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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    656: PetscErrorCode ParabolicInitialize(AppCtx *user)
    657: {
    658:   PetscInt    m, n, i, j, k, linear_index, istart, iend, iblock, lo, hi, lo2, hi2;
    659:   Vec         XX, YY, ZZ, XXwork, YYwork, ZZwork, UTwork, yi, di, bc;
    660:   PetscReal  *x, *y, *z;
    661:   PetscReal   h, stime;
    662:   PetscScalar hinv, neg_hinv, half = 0.5, sqrt_beta;
    663:   PetscInt    im, indx1, indx2, indy1, indy2, indz1, indz2, nx, ny, nz;
    664:   PetscReal   xri, yri, zri, xim, yim, zim, dx1, dx2, dy1, dy2, dz1, dz2, Dx, Dy, Dz;
    665:   PetscScalar v, vx, vy, vz;
    666:   IS          is_from_y, is_to_yi, is_from_d, is_to_di;
    667:   /* Data locations */
    668:   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,
    669:                         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,
    670:                         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};

    672:   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,
    673:                         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,
    674:                         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};

    676:   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,
    677:                         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,
    678:                         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};

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    1004:   /* Assemble RHS of forward problem */
    1005:   PetscCall(VecCopy(bc, user->yiwork[0]));
    1006:   for (i = 1; i < user->nt; i++) PetscCall(VecSet(user->yiwork[i], 0.0));
    1007:   PetscCall(Gather_i(user->q, user->yiwork, user->yi_scatter, user->nt));
    1008:   PetscCall(VecDestroy(&bc));

    1010:   /* Compute true state function yt given ut */
    1011:   PetscCall(VecCreate(PETSC_COMM_WORLD, &user->ytrue));
    1012:   PetscCall(VecSetSizes(user->ytrue, PETSC_DECIDE, n * user->nt));
    1013:   PetscCall(VecSetFromOptions(user->ytrue));

    1015:   /* First compute Av_u = Av*exp(-u) */
    1016:   PetscCall(VecSet(user->uwork, 0));
    1017:   PetscCall(VecAXPY(user->uwork, -1.0, user->utrue)); /*  Note: user->utrue */
    1018:   PetscCall(VecExp(user->uwork));
    1019:   PetscCall(MatMult(user->Av, user->uwork, user->Av_u));

    1021:   /* Symbolic DSG = Div * Grad */
    1022:   PetscCall(MatProductCreate(user->Div, user->Grad, NULL, &user->DSG));
    1023:   PetscCall(MatProductSetType(user->DSG, MATPRODUCT_AB));
    1024:   PetscCall(MatProductSetAlgorithm(user->DSG, "default"));
    1025:   PetscCall(MatProductSetFill(user->DSG, PETSC_DETERMINE));
    1026:   PetscCall(MatProductSetFromOptions(user->DSG));
    1027:   PetscCall(MatProductSymbolic(user->DSG));

    1029:   user->dsg_formed = PETSC_TRUE;

    1031:   /* Next form DSG = Div*Grad */
    1032:   PetscCall(MatCopy(user->Div, user->Divwork, SAME_NONZERO_PATTERN));
    1033:   PetscCall(MatDiagonalScale(user->Divwork, NULL, user->Av_u));
    1034:   if (user->dsg_formed) {
    1035:     PetscCall(MatProductNumeric(user->DSG));
    1036:   } else {
    1037:     PetscCall(MatMatMult(user->Div, user->Grad, MAT_INITIAL_MATRIX, PETSC_DETERMINE, &user->DSG));
    1038:     user->dsg_formed = PETSC_TRUE;
    1039:   }
    1040:   /* B = speye(nx^3) + ht*DSG; */
    1041:   PetscCall(MatScale(user->DSG, user->ht));
    1042:   PetscCall(MatShift(user->DSG, 1.0));

    1044:   /* Now solve for ytrue */
    1045:   PetscCall(StateMatInvMult(user->Js, user->q, user->ytrue));

    1047:   /* Initial guess y0 for state given u0 */

    1049:   /* First compute Av_u = Av*exp(-u) */
    1050:   PetscCall(VecSet(user->uwork, 0));
    1051:   PetscCall(VecAXPY(user->uwork, -1.0, user->u)); /*  Note: user->u */
    1052:   PetscCall(VecExp(user->uwork));
    1053:   PetscCall(MatMult(user->Av, user->uwork, user->Av_u));

    1055:   /* Next form DSG = Div*S*Grad */
    1056:   PetscCall(MatCopy(user->Div, user->Divwork, SAME_NONZERO_PATTERN));
    1057:   PetscCall(MatDiagonalScale(user->Divwork, NULL, user->Av_u));
    1058:   if (user->dsg_formed) {
    1059:     PetscCall(MatProductNumeric(user->DSG));
    1060:   } else {
    1061:     PetscCall(MatMatMult(user->Div, user->Grad, MAT_INITIAL_MATRIX, PETSC_DETERMINE, &user->DSG));

    1063:     user->dsg_formed = PETSC_TRUE;
    1064:   }
    1065:   /* B = speye(nx^3) + ht*DSG; */
    1066:   PetscCall(MatScale(user->DSG, user->ht));
    1067:   PetscCall(MatShift(user->DSG, 1.0));

    1069:   /* Now solve for y */
    1070:   PetscCall(StateMatInvMult(user->Js, user->q, user->y));

    1072:   /* Construct projection matrix Q, a block diagonal matrix consisting of nt copies of Qblock along the diagonal */
    1073:   PetscCall(MatCreate(PETSC_COMM_WORLD, &user->Qblock));
    1074:   PetscCall(MatSetSizes(user->Qblock, PETSC_DECIDE, PETSC_DECIDE, user->ndata, n));
    1075:   PetscCall(MatSetFromOptions(user->Qblock));
    1076:   PetscCall(MatMPIAIJSetPreallocation(user->Qblock, 8, NULL, 8, NULL));
    1077:   PetscCall(MatSeqAIJSetPreallocation(user->Qblock, 8, NULL));

    1079:   for (i = 0; i < user->mx; i++) {
    1080:     x[i] = h * (i + 0.5);
    1081:     y[i] = h * (i + 0.5);
    1082:     z[i] = h * (i + 0.5);
    1083:   }

    1085:   PetscCall(MatGetOwnershipRange(user->Qblock, &istart, &iend));
    1086:   nx = user->mx;
    1087:   ny = user->mx;
    1088:   nz = user->mx;
    1089:   for (i = istart; i < iend; i++) {
    1090:     xri = xr[i];
    1091:     im  = 0;
    1092:     xim = x[im];
    1093:     while (xri > xim && im < nx) {
    1094:       im  = im + 1;
    1095:       xim = x[im];
    1096:     }
    1097:     indx1 = im - 1;
    1098:     indx2 = im;
    1099:     dx1   = xri - x[indx1];
    1100:     dx2   = x[indx2] - xri;

    1102:     yri = yr[i];
    1103:     im  = 0;
    1104:     yim = y[im];
    1105:     while (yri > yim && im < ny) {
    1106:       im  = im + 1;
    1107:       yim = y[im];
    1108:     }
    1109:     indy1 = im - 1;
    1110:     indy2 = im;
    1111:     dy1   = yri - y[indy1];
    1112:     dy2   = y[indy2] - yri;

    1114:     zri = zr[i];
    1115:     im  = 0;
    1116:     zim = z[im];
    1117:     while (zri > zim && im < nz) {
    1118:       im  = im + 1;
    1119:       zim = z[im];
    1120:     }
    1121:     indz1 = im - 1;
    1122:     indz2 = im;
    1123:     dz1   = zri - z[indz1];
    1124:     dz2   = z[indz2] - zri;

    1126:     Dx = x[indx2] - x[indx1];
    1127:     Dy = y[indy2] - y[indy1];
    1128:     Dz = z[indz2] - z[indz1];

    1130:     j = indx1 + indy1 * nx + indz1 * nx * ny;
    1131:     v = (1 - dx1 / Dx) * (1 - dy1 / Dy) * (1 - dz1 / Dz);
    1132:     PetscCall(MatSetValues(user->Qblock, 1, &i, 1, &j, &v, INSERT_VALUES));

    1134:     j = indx1 + indy1 * nx + indz2 * nx * ny;
    1135:     v = (1 - dx1 / Dx) * (1 - dy1 / Dy) * (1 - dz2 / Dz);
    1136:     PetscCall(MatSetValues(user->Qblock, 1, &i, 1, &j, &v, INSERT_VALUES));

    1138:     j = indx1 + indy2 * nx + indz1 * nx * ny;
    1139:     v = (1 - dx1 / Dx) * (1 - dy2 / Dy) * (1 - dz1 / Dz);
    1140:     PetscCall(MatSetValues(user->Qblock, 1, &i, 1, &j, &v, INSERT_VALUES));

    1142:     j = indx1 + indy2 * nx + indz2 * nx * ny;
    1143:     v = (1 - dx1 / Dx) * (1 - dy2 / Dy) * (1 - dz2 / Dz);
    1144:     PetscCall(MatSetValues(user->Qblock, 1, &i, 1, &j, &v, INSERT_VALUES));

    1146:     j = indx2 + indy1 * nx + indz1 * nx * ny;
    1147:     v = (1 - dx2 / Dx) * (1 - dy1 / Dy) * (1 - dz1 / Dz);
    1148:     PetscCall(MatSetValues(user->Qblock, 1, &i, 1, &j, &v, INSERT_VALUES));

    1150:     j = indx2 + indy1 * nx + indz2 * nx * ny;
    1151:     v = (1 - dx2 / Dx) * (1 - dy1 / Dy) * (1 - dz2 / Dz);
    1152:     PetscCall(MatSetValues(user->Qblock, 1, &i, 1, &j, &v, INSERT_VALUES));

    1154:     j = indx2 + indy2 * nx + indz1 * nx * ny;
    1155:     v = (1 - dx2 / Dx) * (1 - dy2 / Dy) * (1 - dz1 / Dz);
    1156:     PetscCall(MatSetValues(user->Qblock, 1, &i, 1, &j, &v, INSERT_VALUES));

    1158:     j = indx2 + indy2 * nx + indz2 * nx * ny;
    1159:     v = (1 - dx2 / Dx) * (1 - dy2 / Dy) * (1 - dz2 / Dz);
    1160:     PetscCall(MatSetValues(user->Qblock, 1, &i, 1, &j, &v, INSERT_VALUES));
    1161:   }
    1162:   PetscCall(MatAssemblyBegin(user->Qblock, MAT_FINAL_ASSEMBLY));
    1163:   PetscCall(MatAssemblyEnd(user->Qblock, MAT_FINAL_ASSEMBLY));

    1165:   PetscCall(MatTranspose(user->Qblock, MAT_INITIAL_MATRIX, &user->QblockT));
    1166:   PetscCall(MatTranspose(user->L, MAT_INITIAL_MATRIX, &user->LT));

    1168:   /* Add noise to the measurement data */
    1169:   PetscCall(VecSet(user->ywork, 1.0));
    1170:   PetscCall(VecAYPX(user->ywork, user->noise, user->ytrue));
    1171:   PetscCall(Scatter_i(user->ywork, user->yiwork, user->yi_scatter, user->nt));
    1172:   for (j = 0; j < user->ns; j++) {
    1173:     i = user->sample_times[j];
    1174:     PetscCall(MatMult(user->Qblock, user->yiwork[i], user->di[j]));
    1175:   }
    1176:   PetscCall(Gather_i(user->d, user->di, user->di_scatter, user->ns));

    1178:   /* Now that initial conditions have been set, let the user pass tolerance options to the KSP solver */
    1179:   PetscCall(KSPSetFromOptions(user->solver));
    1180:   PetscCall(PetscFree(x));
    1181:   PetscCall(PetscFree(y));
    1182:   PetscCall(PetscFree(z));
    1183:   PetscFunctionReturn(PETSC_SUCCESS);
    1184: }

    1186: PetscErrorCode ParabolicDestroy(AppCtx *user)
    1187: {
    1188:   PetscInt i;

    1190:   PetscFunctionBegin;
    1191:   PetscCall(MatDestroy(&user->Qblock));
    1192:   PetscCall(MatDestroy(&user->QblockT));
    1193:   PetscCall(MatDestroy(&user->Div));
    1194:   PetscCall(MatDestroy(&user->Divwork));
    1195:   PetscCall(MatDestroy(&user->Grad));
    1196:   PetscCall(MatDestroy(&user->Av));
    1197:   PetscCall(MatDestroy(&user->Avwork));
    1198:   PetscCall(MatDestroy(&user->AvT));
    1199:   PetscCall(MatDestroy(&user->DSG));
    1200:   PetscCall(MatDestroy(&user->L));
    1201:   PetscCall(MatDestroy(&user->LT));
    1202:   PetscCall(KSPDestroy(&user->solver));
    1203:   PetscCall(MatDestroy(&user->Js));
    1204:   PetscCall(MatDestroy(&user->Jd));
    1205:   PetscCall(MatDestroy(&user->JsInv));
    1206:   PetscCall(MatDestroy(&user->JsBlock));
    1207:   PetscCall(MatDestroy(&user->JsBlockPrec));
    1208:   PetscCall(VecDestroy(&user->u));
    1209:   PetscCall(VecDestroy(&user->uwork));
    1210:   PetscCall(VecDestroy(&user->utrue));
    1211:   PetscCall(VecDestroy(&user->y));
    1212:   PetscCall(VecDestroy(&user->ywork));
    1213:   PetscCall(VecDestroy(&user->ytrue));
    1214:   PetscCall(VecDestroyVecs(user->nt, &user->yi));
    1215:   PetscCall(VecDestroyVecs(user->nt, &user->yiwork));
    1216:   PetscCall(VecDestroyVecs(user->ns, &user->di));
    1217:   PetscCall(PetscFree(user->yi));
    1218:   PetscCall(PetscFree(user->yiwork));
    1219:   PetscCall(PetscFree(user->di));
    1220:   PetscCall(VecDestroy(&user->c));
    1221:   PetscCall(VecDestroy(&user->cwork));
    1222:   PetscCall(VecDestroy(&user->ur));
    1223:   PetscCall(VecDestroy(&user->q));
    1224:   PetscCall(VecDestroy(&user->d));
    1225:   PetscCall(VecDestroy(&user->dwork));
    1226:   PetscCall(VecDestroy(&user->lwork));
    1227:   PetscCall(VecDestroy(&user->S));
    1228:   PetscCall(VecDestroy(&user->Swork));
    1229:   PetscCall(VecDestroy(&user->Av_u));
    1230:   PetscCall(VecDestroy(&user->Twork));
    1231:   PetscCall(VecDestroy(&user->Rwork));
    1232:   PetscCall(VecDestroy(&user->js_diag));
    1233:   PetscCall(ISDestroy(&user->s_is));
    1234:   PetscCall(ISDestroy(&user->d_is));
    1235:   PetscCall(VecScatterDestroy(&user->state_scatter));
    1236:   PetscCall(VecScatterDestroy(&user->design_scatter));
    1237:   for (i = 0; i < user->nt; i++) PetscCall(VecScatterDestroy(&user->yi_scatter[i]));
    1238:   for (i = 0; i < user->ns; i++) PetscCall(VecScatterDestroy(&user->di_scatter[i]));
    1239:   PetscCall(PetscFree(user->yi_scatter));
    1240:   PetscCall(PetscFree(user->di_scatter));
    1241:   PetscCall(PetscFree(user->sample_times));
    1242:   PetscFunctionReturn(PETSC_SUCCESS);
    1243: }

    1245: PetscErrorCode ParabolicMonitor(Tao tao, void *ptr)
    1246: {
    1247:   Vec       X;
    1248:   PetscReal unorm, ynorm;
    1249:   AppCtx   *user = (AppCtx *)ptr;

    1251:   PetscFunctionBegin;
    1252:   PetscCall(TaoGetSolution(tao, &X));
    1253:   PetscCall(Scatter(X, user->ywork, user->state_scatter, user->uwork, user->design_scatter));
    1254:   PetscCall(VecAXPY(user->ywork, -1.0, user->ytrue));
    1255:   PetscCall(VecAXPY(user->uwork, -1.0, user->utrue));
    1256:   PetscCall(VecNorm(user->uwork, NORM_2, &unorm));
    1257:   PetscCall(VecNorm(user->ywork, NORM_2, &ynorm));
    1258:   PetscCall(PetscPrintf(MPI_COMM_WORLD, "||u-ut||=%g ||y-yt||=%g\n", (double)unorm, (double)ynorm));
    1259:   PetscFunctionReturn(PETSC_SUCCESS);
    1260: }

    1262: /*TEST

    1264:    build:
    1265:       requires: !complex

    1267:    test:
    1268:       args: -tao_monitor_constraint_norm -tao_type lcl -ns 1 -tao_gatol 1.e-4 -ksp_max_it 30
    1269:       requires: !single

    1271: TEST*/