MatShellSetOperation(*new_shell, MATOP_DUPLICATE, (void (*)(void))StateMatDuplicate));
577:   PetscCall(MatShellSetOperation(*new_shell, MATOP_MULT_TRANSPOSE, (void (*)(void))StateMatMultTranspose));
578:   PetscCall(MatShellSetOperation(*new_shell, MATOP_GET_DIAGONAL, (void (*)(void))StateMatGetDiagonal));
579:   PetscFunctionReturn(PETSC_SUCCESS);
580: }

582: PetscErrorCode StateMatGetDiagonal(Mat J_shell, Vec X)
583: {
584:   AppCtx *user;

586:   PetscFunctionBegin;
587:   PetscCall(MatShellGetContext(J_shell, &user));
588:   PetscCall(VecCopy(user->js_diag, X));
589:   PetscFunctionReturn(PETSC_SUCCESS);
590: }

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

603:   PetscFunctionBegin;
604:   PetscCall(Scatter(X, user->y, user->state_scatter, user->u, user->design_scatter));
605:   PetscCall(Scatter_i(user->y, user->yi, user->yi_scatter, user->nt));
606:   PetscCall(MatMult(user->JsBlock, user->yi[0], user->yiwork[0]));
607:   for (i = 1; i < user->nt; i++) {
608:     PetscCall(MatMult(user->JsBlock, user->yi[i], user->yiwork[i]));
609:     PetscCall(VecAXPY(user->yiwork[i], -1.0, user->yi[i - 1]));
610:   }
611:   PetscCall(Gather_i(C, user->yiwork, user->yi_scatter, user->nt));
612:   PetscCall(VecAXPY(C, -1.0, user->q));
613:   PetscFunctionReturn(PETSC_SUCCESS);
614: }

616: PetscErrorCode Scatter(Vec x, Vec state, VecScatter s_scat, Vec design, VecScatter d_scat)
617: {
618:   PetscFunctionBegin;
619:   PetscCall(VecScatterBegin(s_scat, x, state, INSERT_VALUES, SCATTER_FORWARD));
620:   PetscCall(VecScatterEnd(s_scat, x, state, INSERT_VALUES, SCATTER_FORWARD));
621:   PetscCall(VecScatterBegin(d_scat, x, design, INSERT_VALUES, SCATTER_FORWARD));
622:   PetscCall(VecScatterEnd(d_scat, x, design, INSERT_VALUES, SCATTER_FORWARD));
623:   PetscFunctionReturn(PETSC_SUCCESS);
624: }

626: PetscErrorCode Scatter_i(Vec y, Vec *yi, VecScatter *scat, PetscInt nt)
627: {
628:   PetscInt i;

630:   PetscFunctionBegin;
631:   for (i = 0; i < nt; i++) {
632:     PetscCall(VecScatterBegin(scat[i], y, yi[i], INSERT_VALUES, SCATTER_FORWARD));
633:     PetscCall(VecScatterEnd(scat[i], y, yi[i], INSERT_VALUES, SCATTER_FORWARD));
634:   }
635:   PetscFunctionReturn(PETSC_SUCCESS);
636: }

638: PetscErrorCode Gather(Vec x, Vec state, VecScatter s_scat, Vec design, VecScatter d_scat)
639: {
640:   PetscFunctionBegin;
641:   PetscCall(VecScatterBegin(s_scat, state, x, INSERT_VALUES, SCATTER_REVERSE));
642:   PetscCall(VecScatterEnd(s_scat, state, x, INSERT_VALUES, SCATTER_REVERSE));
643:   PetscCall(VecScatterBegin(d_scat, design, x, INSERT_VALUES, SCATTER_REVERSE));
644:   PetscCall(VecScatterEnd(d_scat, design, x, INSERT_VALUES, SCATTER_REVERSE));
645:   PetscFunctionReturn(PETSC_SUCCESS);
646: }

648: PetscErrorCode Gather_i(Vec y, Vec *yi, VecScatter *scat, PetscInt nt)
649: {
650:   PetscInt i;

652:   PetscFunctionBegin;
653:   for (i = 0; i < nt; i++) {
654:     PetscCall(VecScatterBegin(scat[i], yi[i], y, INSERT_VALUES, SCATTER_REVERSE));
655:     PetscCall(VecScatterEnd(scat[i], yi[i], y, INSERT_VALUES, SCATTER_REVERSE));
656:   }
657:   PetscFunctionReturn(PETSC_SUCCESS);
658: }

660: PetscErrorCode ParabolicInitialize(AppCtx *user)
661: {
662:   PetscInt    m, n, i, j, k, linear_index, istart, iend, iblock, lo, hi, lo2, hi2;
663:   Vec         XX, YY, ZZ, XXwork, YYwork, ZZwork, UTwork, yi, di, bc;
664:   PetscReal  *x, *y, *z;
665:   PetscReal   h, stime;
666:   PetscScalar hinv, neg_hinv, half = 0.5, sqrt_beta;
667:   PetscInt    im, indx1, indx2, indy1, indy2, indz1, indz2, nx, ny, nz;
668:   PetscReal   xri, yri, zri, xim, yim, zim, dx1, dx2, dy1, dy2, dz1, dz2, Dx, Dy, Dz;
669:   PetscScalar v, vx, vy, vz;
670:   IS          is_from_y, is_to_yi, is_from_d, is_to_di;
671:   /* Data locations */
672:   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,
673:                         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,
674:                         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};

676:   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,
677:                         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,
678:                         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};

680:   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,
681:                         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,
682:                         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};

684:   PetscFunctionBegin;
685:   PetscCall(PetscMalloc1(user->mx, &x));
686:   PetscCall(PetscMalloc1(user->mx, &y));
687:   PetscCall(PetscMalloc1(user->mx, &z));
688:   user->jformed    = PETSC_FALSE;
689:   user->dsg_formed = PETSC_FALSE;

691:   n         = user->mx * user->mx * user->mx;
692:   m         = 3 * user->mx * user->mx * (user->mx - 1);
693:   sqrt_beta = PetscSqrtScalar(user->beta);

695:   user->ksp_its         = 0;
696:   user->ksp_its_initial = 0;

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

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

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

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

723:   PetscCall(VecGetOwnershipRange(XX, &istart, &iend));
724:   for (linear_index = istart; linear_index < iend; linear_index++) {
725:     i  = linear_index % user->mx;
726:     j  = ((linear_index - i) / user->mx) % user->mx;
727:     k  = ((linear_index - i) / user->mx - j) / user->mx;
728:     vx = h * (i + 0.5);
729:     vy = h * (j + 0.5);
730:     vz = h * (k + 0.5);
731:     PetscCall(VecSetValues(XX, 1, &linear_index, &vx, INSERT_VALUES));
732:     PetscCall(VecSetValues(YY, 1, &linear_index, &vy, INSERT_VALUES));
733:     PetscCall(VecSetValues(ZZ, 1, &linear_index, &vz, INSERT_VALUES));
734:     if ((vx < 0.6) && (vx > 0.4) && (vy < 0.6) && (vy > 0.4) && (vy < 0.6) && (vz < 0.6) && (vz > 0.4)) {
735:       v = 1000.0;
736:       PetscCall(VecSetValues(bc, 1, &linear_index, &v, INSERT_VALUES));
737:     }
738:   }

740:   PetscCall(VecAssemblyBegin(XX));
741:   PetscCall(VecAssemblyEnd(XX));
742:   PetscCall(VecAssemblyBegin(YY));
743:   PetscCall(VecAssemblyEnd(YY));
744:   PetscCall(VecAssemblyBegin(ZZ));
745:   PetscCall(VecAssemblyEnd(ZZ));
746:   PetscCall(VecAssemblyBegin(bc));
747:   PetscCall(VecAssemblyEnd(bc));

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

755:   PetscCall(VecShift(XXwork, -0.5));
756:   PetscCall(VecShift(YYwork, -0.5));
757:   PetscCall(VecShift(ZZwork, -0.5));

759:   PetscCall(VecPointwiseMult(XXwork, XXwork, XXwork));
760:   PetscCall(VecPointwiseMult(YYwork, YYwork, YYwork));
761:   PetscCall(VecPointwiseMult(ZZwork, ZZwork, ZZwork));

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

770:   PetscCall(VecDestroy(&XX));
771:   PetscCall(VecDestroy(&YY));
772:   PetscCall(VecDestroy(&ZZ));
773:   PetscCall(VecDestroy(&XXwork));
774:   PetscCall(VecDestroy(&YYwork));
775:   PetscCall(VecDestroy(&ZZwork));
776:   PetscCall(VecDestroy(&UTwork));

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

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

790:   for (i = istart; i < iend; i++) {
791:     if (i < m / 3) {
792:       iblock = i / (user->mx - 1);
793:       j      = iblock * user->mx + (i % (user->mx - 1));
794:       PetscCall(MatSetValues(user->Grad, 1, &i, 1, &j, &neg_hinv, INSERT_VALUES));
795:       j = j + 1;
796:       PetscCall(MatSetValues(user->Grad, 1, &i, 1, &j, &hinv, INSERT_VALUES));
797:     }
798:     if (i >= m / 3 && i < 2 * m / 3) {
799:       iblock = (i - m / 3) / (user->mx * (user->mx - 1));
800:       j      = iblock * user->mx * user->mx + ((i - m / 3) % (user->mx * (user->mx - 1)));
801:       PetscCall(MatSetValues(user->Grad, 1, &i, 1, &j, &neg_hinv, INSERT_VALUES));
802:       j = j + user->mx;
803:       PetscCall(MatSetValues(user->Grad, 1, &i, 1, &j, &hinv, INSERT_VALUES));
804:     }
805:     if (i >= 2 * m / 3) {
806:       j = i - 2 * m / 3;
807:       PetscCall(MatSetValues(user->Grad, 1, &i, 1, &j, &neg_hinv, INSERT_VALUES));
808:       j = j + user->mx * user->mx;
809:       PetscCall(MatSetValues(user->Grad, 1, &i, 1, &j, &hinv, INSERT_VALUES));
810:     }
811:   }

813:   PetscCall(MatAssemblyBegin(user->Grad, MAT_FINAL_ASSEMBLY));
814:   PetscCall(MatAssemblyEnd(user->Grad, MAT_FINAL_ASSEMBLY));

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

824:   for (i = istart; i < iend; i++) {
825:     if (i < m / 3) {
826:       iblock = i / (user->mx - 1);
827:       j      = iblock * user->mx + (i % (user->mx - 1));
828:       PetscCall(MatSetValues(user->Av, 1, &i, 1, &j, &half, INSERT_VALUES));
829:       j = j + 1;
830:       PetscCall(MatSetValues(user->Av, 1, &i, 1, &j, &half, INSERT_VALUES));
831:     }
832:     if (i >= m / 3 && i < 2 * m / 3) {
833:       iblock = (i - m / 3) / (user->mx * (user->mx - 1));
834:       j      = iblock * user->mx * user->mx + ((i - m / 3) % (user->mx * (user->mx - 1)));
835:       PetscCall(MatSetValues(user->Av, 1, &i, 1, &j, &half, INSERT_VALUES));
836:       j = j + user->mx;
837:       PetscCall(MatSetValues(user->Av, 1, &i, 1, &j, &half, INSERT_VALUES));
838:     }
839:     if (i >= 2 * m / 3) {
840:       j = i - 2 * m / 3;
841:       PetscCall(MatSetValues(user->Av, 1, &i, 1, &j, &half, INSERT_VALUES));
842:       j = j + user->mx * user->mx;
843:       PetscCall(MatSetValues(user->Av, 1, &i, 1, &j, &half, INSERT_VALUES));
844:     }
845:   }

847:   PetscCall(MatAssemblyBegin(user->Av, MAT_FINAL_ASSEMBLY));
848:   PetscCall(MatAssemblyEnd(user->Av, MAT_FINAL_ASSEMBLY));

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

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

860:   for (i = istart; i < iend; i++) {
861:     if (i < m / 3) {
862:       iblock = i / (user->mx - 1);
863:       j      = iblock * user->mx + (i % (user->mx - 1));
864:       PetscCall(MatSetValues(user->L, 1, &i, 1, &j, &neg_hinv, INSERT_VALUES));
865:       j = j + 1;
866:       PetscCall(MatSetValues(user->L, 1, &i, 1, &j, &hinv, INSERT_VALUES));
867:     }
868:     if (i >= m / 3 && i < 2 * m / 3) {
869:       iblock = (i - m / 3) / (user->mx * (user->mx - 1));
870:       j      = iblock * user->mx * user->mx + ((i - m / 3) % (user->mx * (user->mx - 1)));
871:       PetscCall(MatSetValues(user->L, 1, &i, 1, &j, &neg_hinv, INSERT_VALUES));
872:       j = j + user->mx;
873:       PetscCall(MatSetValues(user->L, 1, &i, 1, &j, &hinv, INSERT_VALUES));
874:     }
875:     if (i >= 2 * m / 3 && i < m) {
876:       j = i - 2 * m / 3;
877:       PetscCall(MatSetValues(user->L, 1, &i, 1, &j, &neg_hinv, INSERT_VALUES));
878:       j = j + user->mx * user->mx;
879:       PetscCall(MatSetValues(user->L, 1, &i, 1, &j, &hinv, INSERT_VALUES));
880:     }
881:     if (i >= m) {
882:       j = i - m;
883:       PetscCall(MatSetValues(user->L, 1, &i, 1, &j, &sqrt_beta, INSERT_VALUES));
884:     }
885:   }

887:   PetscCall(MatAssemblyBegin(user->L, MAT_FINAL_ASSEMBLY));
888:   PetscCall(MatAssemblyEnd(user->L, MAT_FINAL_ASSEMBLY));

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

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

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

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

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

907:   PetscCall(VecCreate(PETSC_COMM_WORLD, &user->d));
908:   PetscCall(VecSetSizes(user->d, PETSC_DECIDE, user->ndata * user->nt));
909:   PetscCall(VecSetFromOptions(user->d));

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

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

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

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

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

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

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

964:   PetscCall(PCShellSetApply(user->prec, StateMatBlockPrecMult));
965:   PetscCall(PCShellSetApplyTranspose(user->prec, StateMatBlockPrecMult));
966:   PetscCall(PCShellSetContext(user->prec, user));

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

976:   PetscCall(VecGetOwnershipRange(user->y, &lo2, &hi2));
977:   istart = 0;
978:   for (i = 0; i < user->nt; i++) {
979:     PetscCall(VecGetOwnershipRange(user->yi[i], &lo, &hi));
980:     PetscCall(ISCreateStride(PETSC_COMM_SELF, hi - lo, lo, 1, &is_to_yi));
981:     PetscCall(ISCreateStride(PETSC_COMM_SELF, hi - lo, lo2 + istart, 1, &is_from_y));
982:     PetscCall(VecScatterCreate(user->y, is_from_y, user->yi[i], is_to_yi, &user->yi_scatter[i]));
983:     istart = istart + hi - lo;
984:     PetscCall(ISDestroy(&is_to_yi));
985:     PetscCall(ISDestroy(&is_from_y));
986:   }
987:   PetscCall(VecDestroy(&yi));

989:   /* Create scatter from d to d_1,d_2,...,d_ns */
990:   PetscCall(PetscMalloc1(user->ns * user->ndata, &user->di_scatter));
991:   PetscCall(VecCreate(PETSC_COMM_WORLD, &di));
992:   PetscCall(VecSetSizes(di, PETSC_DECIDE, user->ndata));
993:   PetscCall(VecSetFromOptions(di));
994:   PetscCall(VecDuplicateVecs(di, user->ns, &user->di));
995:   PetscCall(VecGetOwnershipRange(user->d, &lo2, &hi2));
996:   istart = 0;
997:   for (i = 0; i < user->ns; i++) {
998:     PetscCall(VecGetOwnershipRange(user->di[i], &lo, &hi));
999:     PetscCall(ISCreateStride(PETSC_COMM_SELF, hi - lo, lo, 1, &is_to_di));
1000:     PetscCall(ISCreateStride(PETSC_COMM_SELF, hi - lo, lo2 + istart, 1, &is_from_d));
1001:     PetscCall(VecScatterCreate(user->d, is_from_d, user->di[i], is_to_di, &user->di_scatter[i]));
1002:     istart = istart + hi - lo;
1003:     PetscCall(ISDestroy(&is_to_di));
1004:     PetscCall(ISDestroy(&is_from_d));
1005:   }
1006:   PetscCall(VecDestroy(&di));

1008:   /* Assemble RHS of forward problem */
1009:   PetscCall(VecCopy(bc, user->yiwork[0]));
1010:   for (i = 1; i < user->nt; i++) PetscCall(VecSet(user->yiwork[i], 0.0));
1011:   PetscCall(Gather_i(user->q, user->yiwork, user->yi_scatter, user->nt));
1012:   PetscCall(VecDestroy(&bc));

1014:   /* Compute true state function yt given ut */
1015:   PetscCall(VecCreate(PETSC_COMM_WORLD, &user->ytrue));
1016:   PetscCall(VecSetSizes(user->ytrue, PETSC_DECIDE, n * user->nt));
1017:   PetscCall(VecSetFromOptions(user->ytrue));

1019:   /* First compute Av_u = Av*exp(-u) */
1020:   PetscCall(VecSet(user->uwork, 0));
1021:   PetscCall(VecAXPY(user->uwork, -1.0, user->utrue)); /*  Note: user->utrue */
1022:   PetscCall(VecExp(user->uwork));
1023:   PetscCall(MatMult(user->Av, user->uwork, user->Av_u));

1025:   /* Symbolic DSG = Div * Grad */
1026:   PetscCall(MatProductCreate(user->Div, user->Grad, NULL, &user->DSG));
1027:   PetscCall(MatProductSetType(user->DSG, MATPRODUCT_AB));
1028:   PetscCall(MatProductSetAlgorithm(user->DSG, "default"));
1029:   PetscCall(MatProductSetFill(user->DSG, PETSC_DEFAULT));
1030:   PetscCall(MatProductSetFromOptions(user->DSG));
1031:   PetscCall(MatProductSymbolic(user->DSG));

1033:   user->dsg_formed = PETSC_TRUE;

1035:   /* Next form DSG = Div*Grad */
1036:   PetscCall(MatCopy(user->Div, user->Divwork, SAME_NONZERO_PATTERN));
1037:   PetscCall(MatDiagonalScale(user->Divwork, NULL, user->Av_u));
1038:   if (user->dsg_formed) {
1039:     PetscCall(MatProductNumeric(user->DSG));
1040:   } else {
1041:     PetscCall(MatMatMult(user->Div, user->Grad, MAT_INITIAL_MATRIX, PETSC_DEFAULT, &user->DSG));
1042:     user->dsg_formed = PETSC_TRUE;
1043:   }
1044:   /* B = speye(nx^3) + ht*DSG; */
1045:   PetscCall(MatScale(user->DSG, user->ht));
1046:   PetscCall(MatShift(user->DSG, 1.0));

1048:   /* Now solve for ytrue */
1049:   PetscCall(StateMatInvMult(user->Js, user->q, user->ytrue));

1051:   /* Initial guess y0 for state given u0 */

1053:   /* First compute Av_u = Av*exp(-u) */
1054:   PetscCall(VecSet(user->uwork, 0));
1055:   PetscCall(VecAXPY(user->uwork, -1.0, user->u)); /*  Note: user->u */
1056:   PetscCall(VecExp(user->uwork));
1057:   PetscCall(MatMult(user->Av, user->uwork, user->Av_u));

1059:   /* Next form DSG = Div*S*Grad */
1060:   PetscCall(MatCopy(user->Div, user->Divwork, SAME_NONZERO_PATTERN));
1061:   PetscCall(MatDiagonalScale(user->Divwork, NULL, user->Av_u));
1062:   if (user->dsg_formed) {
1063:     PetscCall(MatProductNumeric(user->DSG));
1064:   } else {
1065:     PetscCall(MatMatMult(user->Div, user->Grad, MAT_INITIAL_MATRIX, PETSC_DEFAULT, &user->DSG));

1067:     user->dsg_formed = PETSC_TRUE;
1068:   }
1069:   /* B = speye(nx^3) + ht*DSG; */
1070:   PetscCall(MatScale(user->DSG, user->ht));
1071:   PetscCall(MatShift(user->DSG, 1.0));

1073:   /* Now solve for y */
1074:   PetscCall(StateMatInvMult(user->Js, user->q, user->y));

1076:   /* Construct projection matrix Q, a block diagonal matrix consisting of nt copies of Qblock along the diagonal */
1077:   PetscCall(MatCreate(PETSC_COMM_WORLD, &user->Qblock));
1078:   PetscCall(MatSetSizes(user->Qblock, PETSC_DECIDE, PETSC_DECIDE, user->ndata, n));
1079:   PetscCall(MatSetFromOptions(user->Qblock));
1080:   PetscCall(MatMPIAIJSetPreallocation(user->Qblock, 8, NULL, 8, NULL));
1081:   PetscCall(MatSeqAIJSetPreallocation(user->Qblock, 8, NULL));

1083:   for (i = 0; i < user->mx; i++) {
1084:     x[i] = h * (i + 0.5);
1085:     y[i] = h * (i + 0.5);
1086:     z[i] = h * (i + 0.5);
1087:   }

1089:   PetscCall(MatGetOwnershipRange(user->Qblock, &istart, &iend));
1090:   nx = user->mx;
1091:   ny = user->mx;
1092:   nz = user->mx;
1093:   for (i = istart; i < iend; i++) {
1094:     xri = xr[i];
1095:     im  = 0;
1096:     xim = x[im];
1097:     while (xri > xim && im < nx) {
1098:       im  = im + 1;
1099:       xim = x[im];
1100:     }
1101:     indx1 = im - 1;
1102:     indx2 = im;
1103:     dx1   = xri - x[indx1];
1104:     dx2   = x[indx2] - xri;

1106:     yri = yr[i];
1107:     im  = 0;
1108:     yim = y[im];
1109:     while (yri > yim && im < ny) {
1110:       im  = im + 1;
1111:       yim = y[im];
1112:     }
1113:     indy1 = im - 1;
1114:     indy2 = im;
1115:     dy1   = yri - y[indy1];
1116:     dy2   = y[indy2] - yri;

1118:     zri = zr[i];
1119:     im  = 0;
1120:     zim = z[im];
1121:     while (zri > zim && im < nz) {
1122:       im  = im + 1;
1123:       zim = z[im];
1124:     }
1125:     indz1 = im - 1;
1126:     indz2 = im;
1127:     dz1   = zri - z[indz1];
1128:     dz2   = z[indz2] - zri;

1130:     Dx = x[indx2] - x[indx1];
1131:     Dy = y[indy2] - y[indy1];
1132:     Dz = z[indz2] - z[indz1];

1134:     j = indx1 + indy1 * nx + indz1 * nx * ny;
1135:     v = (1 - dx1 / Dx) * (1 - dy1 / Dy) * (1 - dz1 / Dz);
1136:     PetscCall(MatSetValues(user->Qblock, 1, &i, 1, &j, &v, INSERT_VALUES));

1138:     j = indx1 + indy1 * nx + indz2 * nx * ny;
1139:     v = (1 - dx1 / Dx) * (1 - dy1 / Dy) * (1 - dz2 / Dz);
1140:     PetscCall(MatSetValues(user->Qblock, 1, &i, 1, &j, &v, INSERT_VALUES));

1142:     j = indx1 + indy2 * nx + indz1 * nx * ny;
1143:     v = (1 - dx1 / Dx) * (1 - dy2 / Dy) * (1 - dz1 / Dz);
1144:     PetscCall(MatSetValues(user->Qblock, 1, &i, 1, &j, &v, INSERT_VALUES));

1146:     j = indx1 + indy2 * nx + indz2 * nx * ny;
1147:     v = (1 - dx1 / Dx) * (1 - dy2 / Dy) * (1 - dz2 / Dz);
1148:     PetscCall(MatSetValues(user->Qblock, 1, &i, 1, &j, &v, INSERT_VALUES));

1150:     j = indx2 + indy1 * nx + indz1 * nx * ny;
1151:     v = (1 - dx2 / Dx) * (1 - dy1 / Dy) * (1 - dz1 / Dz);
1152:     PetscCall(MatSetValues(user->Qblock, 1, &i, 1, &j, &v, INSERT_VALUES));

1154:     j = indx2 + indy1 * nx + indz2 * nx * ny;
1155:     v = (1 - dx2 / Dx) * (1 - dy1 / Dy) * (1 - dz2 / Dz);
1156:     PetscCall(MatSetValues(user->Qblock, 1, &i, 1, &j, &v, INSERT_VALUES));

1158:     j = indx2 + indy2 * nx + indz1 * nx * ny;
1159:     v = (1 - dx2 / Dx) * (1 - dy2 / Dy) * (1 - dz1 / Dz);
1160:     PetscCall(MatSetValues(user->Qblock, 1, &i, 1, &j, &v, INSERT_VALUES));

1162:     j = indx2 + indy2 * nx + indz2 * nx * ny;
1163:     v = (1 - dx2 / Dx) * (1 - dy2 / Dy) * (1 - dz2 / Dz);
1164:     PetscCall(MatSetValues(user->Qblock, 1, &i, 1, &j, &v, INSERT_VALUES));
1165:   }
1166:   PetscCall(MatAssemblyBegin(user->Qblock, MAT_FINAL_ASSEMBLY));
1167:   PetscCall(MatAssemblyEnd(user->Qblock, MAT_FINAL_ASSEMBLY));

1169:   PetscCall(MatTranspose(user->Qblock, MAT_INITIAL_MATRIX, &user->QblockT));
1170:   PetscCall(MatTranspose(user->L, MAT_INITIAL_MATRIX, &user->LT));

1172:   /* Add noise to the measurement data */
1173:   PetscCall(VecSet(user->ywork, 1.0));
1174:   PetscCall(VecAYPX(user->ywork, user->noise, user->ytrue));
1175:   PetscCall(Scatter_i(user->ywork, user->yiwork, user->yi_scatter, user->nt));
1176:   for (j = 0; j < user->ns; j++) {
1177:     i = user->sample_times[j];
1178:     PetscCall(MatMult(user->Qblock, user->yiwork[i], user->di[j]));
1179:   }
1180:   PetscCall(Gather_i(user->d, user->di, user->di_scatter, user->ns));

1182:   /* Now that initial conditions have been set, let the user pass tolerance options to the KSP solver */
1183:   PetscCall(KSPSetFromOptions(user->solver));
1184:   PetscCall(PetscFree(x));
1185:   PetscCall(PetscFree(y));
1186:   PetscCall(PetscFree(z));
1187:   PetscFunctionReturn(PETSC_SUCCESS);
1188: }

1190: PetscErrorCode ParabolicDestroy(AppCtx *user)
1191: {
1192:   PetscInt i;

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

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

1255:   PetscFunctionBegin;
1256:   PetscCall(TaoGetSolution(tao, &X));
1257:   PetscCall(Scatter(X, user->ywork, user->state_scatter, user->uwork, user->design_scatter));
1258:   PetscCall(VecAXPY(user->ywork, -1.0, user->ytrue));
1259:   PetscCall(VecAXPY(user->uwork, -1.0, user->utrue));
1260:   PetscCall(VecNorm(user->uwork, NORM_2, &unorm));
1261:   PetscCall(VecNorm(user->ywork, NORM_2, &ynorm));
1262:   PetscCall(PetscPrintf(MPI_COMM_WORLD, "||u-ut||=%g ||y-yt||=%g\n", (double)unorm, (double)ynorm));
1263:   PetscFunctionReturn(PETSC_SUCCESS);
1264: }

1266: /*TEST

1268:    build:
1269:       requires: !complex

1271:    test:
1272:       args: -tao_cmonitor -tao_type lcl -ns 1 -tao_gatol 1.e-4 -ksp_max_it 30
1273:       requires: !single

1275: TEST*/