Date: 2022-03-04T17:39:58+00:00

Actual source code: parabolic.c

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

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

     20: typedef struct {
     21:   PetscInt n; /*  Number of variables */
     22:   PetscInt m; /*  Number of constraints per time step */
     23:   PetscInt mx; /*  grid points in each direction */
     24:   PetscInt nt; /*  Number of time steps; as of now, must be divisible by 8 */
     25:   PetscInt ndata; /*  Number of data points per sample */
     26:   PetscInt ns; /*  Number of samples */
     27:   PetscInt *sample_times; /*  Times of samples */
     28:   IS       s_is;
     29:   IS       d_is;

     31:   VecScatter state_scatter;
     32:   VecScatter design_scatter;
     33:   VecScatter *yi_scatter;
     34:   VecScatter *di_scatter;

     36:   Mat       Js,Jd,JsBlockPrec,JsInv,JsBlock;
     37:   PetscBool jformed,dsg_formed;

     39:   PetscReal alpha; /*  Regularization parameter */
     40:   PetscReal beta; /*  Weight attributed to ||u||^2 in regularization functional */
     41:   PetscReal noise; /*  Amount of noise to add to data */
     42:   PetscReal ht; /*  Time step */

     44:   Mat Qblock,QblockT;
     45:   Mat L,LT;
     46:   Mat Div,Divwork;
     47:   Mat Grad;
     48:   Mat Av,Avwork,AvT;
     49:   Mat DSG;
     50:   Vec q;
     51:   Vec ur; /*  reference */

     53:   Vec d;
     54:   Vec dwork;
     55:   Vec *di;

     57:   Vec y; /*  state variables */
     58:   Vec ywork;

     60:   Vec ytrue;
     61:   Vec *yi,*yiwork;

     63:   Vec u; /*  design variables */
     64:   Vec uwork;

     66:   Vec utrue;
     67:   Vec js_diag;
     68:   Vec c; /*  constraint vector */
     69:   Vec cwork;

     71:   Vec lwork;
     72:   Vec S;
     73:   Vec Rwork,Swork,Twork;
     74:   Vec Av_u;

     76:   KSP solver;
     77:   PC prec;

     79:   PetscInt ksp_its;
     80:   PetscInt ksp_its_initial;
     81: } AppCtx;

     83: PetscErrorCode FormFunction(Tao, Vec, PetscReal*, void*);
     84: PetscErrorCode FormGradient(Tao, Vec, Vec, void*);
     85: PetscErrorCode FormFunctionGradient(Tao, Vec, PetscReal*, Vec, void*);
     86: PetscErrorCode FormJacobianState(Tao, Vec, Mat, Mat, Mat, void*);
     87: PetscErrorCode FormJacobianDesign(Tao, Vec, Mat, void*);
     88: PetscErrorCode FormConstraints(Tao, Vec, Vec, void*);
     89: PetscErrorCode FormHessian(Tao, Vec, Mat, Mat, void*);
     90: PetscErrorCode Gather(Vec x, Vec state, VecScatter s_scat, Vec design, VecScatter d_scat);
     91: PetscErrorCode Scatter(Vec x, Vec state, VecScatter s_scat, Vec design, VecScatter d_scat);
     92: PetscErrorCode ParabolicInitialize(AppCtx *user);
     93: PetscErrorCode ParabolicDestroy(AppCtx *user);
     94: PetscErrorCode ParabolicMonitor(Tao, void*);

     96: PetscErrorCode StateMatMult(Mat,Vec,Vec);
     97: PetscErrorCode StateMatBlockMult(Mat,Vec,Vec);
     98: PetscErrorCode StateMatMultTranspose(Mat,Vec,Vec);
     99: PetscErrorCode StateMatGetDiagonal(Mat,Vec);
    100: PetscErrorCode StateMatDuplicate(Mat,MatDuplicateOption,Mat*);
    101: PetscErrorCode StateMatInvMult(Mat,Vec,Vec);
    102: PetscErrorCode StateMatInvTransposeMult(Mat,Vec,Vec);
    103: PetscErrorCode StateMatBlockPrecMult(PC,Vec,Vec);

    105: PetscErrorCode DesignMatMult(Mat,Vec,Vec);
    106: PetscErrorCode DesignMatMultTranspose(Mat,Vec,Vec);

    108: PetscErrorCode Gather_i(Vec,Vec*,VecScatter*,PetscInt);
    109: PetscErrorCode Scatter_i(Vec,Vec*,VecScatter*,PetscInt);

    111: static  char help[]="";

    113: int main(int argc, char **argv)
    114: {
    115:   PetscErrorCode     ierr;
    116:   Vec                x,x0;
    117:   Tao                tao;
    118:   AppCtx             user;
    119:   IS                 is_allstate,is_alldesign;
    120:   PetscInt           lo,hi,hi2,lo2,ksp_old;
    121:   PetscInt           ntests = 1;
    122:   PetscInt           i;
    123: #if defined(PETSC_USE_LOG)
    124:   PetscLogStage      stages[1];
    125: #endif

    127:   PetscInitialize(&argc, &argv, (char*)0,help);if (ierr) return ierr;
    128:   user.mx = 8;
    129:   PetscOptionsBegin(PETSC_COMM_WORLD,NULL,"parabolic example",NULL);
    130:   PetscOptionsInt("-mx","Number of grid points in each direction","",user.mx,&user.mx,NULL);
    131:   user.nt = 8;
    132:   PetscOptionsInt("-nt","Number of time steps","",user.nt,&user.nt,NULL);
    133:   user.ndata = 64;
    134:   PetscOptionsInt("-ndata","Numbers of data points per sample","",user.ndata,&user.ndata,NULL);
    135:   user.ns = 8;
    136:   PetscOptionsInt("-ns","Number of samples","",user.ns,&user.ns,NULL);
    137:   user.alpha = 1.0;
    138:   PetscOptionsReal("-alpha","Regularization parameter","",user.alpha,&user.alpha,NULL);
    139:   user.beta = 0.01;
    140:   PetscOptionsReal("-beta","Weight attributed to ||u||^2 in regularization functional","",user.beta,&user.beta,NULL);
    141:   user.noise = 0.01;
    142:   PetscOptionsReal("-noise","Amount of noise to add to data","",user.noise,&user.noise,NULL);
    143:   PetscOptionsInt("-ntests","Number of times to repeat TaoSolve","",ntests,&ntests,NULL);
    144:   PetscOptionsEnd();

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

    150:   VecCreate(PETSC_COMM_WORLD,&user.u);
    151:   VecCreate(PETSC_COMM_WORLD,&user.y);
    152:   VecCreate(PETSC_COMM_WORLD,&user.c);
    153:   VecSetSizes(user.u,PETSC_DECIDE,user.n-user.m*user.nt);
    154:   VecSetSizes(user.y,PETSC_DECIDE,user.m*user.nt);
    155:   VecSetSizes(user.c,PETSC_DECIDE,user.m*user.nt);
    156:   VecSetFromOptions(user.u);
    157:   VecSetFromOptions(user.y);
    158:   VecSetFromOptions(user.c);

    160:   /* Create scatters for reduced spaces.
    161:      If the state vector y and design vector u are partitioned as
    162:      [y_1; y_2; ...; y_np] and [u_1; u_2; ...; u_np] (with np = # of processors),
    163:      then the solution vector x is organized as
    164:      [y_1; u_1; y_2; u_2; ...; y_np; u_np].
    165:      The index sets user.s_is and user.d_is correspond to the indices of the
    166:      state and design variables owned by the current processor.
    167:   */
    168:   VecCreate(PETSC_COMM_WORLD,&x);

    170:   VecGetOwnershipRange(user.y,&lo,&hi);
    171:   VecGetOwnershipRange(user.u,&lo2,&hi2);

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

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

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

    182:   VecScatterCreate(x,user.s_is,user.y,is_allstate,&user.state_scatter);
    183:   VecScatterCreate(x,user.d_is,user.u,is_alldesign,&user.design_scatter);
    184:   ISDestroy(&is_alldesign);
    185:   ISDestroy(&is_allstate);

    187:   /* Create TAO solver and set desired solution method */
    188:   TaoCreate(PETSC_COMM_WORLD,&tao);
    189:   TaoSetType(tao,TAOLCL);

    191:   /* Set up initial vectors and matrices */
    192:   ParabolicInitialize(&user);

    194:   Gather(x,user.y,user.state_scatter,user.u,user.design_scatter);
    195:   VecDuplicate(x,&x0);
    196:   VecCopy(x,x0);

    198:   /* Set solution vector with an initial guess */
    199:   TaoSetInitialVector(tao,x);
    200:   TaoSetObjectiveRoutine(tao, FormFunction, &user);
    201:   TaoSetGradientRoutine(tao, FormGradient, &user);
    202:   TaoSetConstraintsRoutine(tao, user.c, FormConstraints, &user);

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

    207:   TaoSetFromOptions(tao);
    208:   TaoSetStateDesignIS(tao,user.s_is,user.d_is);

    210:  /* SOLVE THE APPLICATION */
    211:   PetscLogStageRegister("Trials",&stages[0]);
    212:   PetscLogStagePush(stages[0]);
    213:   user.ksp_its_initial = user.ksp_its;
    214:   ksp_old = user.ksp_its;
    215:   for (i=0; i<ntests; i++) {
    216:     TaoSolve(tao);
    217:     PetscPrintf(PETSC_COMM_WORLD,"KSP Iterations = %D\n",user.ksp_its-ksp_old);
    218:     VecCopy(x0,x);
    219:     TaoSetInitialVector(tao,x);
    220:   }
    221:   PetscLogStagePop();
    222:   PetscBarrier((PetscObject)x);
    223:   PetscPrintf(PETSC_COMM_WORLD,"KSP iterations within initialization: ");
    224:   PetscPrintf(PETSC_COMM_WORLD,"%D\n",user.ksp_its_initial);
    225:   PetscPrintf(PETSC_COMM_WORLD,"Total KSP iterations over %D trial(s): ",ntests);
    226:   PetscPrintf(PETSC_COMM_WORLD,"%D\n",user.ksp_its);
    227:   PetscPrintf(PETSC_COMM_WORLD,"KSP iterations per trial: ");
    228:   PetscPrintf(PETSC_COMM_WORLD,"%D\n",(user.ksp_its-user.ksp_its_initial)/ntests);

    230:   TaoDestroy(&tao);
    231:   VecDestroy(&x);
    232:   VecDestroy(&x0);
    233:   ParabolicDestroy(&user);

    235:   PetscFinalize();
    236:   return ierr;
    237: }
    238: /* ------------------------------------------------------------------- */
    239: /*
    240:    dwork = Qy - d
    241:    lwork = L*(u-ur)
    242:    f = 1/2 * (dwork.dork + alpha*lwork.lwork)
    243: */
    244: PetscErrorCode FormFunction(Tao tao,Vec X,PetscReal *f,void *ptr)
    245: {
    247:   PetscReal      d1=0,d2=0;
    248:   PetscInt       i,j;
    249:   AppCtx         *user = (AppCtx*)ptr;

    252:   Scatter(X,user->y,user->state_scatter,user->u,user->design_scatter);
    253:   Scatter_i(user->y,user->yi,user->yi_scatter,user->nt);
    254:   for (j=0; j<user->ns; j++) {
    255:     i = user->sample_times[j];
    256:     MatMult(user->Qblock,user->yi[i],user->di[j]);
    257:   }
    258:   Gather_i(user->dwork,user->di,user->di_scatter,user->ns);
    259:   VecAXPY(user->dwork,-1.0,user->d);
    260:   VecDot(user->dwork,user->dwork,&d1);

    262:   VecWAXPY(user->uwork,-1.0,user->ur,user->u);
    263:   MatMult(user->L,user->uwork,user->lwork);
    264:   VecDot(user->lwork,user->lwork,&d2);

    266:   *f = 0.5 * (d1 + user->alpha*d2);
    267:   return(0);
    268: }

    270: /* ------------------------------------------------------------------- */
    271: /*
    272:     state: g_s = Q' *(Qy - d)
    273:     design: g_d = alpha*L'*L*(u-ur)
    274: */
    275: PetscErrorCode FormGradient(Tao tao,Vec X,Vec G,void *ptr)
    276: {
    278:   PetscInt       i,j;
    279:   AppCtx         *user = (AppCtx*)ptr;

    282:   Scatter(X,user->y,user->state_scatter,user->u,user->design_scatter);
    283:   Scatter_i(user->y,user->yi,user->yi_scatter,user->nt);
    284:   for (j=0; j<user->ns; j++) {
    285:     i = user->sample_times[j];
    286:     MatMult(user->Qblock,user->yi[i],user->di[j]);
    287:   }
    288:   Gather_i(user->dwork,user->di,user->di_scatter,user->ns);
    289:   VecAXPY(user->dwork,-1.0,user->d);
    290:   Scatter_i(user->dwork,user->di,user->di_scatter,user->ns);
    291:   VecSet(user->ywork,0.0);
    292:   Scatter_i(user->ywork,user->yiwork,user->yi_scatter,user->nt);
    293:   for (j=0; j<user->ns; j++) {
    294:     i = user->sample_times[j];
    295:     MatMult(user->QblockT,user->di[j],user->yiwork[i]);
    296:   }
    297:   Gather_i(user->ywork,user->yiwork,user->yi_scatter,user->nt);

    299:   VecWAXPY(user->uwork,-1.0,user->ur,user->u);
    300:   MatMult(user->L,user->uwork,user->lwork);
    301:   MatMult(user->LT,user->lwork,user->uwork);
    302:   VecScale(user->uwork, user->alpha);
    303:   Gather(G,user->ywork,user->state_scatter,user->uwork,user->design_scatter);
    304:   return(0);
    305: }

    307: PetscErrorCode FormFunctionGradient(Tao tao, Vec X, PetscReal *f, Vec G, void *ptr)
    308: {
    310:   PetscReal      d1,d2;
    311:   PetscInt       i,j;
    312:   AppCtx         *user = (AppCtx*)ptr;

    315:   Scatter(X,user->y,user->state_scatter,user->u,user->design_scatter);
    316:   Scatter_i(user->y,user->yi,user->yi_scatter,user->nt);
    317:   for (j=0; j<user->ns; j++) {
    318:     i = user->sample_times[j];
    319:     MatMult(user->Qblock,user->yi[i],user->di[j]);
    320:   }
    321:   Gather_i(user->dwork,user->di,user->di_scatter,user->ns);
    322:   VecAXPY(user->dwork,-1.0,user->d);
    323:   VecDot(user->dwork,user->dwork,&d1);
    324:   Scatter_i(user->dwork,user->di,user->di_scatter,user->ns);
    325:   VecSet(user->ywork,0.0);
    326:   Scatter_i(user->ywork,user->yiwork,user->yi_scatter,user->nt);
    327:   for (j=0; j<user->ns; j++) {
    328:     i = user->sample_times[j];
    329:     MatMult(user->QblockT,user->di[j],user->yiwork[i]);
    330:   }
    331:   Gather_i(user->ywork,user->yiwork,user->yi_scatter,user->nt);

    333:   VecWAXPY(user->uwork,-1.0,user->ur,user->u);
    334:   MatMult(user->L,user->uwork,user->lwork);
    335:   VecDot(user->lwork,user->lwork,&d2);
    336:   MatMult(user->LT,user->lwork,user->uwork);
    337:   VecScale(user->uwork, user->alpha);
    338:   *f = 0.5 * (d1 + user->alpha*d2);

    340:   Gather(G,user->ywork,user->state_scatter,user->uwork,user->design_scatter);
    341:   return(0);
    342: }

    344: /* ------------------------------------------------------------------- */
    345: /* A
    346: MatShell object
    347: */
    348: PetscErrorCode FormJacobianState(Tao tao, Vec X, Mat J, Mat JPre, Mat JInv, void *ptr)
    349: {
    351:   AppCtx         *user = (AppCtx*)ptr;

    354:   Scatter(X,user->y,user->state_scatter,user->u,user->design_scatter);
    355:   VecSet(user->uwork,0);
    356:   VecAXPY(user->uwork,-1.0,user->u);
    357:   VecExp(user->uwork);
    358:   MatMult(user->Av,user->uwork,user->Av_u);
    359:   VecCopy(user->Av_u,user->Swork);
    360:   VecReciprocal(user->Swork);
    361:   MatCopy(user->Div,user->Divwork,SAME_NONZERO_PATTERN);
    362:   MatDiagonalScale(user->Divwork,NULL,user->Swork);
    363:   if (user->dsg_formed) {
    364:     MatProductNumeric(user->DSG);
    365:   } else {
    366:     MatMatMult(user->Divwork,user->Grad,MAT_INITIAL_MATRIX,PETSC_DEFAULT,&user->DSG);
    367:     user->dsg_formed = PETSC_TRUE;
    368:   }

    370:   /* B = speye(nx^3) + ht*DSG; */
    371:   MatScale(user->DSG,user->ht);
    372:   MatShift(user->DSG,1.0);
    373:   return(0);
    374: }

    376: /* ------------------------------------------------------------------- */
    377: /* B */
    378: PetscErrorCode FormJacobianDesign(Tao tao, Vec X, Mat J, void *ptr)
    379: {
    381:   AppCtx         *user = (AppCtx*)ptr;

    384:   Scatter(X,user->y,user->state_scatter,user->u,user->design_scatter);
    385:   return(0);
    386: }

    388: PetscErrorCode StateMatMult(Mat J_shell, Vec X, Vec Y)
    389: {
    391:   PetscInt       i;
    392:   AppCtx         *user;

    395:   MatShellGetContext(J_shell,&user);
    396:   Scatter_i(X,user->yi,user->yi_scatter,user->nt);
    397:   MatMult(user->JsBlock,user->yi[0],user->yiwork[0]);
    398:   for (i=1; i<user->nt; i++) {
    399:     MatMult(user->JsBlock,user->yi[i],user->yiwork[i]);
    400:     VecAXPY(user->yiwork[i],-1.0,user->yi[i-1]);
    401:   }
    402:   Gather_i(Y,user->yiwork,user->yi_scatter,user->nt);
    403:   return(0);
    404: }

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

    413:   MatShellGetContext(J_shell,&user);
    414:   Scatter_i(X,user->yi,user->yi_scatter,user->nt);
    415:   for (i=0; i<user->nt-1; i++) {
    416:     MatMult(user->JsBlock,user->yi[i],user->yiwork[i]);
    417:     VecAXPY(user->yiwork[i],-1.0,user->yi[i+1]);
    418:   }
    419:   i = user->nt-1;
    420:   MatMult(user->JsBlock,user->yi[i],user->yiwork[i]);
    421:   Gather_i(Y,user->yiwork,user->yi_scatter,user->nt);
    422:   return(0);
    423: }

    425: PetscErrorCode StateMatBlockMult(Mat J_shell, Vec X, Vec Y)
    426: {
    428:   AppCtx         *user;

    431:   MatShellGetContext(J_shell,&user);
    432:   MatMult(user->Grad,X,user->Swork);
    433:   VecPointwiseDivide(user->Swork,user->Swork,user->Av_u);
    434:   MatMult(user->Div,user->Swork,Y);
    435:   VecAYPX(Y,user->ht,X);
    436:   return(0);
    437: }

    439: PetscErrorCode DesignMatMult(Mat J_shell, Vec X, Vec Y)
    440: {
    442:   PetscInt       i;
    443:   AppCtx         *user;

    446:   MatShellGetContext(J_shell,&user);

    448:   /* sdiag(1./v) */
    449:   VecSet(user->uwork,0);
    450:   VecAXPY(user->uwork,-1.0,user->u);
    451:   VecExp(user->uwork);

    453:   /* sdiag(1./((Av*(1./v)).^2)) */
    454:   MatMult(user->Av,user->uwork,user->Swork);
    455:   VecPointwiseMult(user->Swork,user->Swork,user->Swork);
    456:   VecReciprocal(user->Swork);

    458:   /* (Av * (sdiag(1./v) * b)) */
    459:   VecPointwiseMult(user->uwork,user->uwork,X);
    460:   MatMult(user->Av,user->uwork,user->Twork);

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

    465:   Scatter_i(user->y,user->yi,user->yi_scatter,user->nt);
    466:   for (i=0; i<user->nt; i++) {
    467:     /* (sdiag(Grad*y(:,i)) */
    468:     MatMult(user->Grad,user->yi[i],user->Twork);

    470:     /* ht * Div * (sdiag(Grad*y(:,i)) * (sdiag(1./((Av*(1./v)).^2)) * (Av * (sdiag(1./v) * b)))) */
    471:     VecPointwiseMult(user->Twork,user->Twork,user->Swork);
    472:     MatMult(user->Div,user->Twork,user->yiwork[i]);
    473:     VecScale(user->yiwork[i],user->ht);
    474:   }
    475:   Gather_i(Y,user->yiwork,user->yi_scatter,user->nt);

    477:   return(0);
    478: }

    480: PetscErrorCode DesignMatMultTranspose(Mat J_shell, Vec X, Vec Y)
    481: {
    483:   PetscInt       i;
    484:   AppCtx         *user;

    487:   MatShellGetContext(J_shell,&user);

    489:   /* sdiag(1./((Av*(1./v)).^2)) */
    490:   VecSet(user->uwork,0);
    491:   VecAXPY(user->uwork,-1.0,user->u);
    492:   VecExp(user->uwork);
    493:   MatMult(user->Av,user->uwork,user->Rwork);
    494:   VecPointwiseMult(user->Rwork,user->Rwork,user->Rwork);
    495:   VecReciprocal(user->Rwork);

    497:   VecSet(Y,0.0);
    498:   Scatter_i(user->y,user->yi,user->yi_scatter,user->nt);
    499:   Scatter_i(X,user->yiwork,user->yi_scatter,user->nt);
    500:   for (i=0; i<user->nt; i++) {
    501:     /* (Div' * b(:,i)) */
    502:     MatMult(user->Grad,user->yiwork[i],user->Swork);

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

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

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

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

    516:     /* sdiag(1./v) * (Av' * (sdiag(1./((Av*(1./v)).^2)) * (sdiag(Grad*y(:,i)) * (Div' * b(:,i))))) */
    517:     VecPointwiseMult(user->yiwork[i],user->uwork,user->yiwork[i]);
    518:     VecAXPY(Y,user->ht,user->yiwork[i]);
    519:   }
    520:   return(0);
    521: }

    523: PetscErrorCode StateMatBlockPrecMult(PC PC_shell, Vec X, Vec Y)
    524: {
    526:   AppCtx         *user;

    529:   PCShellGetContext(PC_shell,&user);

    531:   if (user->dsg_formed) {
    532:     MatSOR(user->DSG,X,1.0,(MatSORType)(SOR_ZERO_INITIAL_GUESS | SOR_LOCAL_SYMMETRIC_SWEEP),0.0,1,1,Y);
    533:   } else SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"DSG not formed");
    534:   return(0);
    535: }

    537: PetscErrorCode StateMatInvMult(Mat J_shell, Vec X, Vec Y)
    538: {
    540:   AppCtx         *user;
    541:   PetscInt       its,i;

    544:   MatShellGetContext(J_shell,&user);

    546:   if (Y == user->ytrue) {
    547:     /* First solve is done with true solution to set up problem */
    548:     KSPSetTolerances(user->solver,1e-8,PETSC_DEFAULT,PETSC_DEFAULT,PETSC_DEFAULT);
    549:   } else {
    550:     KSPSetTolerances(user->solver,PETSC_DEFAULT,PETSC_DEFAULT,PETSC_DEFAULT,PETSC_DEFAULT);
    551:   }

    553:   Scatter_i(X,user->yi,user->yi_scatter,user->nt);
    554:   KSPSolve(user->solver,user->yi[0],user->yiwork[0]);
    555:   KSPGetIterationNumber(user->solver,&its);
    556:   user->ksp_its = user->ksp_its + its;

    558:   for (i=1; i<user->nt; i++) {
    559:     VecAXPY(user->yi[i],1.0,user->yiwork[i-1]);
    560:     KSPSolve(user->solver,user->yi[i],user->yiwork[i]);
    561:     KSPGetIterationNumber(user->solver,&its);
    562:     user->ksp_its = user->ksp_its + its;
    563:   }
    564:   Gather_i(Y,user->yiwork,user->yi_scatter,user->nt);
    565:   return(0);
    566: }

    568: PetscErrorCode StateMatInvTransposeMult(Mat J_shell, Vec X, Vec Y)
    569: {
    571:   AppCtx         *user;
    572:   PetscInt       its,i;

    575:   MatShellGetContext(J_shell,&user);

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

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

    582:   KSPGetIterationNumber(user->solver,&its);
    583:   user->ksp_its = user->ksp_its + its;

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

    589:     KSPGetIterationNumber(user->solver,&its);
    590:     user->ksp_its = user->ksp_its + its;

    592:   }

    594:   Gather_i(Y,user->yiwork,user->yi_scatter,user->nt);
    595:   return(0);
    596: }

    598: PetscErrorCode StateMatDuplicate(Mat J_shell, MatDuplicateOption opt, Mat *new_shell)
    599: {
    601:   AppCtx         *user;

    604:   MatShellGetContext(J_shell,&user);

    606:   MatCreateShell(PETSC_COMM_WORLD,PETSC_DETERMINE,PETSC_DETERMINE,user->m,user->m,user,new_shell);
    607:   MatShellSetOperation(*new_shell,MATOP_MULT,(void(*)(void))StateMatMult);
    608:   MatShellSetOperation(*new_shell,MATOP_DUPLICATE,(void(*)(void))StateMatDuplicate);
    609:   MatShellSetOperation(*new_shell,MATOP_MULT_TRANSPOSE,(void(*)(void))StateMatMultTranspose);
    610:   MatShellSetOperation(*new_shell,MATOP_GET_DIAGONAL,(void(*)(void))StateMatGetDiagonal);
    611:   return(0);
    612: }

    614: PetscErrorCode StateMatGetDiagonal(Mat J_shell, Vec X)
    615: {
    617:   AppCtx         *user;

    620:   MatShellGetContext(J_shell,&user);
    621:    VecCopy(user->js_diag,X);
    622:   return(0);

    624: }

    626: PetscErrorCode FormConstraints(Tao tao, Vec X, Vec C, void *ptr)
    627: {
    628:   /* con = Ay - q, A = [B  0  0 ... 0;
    629:                        -I  B  0 ... 0;
    630:                         0 -I  B ... 0;
    631:                              ...     ;
    632:                         0    ... -I B]
    633:      B = ht * Div * Sigma * Grad + eye */
    635:   PetscInt       i;
    636:   AppCtx         *user = (AppCtx*)ptr;

    639:   Scatter(X,user->y,user->state_scatter,user->u,user->design_scatter);
    640:   Scatter_i(user->y,user->yi,user->yi_scatter,user->nt);
    641:   MatMult(user->JsBlock,user->yi[0],user->yiwork[0]);
    642:   for (i=1; i<user->nt; i++) {
    643:     MatMult(user->JsBlock,user->yi[i],user->yiwork[i]);
    644:     VecAXPY(user->yiwork[i],-1.0,user->yi[i-1]);
    645:   }
    646:   Gather_i(C,user->yiwork,user->yi_scatter,user->nt);
    647:   VecAXPY(C,-1.0,user->q);
    648:   return(0);
    649: }

    651: PetscErrorCode Scatter(Vec x, Vec state, VecScatter s_scat, Vec design, VecScatter d_scat)
    652: {

    656:   VecScatterBegin(s_scat,x,state,INSERT_VALUES,SCATTER_FORWARD);
    657:   VecScatterEnd(s_scat,x,state,INSERT_VALUES,SCATTER_FORWARD);
    658:   VecScatterBegin(d_scat,x,design,INSERT_VALUES,SCATTER_FORWARD);
    659:   VecScatterEnd(d_scat,x,design,INSERT_VALUES,SCATTER_FORWARD);
    660:   return(0);
    661: }

    663: PetscErrorCode Scatter_i(Vec y, Vec *yi, VecScatter *scat, PetscInt nt)
    664: {
    666:   PetscInt       i;

    669:   for (i=0; i<nt; i++) {
    670:     VecScatterBegin(scat[i],y,yi[i],INSERT_VALUES,SCATTER_FORWARD);
    671:     VecScatterEnd(scat[i],y,yi[i],INSERT_VALUES,SCATTER_FORWARD);
    672:   }
    673:   return(0);
    674: }

    676: PetscErrorCode Gather(Vec x, Vec state, VecScatter s_scat, Vec design, VecScatter d_scat)
    677: {

    681:   VecScatterBegin(s_scat,state,x,INSERT_VALUES,SCATTER_REVERSE);
    682:   VecScatterEnd(s_scat,state,x,INSERT_VALUES,SCATTER_REVERSE);
    683:   VecScatterBegin(d_scat,design,x,INSERT_VALUES,SCATTER_REVERSE);
    684:   VecScatterEnd(d_scat,design,x,INSERT_VALUES,SCATTER_REVERSE);
    685:   return(0);
    686: }

    688: PetscErrorCode Gather_i(Vec y, Vec *yi, VecScatter *scat, PetscInt nt)
    689: {
    691:   PetscInt       i;

    694:   for (i=0; i<nt; i++) {
    695:     VecScatterBegin(scat[i],yi[i],y,INSERT_VALUES,SCATTER_REVERSE);
    696:     VecScatterEnd(scat[i],yi[i],y,INSERT_VALUES,SCATTER_REVERSE);
    697:   }
    698:   return(0);
    699: }

    701: PetscErrorCode ParabolicInitialize(AppCtx *user)
    702: {
    704:   PetscInt       m,n,i,j,k,linear_index,istart,iend,iblock,lo,hi,lo2,hi2;
    705:   Vec            XX,YY,ZZ,XXwork,YYwork,ZZwork,UTwork,yi,di,bc;
    706:   PetscReal      *x, *y, *z;
    707:   PetscReal      h,stime;
    708:   PetscScalar    hinv,neg_hinv,half = 0.5,sqrt_beta;
    709:   PetscInt       im,indx1,indx2,indy1,indy2,indz1,indz2,nx,ny,nz;
    710:   PetscReal      xri,yri,zri,xim,yim,zim,dx1,dx2,dy1,dy2,dz1,dz2,Dx,Dy,Dz;
    711:   PetscScalar    v,vx,vy,vz;
    712:   IS             is_from_y,is_to_yi,is_from_d,is_to_di;
    713:   /* Data locations */
    714:   PetscScalar xr[64] = {0.4970,     0.8498,     0.7814,     0.6268,     0.7782,     0.6402,     0.3617,     0.3160,
    715:                         0.3610,     0.5298,     0.6987,     0.3331,     0.7962,     0.5596,     0.3866,     0.6774,
    716:                         0.5407,     0.4518,     0.6702,     0.6061,     0.7580,     0.8997,     0.5198,     0.8326,
    717:                         0.2138,     0.9198,     0.3000,     0.2833,     0.8288,     0.7076,     0.1820,     0.0728,
    718:                         0.8447,     0.2367,     0.3239,     0.6413,     0.3114,     0.4731,     0.1192,     0.9273,
    719:                         0.5724,     0.4331,     0.5136,     0.3547,     0.4413,     0.2602,     0.5698,     0.7278,
    720:                         0.5261,     0.6230,     0.2454,     0.3948,     0.7479,     0.6582,     0.4660,     0.5594,
    721:                         0.7574,     0.1143,     0.5900,     0.1065,     0.4260,     0.3294,     0.8276,     0.0756};

    723:   PetscScalar yr[64] = {0.7345,     0.9120,     0.9288,     0.7528,     0.4463,     0.4985,     0.2497,     0.6256,
    724:                         0.3425,     0.9026,     0.6983,     0.4230,     0.7140,     0.2970,     0.4474,     0.8792,
    725:                         0.6604,     0.2485,     0.7968,     0.6127,     0.1796,     0.2437,     0.5938,     0.6137,
    726:                         0.3867,     0.5658,     0.4575,     0.1009,     0.0863,     0.3361,     0.0738,     0.3985,
    727:                         0.6602,     0.1437,     0.0934,     0.5983,     0.5950,     0.0763,     0.0768,     0.2288,
    728:                         0.5761,     0.1129,     0.3841,     0.6150,     0.6904,     0.6686,     0.1361,     0.4601,
    729:                         0.4491,     0.3716,     0.1969,     0.6537,     0.6743,     0.6991,     0.4811,     0.5480,
    730:                         0.1684,     0.4569,     0.6889,     0.8437,     0.3015,     0.2854,     0.8199,     0.2658};

    732:   PetscScalar zr[64] = {0.7668,     0.8573,     0.2654,     0.2719,     0.1060,     0.1311,     0.6232,     0.2295,
    733:                         0.8009,     0.2147,     0.2119,     0.9325,     0.4473,     0.3600,     0.3374,     0.3819,
    734:                         0.4066,     0.5801,     0.1673,     0.0959,     0.4638,     0.8236,     0.8800,     0.2939,
    735:                         0.2028,     0.8262,     0.2706,     0.6276,     0.9085,     0.6443,     0.8241,     0.0712,
    736:                         0.1824,     0.7789,     0.4389,     0.8415,     0.7055,     0.6639,     0.3653,     0.2078,
    737:                         0.1987,     0.2297,     0.4321,     0.8115,     0.4915,     0.7764,     0.4657,     0.4627,
    738:                         0.4569,     0.4232,     0.8514,     0.0674,     0.3227,     0.1055,     0.6690,     0.6313,
    739:                         0.9226,     0.5461,     0.4126,     0.2364,     0.6096,     0.7042,     0.3914,     0.0711};

    742:   PetscMalloc1(user->mx,&x);
    743:   PetscMalloc1(user->mx,&y);
    744:   PetscMalloc1(user->mx,&z);
    745:   user->jformed = PETSC_FALSE;
    746:   user->dsg_formed = PETSC_FALSE;

    748:   n = user->mx * user->mx * user->mx;
    749:   m = 3 * user->mx * user->mx * (user->mx-1);
    750:   sqrt_beta = PetscSqrtScalar(user->beta);

    752:   user->ksp_its = 0;
    753:   user->ksp_its_initial = 0;

    755:   stime = (PetscReal)user->nt/user->ns;
    756:   PetscMalloc1(user->ns,&user->sample_times);
    757:   for (i=0; i<user->ns; i++) {
    758:     user->sample_times[i] = (PetscInt)(stime*((PetscReal)i+1.0)-0.5);
    759:   }

    761:   VecCreate(PETSC_COMM_WORLD,&XX);
    762:   VecCreate(PETSC_COMM_WORLD,&user->q);
    763:   VecSetSizes(XX,PETSC_DECIDE,n);
    764:   VecSetSizes(user->q,PETSC_DECIDE,n*user->nt);
    765:   VecSetFromOptions(XX);
    766:   VecSetFromOptions(user->q);

    768:   VecDuplicate(XX,&YY);
    769:   VecDuplicate(XX,&ZZ);
    770:   VecDuplicate(XX,&XXwork);
    771:   VecDuplicate(XX,&YYwork);
    772:   VecDuplicate(XX,&ZZwork);
    773:   VecDuplicate(XX,&UTwork);
    774:   VecDuplicate(XX,&user->utrue);
    775:   VecDuplicate(XX,&bc);

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

    782:   VecGetOwnershipRange(XX,&istart,&iend);
    783:   for (linear_index=istart; linear_index<iend; linear_index++) {
    784:     i = linear_index % user->mx;
    785:     j = ((linear_index-i)/user->mx) % user->mx;
    786:     k = ((linear_index-i)/user->mx-j) / user->mx;
    787:     vx = h*(i+0.5);
    788:     vy = h*(j+0.5);
    789:     vz = h*(k+0.5);
    790:     VecSetValues(XX,1,&linear_index,&vx,INSERT_VALUES);
    791:     VecSetValues(YY,1,&linear_index,&vy,INSERT_VALUES);
    792:     VecSetValues(ZZ,1,&linear_index,&vz,INSERT_VALUES);
    793:     if ((vx<0.6) && (vx>0.4) && (vy<0.6) && (vy>0.4) && (vy<0.6) && (vz<0.6) && (vz>0.4)) {
    794:       v = 1000.0;
    795:       VecSetValues(bc,1,&linear_index,&v,INSERT_VALUES);
    796:     }
    797:   }

    799:   VecAssemblyBegin(XX);
    800:   VecAssemblyEnd(XX);
    801:   VecAssemblyBegin(YY);
    802:   VecAssemblyEnd(YY);
    803:   VecAssemblyBegin(ZZ);
    804:   VecAssemblyEnd(ZZ);
    805:   VecAssemblyBegin(bc);
    806:   VecAssemblyEnd(bc);

    808:   /* Compute true parameter function
    809:      ut = 0.5 + exp(-10*((x-0.5)^2+(y-0.5)^2+(z-0.5)^2)) */
    810:   VecCopy(XX,XXwork);
    811:   VecCopy(YY,YYwork);
    812:   VecCopy(ZZ,ZZwork);

    814:   VecShift(XXwork,-0.5);
    815:   VecShift(YYwork,-0.5);
    816:   VecShift(ZZwork,-0.5);

    818:   VecPointwiseMult(XXwork,XXwork,XXwork);
    819:   VecPointwiseMult(YYwork,YYwork,YYwork);
    820:   VecPointwiseMult(ZZwork,ZZwork,ZZwork);

    822:   VecCopy(XXwork,user->utrue);
    823:   VecAXPY(user->utrue,1.0,YYwork);
    824:   VecAXPY(user->utrue,1.0,ZZwork);
    825:   VecScale(user->utrue,-10.0);
    826:   VecExp(user->utrue);
    827:   VecShift(user->utrue,0.5);

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

    837:   /* Initial guess and reference model */
    838:   VecDuplicate(user->utrue,&user->ur);
    839:   VecSet(user->ur,0.0);

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

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

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

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

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

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

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

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

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

    946:   MatAssemblyBegin(user->L,MAT_FINAL_ASSEMBLY);
    947:   MatAssemblyEnd(user->L,MAT_FINAL_ASSEMBLY);

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

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

    954:   /* Build work vectors and matrices */
    955:   VecCreate(PETSC_COMM_WORLD,&user->S);
    956:   VecSetSizes(user->S, PETSC_DECIDE, user->mx*user->mx*(user->mx-1)*3);
    957:   VecSetFromOptions(user->S);

    959:   VecCreate(PETSC_COMM_WORLD,&user->lwork);
    960:   VecSetSizes(user->lwork,PETSC_DECIDE,m+user->mx*user->mx*user->mx);
    961:   VecSetFromOptions(user->lwork);

    963:   MatDuplicate(user->Div,MAT_SHARE_NONZERO_PATTERN,&user->Divwork);
    964:   MatDuplicate(user->Av,MAT_SHARE_NONZERO_PATTERN,&user->Avwork);

    966:   VecCreate(PETSC_COMM_WORLD,&user->d);
    967:   VecSetSizes(user->d,PETSC_DECIDE,user->ndata*user->nt);
    968:   VecSetFromOptions(user->d);

    970:   VecDuplicate(user->S,&user->Swork);
    971:   VecDuplicate(user->S,&user->Av_u);
    972:   VecDuplicate(user->S,&user->Twork);
    973:   VecDuplicate(user->S,&user->Rwork);
    974:   VecDuplicate(user->y,&user->ywork);
    975:   VecDuplicate(user->u,&user->uwork);
    976:   VecDuplicate(user->u,&user->js_diag);
    977:   VecDuplicate(user->c,&user->cwork);
    978:   VecDuplicate(user->d,&user->dwork);

    980:   /* Create matrix-free shell user->Js for computing A*x */
    981:   MatCreateShell(PETSC_COMM_WORLD,PETSC_DETERMINE,PETSC_DETERMINE,user->m*user->nt,user->m*user->nt,user,&user->Js);
    982:   MatShellSetOperation(user->Js,MATOP_MULT,(void(*)(void))StateMatMult);
    983:   MatShellSetOperation(user->Js,MATOP_DUPLICATE,(void(*)(void))StateMatDuplicate);
    984:   MatShellSetOperation(user->Js,MATOP_MULT_TRANSPOSE,(void(*)(void))StateMatMultTranspose);
    985:   MatShellSetOperation(user->Js,MATOP_GET_DIAGONAL,(void(*)(void))StateMatGetDiagonal);

    987:   /* Diagonal blocks of user->Js */
    988:   MatCreateShell(PETSC_COMM_WORLD,PETSC_DETERMINE,PETSC_DETERMINE,user->m,user->m,user,&user->JsBlock);
    989:   MatShellSetOperation(user->JsBlock,MATOP_MULT,(void(*)(void))StateMatBlockMult);
    990:   /* Blocks are symmetric */
    991:   MatShellSetOperation(user->JsBlock,MATOP_MULT_TRANSPOSE,(void(*)(void))StateMatBlockMult);

    993:   /* Create a matrix-free shell user->JsBlockPrec for computing (U+D)\D*(L+D)\x, where JsBlock = L+D+U,
    994:      D is diagonal, L is strictly lower triangular, and U is strictly upper triangular.
    995:      This is an SSOR preconditioner for user->JsBlock. */
    996:   MatCreateShell(PETSC_COMM_WORLD,PETSC_DETERMINE,PETSC_DETERMINE,user->m,user->m,user,&user->JsBlockPrec);
    997:   MatShellSetOperation(user->JsBlockPrec,MATOP_MULT,(void(*)(void))StateMatBlockPrecMult);
    998:   /* JsBlockPrec is symmetric */
    999:   MatShellSetOperation(user->JsBlockPrec,MATOP_MULT_TRANSPOSE,(void(*)(void))StateMatBlockPrecMult);
    1000:   MatSetOption(user->JsBlockPrec,MAT_SYMMETRY_ETERNAL,PETSC_TRUE);

    1002:   /* Create a matrix-free shell user->Jd for computing B*x */
    1003:   MatCreateShell(PETSC_COMM_WORLD,PETSC_DETERMINE,PETSC_DETERMINE,user->m*user->nt,user->m,user,&user->Jd);
    1004:   MatShellSetOperation(user->Jd,MATOP_MULT,(void(*)(void))DesignMatMult);
    1005:   MatShellSetOperation(user->Jd,MATOP_MULT_TRANSPOSE,(void(*)(void))DesignMatMultTranspose);

    1007:   /* User-defined routines for computing user->Js\x and user->Js^T\x*/
    1008:   MatCreateShell(PETSC_COMM_WORLD,PETSC_DETERMINE,PETSC_DETERMINE,user->m*user->nt,user->m*user->nt,user,&user->JsInv);
    1009:   MatShellSetOperation(user->JsInv,MATOP_MULT,(void(*)(void))StateMatInvMult);
    1010:   MatShellSetOperation(user->JsInv,MATOP_MULT_TRANSPOSE,(void(*)(void))StateMatInvTransposeMult);

    1012:   /* Solver options and tolerances */
    1013:   KSPCreate(PETSC_COMM_WORLD,&user->solver);
    1014:   KSPSetType(user->solver,KSPCG);
    1015:   KSPSetOperators(user->solver,user->JsBlock,user->JsBlockPrec);
    1016:   KSPSetInitialGuessNonzero(user->solver,PETSC_FALSE);
    1017:   KSPSetTolerances(user->solver,1e-4,1e-20,1e3,500);
    1018:   KSPSetFromOptions(user->solver);
    1019:   KSPGetPC(user->solver,&user->prec);
    1020:   PCSetType(user->prec,PCSHELL);

    1022:   PCShellSetApply(user->prec,StateMatBlockPrecMult);
    1023:   PCShellSetApplyTranspose(user->prec,StateMatBlockPrecMult);
    1024:   PCShellSetContext(user->prec,user);

    1026:   /* Create scatter from y to y_1,y_2,...,y_nt */
    1027:   PetscMalloc1(user->nt*user->m,&user->yi_scatter);
    1028:   VecCreate(PETSC_COMM_WORLD,&yi);
    1029:   VecSetSizes(yi,PETSC_DECIDE,user->mx*user->mx*user->mx);
    1030:   VecSetFromOptions(yi);
    1031:   VecDuplicateVecs(yi,user->nt,&user->yi);
    1032:   VecDuplicateVecs(yi,user->nt,&user->yiwork);

    1034:   VecGetOwnershipRange(user->y,&lo2,&hi2);
    1035:   istart = 0;
    1036:   for (i=0; i<user->nt; i++) {
    1037:     VecGetOwnershipRange(user->yi[i],&lo,&hi);
    1038:     ISCreateStride(PETSC_COMM_SELF,hi-lo,lo,1,&is_to_yi);
    1039:     ISCreateStride(PETSC_COMM_SELF,hi-lo,lo2+istart,1,&is_from_y);
    1040:     VecScatterCreate(user->y,is_from_y,user->yi[i],is_to_yi,&user->yi_scatter[i]);
    1041:     istart = istart + hi-lo;
    1042:     ISDestroy(&is_to_yi);
    1043:     ISDestroy(&is_from_y);
    1044:   }
    1045:   VecDestroy(&yi);

    1047:   /* Create scatter from d to d_1,d_2,...,d_ns */
    1048:   PetscMalloc1(user->ns*user->ndata,&user->di_scatter);
    1049:   VecCreate(PETSC_COMM_WORLD,&di);
    1050:   VecSetSizes(di,PETSC_DECIDE,user->ndata);
    1051:   VecSetFromOptions(di);
    1052:   VecDuplicateVecs(di,user->ns,&user->di);
    1053:   VecGetOwnershipRange(user->d,&lo2,&hi2);
    1054:   istart = 0;
    1055:   for (i=0; i<user->ns; i++) {
    1056:     VecGetOwnershipRange(user->di[i],&lo,&hi);
    1057:     ISCreateStride(PETSC_COMM_SELF,hi-lo,lo,1,&is_to_di);
    1058:     ISCreateStride(PETSC_COMM_SELF,hi-lo,lo2+istart,1,&is_from_d);
    1059:     VecScatterCreate(user->d,is_from_d,user->di[i],is_to_di,&user->di_scatter[i]);
    1060:     istart = istart + hi-lo;
    1061:     ISDestroy(&is_to_di);
    1062:     ISDestroy(&is_from_d);
    1063:   }
    1064:   VecDestroy(&di);

    1066:   /* Assemble RHS of forward problem */
    1067:   VecCopy(bc,user->yiwork[0]);
    1068:   for (i=1; i<user->nt; i++) {
    1069:     VecSet(user->yiwork[i],0.0);
    1070:   }
    1071:   Gather_i(user->q,user->yiwork,user->yi_scatter,user->nt);
    1072:   VecDestroy(&bc);

    1074:   /* Compute true state function yt given ut */
    1075:   VecCreate(PETSC_COMM_WORLD,&user->ytrue);
    1076:   VecSetSizes(user->ytrue,PETSC_DECIDE,n*user->nt);
    1077:   VecSetFromOptions(user->ytrue);

    1079:   /* First compute Av_u = Av*exp(-u) */
    1080:   VecSet(user->uwork,0);
    1081:   VecAXPY(user->uwork,-1.0,user->utrue); /*  Note: user->utrue */
    1082:   VecExp(user->uwork);
    1083:   MatMult(user->Av,user->uwork,user->Av_u);

    1085:   /* Symbolic DSG = Div * Grad */
    1086:   MatProductCreate(user->Div,user->Grad,NULL,&user->DSG);
    1087:   MatProductSetType(user->DSG,MATPRODUCT_AB);
    1088:   MatProductSetAlgorithm(user->DSG,"default");
    1089:   MatProductSetFill(user->DSG,PETSC_DEFAULT);
    1090:   MatProductSetFromOptions(user->DSG);
    1091:   MatProductSymbolic(user->DSG);

    1093:   user->dsg_formed = PETSC_TRUE;

    1095:   /* Next form DSG = Div*Grad */
    1096:   MatCopy(user->Div,user->Divwork,SAME_NONZERO_PATTERN);
    1097:   MatDiagonalScale(user->Divwork,NULL,user->Av_u);
    1098:   if (user->dsg_formed) {
    1099:     MatProductNumeric(user->DSG);
    1100:   } else {
    1101:     MatMatMult(user->Div,user->Grad,MAT_INITIAL_MATRIX,PETSC_DEFAULT,&user->DSG);
    1102:     user->dsg_formed = PETSC_TRUE;
    1103:   }
    1104:   /* B = speye(nx^3) + ht*DSG; */
    1105:   MatScale(user->DSG,user->ht);
    1106:   MatShift(user->DSG,1.0);

    1108:   /* Now solve for ytrue */
    1109:   StateMatInvMult(user->Js,user->q,user->ytrue);

    1111:   /* Initial guess y0 for state given u0 */

    1113:   /* First compute Av_u = Av*exp(-u) */
    1114:   VecSet(user->uwork,0);
    1115:   VecAXPY(user->uwork,-1.0,user->u); /*  Note: user->u */
    1116:   VecExp(user->uwork);
    1117:   MatMult(user->Av,user->uwork,user->Av_u);

    1119:   /* Next form DSG = Div*S*Grad */
    1120:   MatCopy(user->Div,user->Divwork,SAME_NONZERO_PATTERN);
    1121:   MatDiagonalScale(user->Divwork,NULL,user->Av_u);
    1122:   if (user->dsg_formed) {
    1123:     MatProductNumeric(user->DSG);
    1124:   } else {
    1125:     MatMatMult(user->Div,user->Grad,MAT_INITIAL_MATRIX,PETSC_DEFAULT,&user->DSG);

    1127:     user->dsg_formed = PETSC_TRUE;
    1128:   }
    1129:   /* B = speye(nx^3) + ht*DSG; */
    1130:   MatScale(user->DSG,user->ht);
    1131:   MatShift(user->DSG,1.0);

    1133:   /* Now solve for y */
    1134:   StateMatInvMult(user->Js,user->q,user->y);

    1136:   /* Construct projection matrix Q, a block diagonal matrix consisting of nt copies of Qblock along the diagonal */
    1137:   MatCreate(PETSC_COMM_WORLD,&user->Qblock);
    1138:   MatSetSizes(user->Qblock,PETSC_DECIDE,PETSC_DECIDE,user->ndata,n);
    1139:   MatSetFromOptions(user->Qblock);
    1140:   MatMPIAIJSetPreallocation(user->Qblock,8,NULL,8,NULL);
    1141:   MatSeqAIJSetPreallocation(user->Qblock,8,NULL);

    1143:   for (i=0; i<user->mx; i++) {
    1144:     x[i] = h*(i+0.5);
    1145:     y[i] = h*(i+0.5);
    1146:     z[i] = h*(i+0.5);
    1147:   }

    1149:   MatGetOwnershipRange(user->Qblock,&istart,&iend);
    1150:   nx = user->mx; ny = user->mx; nz = user->mx;
    1151:   for (i=istart; i<iend; i++) {
    1152:     xri = xr[i];
    1153:     im = 0;
    1154:     xim = x[im];
    1155:     while (xri>xim && im<nx) {
    1156:       im = im+1;
    1157:       xim = x[im];
    1158:     }
    1159:     indx1 = im-1;
    1160:     indx2 = im;
    1161:     dx1 = xri - x[indx1];
    1162:     dx2 = x[indx2] - xri;

    1164:     yri = yr[i];
    1165:     im = 0;
    1166:     yim = y[im];
    1167:     while (yri>yim && im<ny) {
    1168:       im = im+1;
    1169:       yim = y[im];
    1170:     }
    1171:     indy1 = im-1;
    1172:     indy2 = im;
    1173:     dy1 = yri - y[indy1];
    1174:     dy2 = y[indy2] - yri;

    1176:     zri = zr[i];
    1177:     im = 0;
    1178:     zim = z[im];
    1179:     while (zri>zim && im<nz) {
    1180:       im = im+1;
    1181:       zim = z[im];
    1182:     }
    1183:     indz1 = im-1;
    1184:     indz2 = im;
    1185:     dz1 = zri - z[indz1];
    1186:     dz2 = z[indz2] - zri;

    1188:     Dx = x[indx2] - x[indx1];
    1189:     Dy = y[indy2] - y[indy1];
    1190:     Dz = z[indz2] - z[indz1];

    1192:     j = indx1 + indy1*nx + indz1*nx*ny;
    1193:     v = (1-dx1/Dx)*(1-dy1/Dy)*(1-dz1/Dz);
    1194:     MatSetValues(user->Qblock,1,&i,1,&j,&v,INSERT_VALUES);

    1196:     j = indx1 + indy1*nx + indz2*nx*ny;
    1197:     v = (1-dx1/Dx)*(1-dy1/Dy)*(1-dz2/Dz);
    1198:     MatSetValues(user->Qblock,1,&i,1,&j,&v,INSERT_VALUES);

    1200:     j = indx1 + indy2*nx + indz1*nx*ny;
    1201:     v = (1-dx1/Dx)*(1-dy2/Dy)*(1-dz1/Dz);
    1202:     MatSetValues(user->Qblock,1,&i,1,&j,&v,INSERT_VALUES);

    1204:     j = indx1 + indy2*nx + indz2*nx*ny;
    1205:     v = (1-dx1/Dx)*(1-dy2/Dy)*(1-dz2/Dz);
    1206:     MatSetValues(user->Qblock,1,&i,1,&j,&v,INSERT_VALUES);

    1208:     j = indx2 + indy1*nx + indz1*nx*ny;
    1209:     v = (1-dx2/Dx)*(1-dy1/Dy)*(1-dz1/Dz);
    1210:     MatSetValues(user->Qblock,1,&i,1,&j,&v,INSERT_VALUES);

    1212:     j = indx2 + indy1*nx + indz2*nx*ny;
    1213:     v = (1-dx2/Dx)*(1-dy1/Dy)*(1-dz2/Dz);
    1214:     MatSetValues(user->Qblock,1,&i,1,&j,&v,INSERT_VALUES);

    1216:     j = indx2 + indy2*nx + indz1*nx*ny;
    1217:     v = (1-dx2/Dx)*(1-dy2/Dy)*(1-dz1/Dz);
    1218:     MatSetValues(user->Qblock,1,&i,1,&j,&v,INSERT_VALUES);

    1220:     j = indx2 + indy2*nx + indz2*nx*ny;
    1221:     v = (1-dx2/Dx)*(1-dy2/Dy)*(1-dz2/Dz);
    1222:     MatSetValues(user->Qblock,1,&i,1,&j,&v,INSERT_VALUES);
    1223:   }
    1224:   MatAssemblyBegin(user->Qblock,MAT_FINAL_ASSEMBLY);
    1225:   MatAssemblyEnd(user->Qblock,MAT_FINAL_ASSEMBLY);

    1227:   MatTranspose(user->Qblock,MAT_INITIAL_MATRIX,&user->QblockT);
    1228:   MatTranspose(user->L,MAT_INITIAL_MATRIX,&user->LT);

    1230:   /* Add noise to the measurement data */
    1231:   VecSet(user->ywork,1.0);
    1232:   VecAYPX(user->ywork,user->noise,user->ytrue);
    1233:   Scatter_i(user->ywork,user->yiwork,user->yi_scatter,user->nt);
    1234:   for (j=0; j<user->ns; j++) {
    1235:     i = user->sample_times[j];
    1236:     MatMult(user->Qblock,user->yiwork[i],user->di[j]);
    1237:   }
    1238:   Gather_i(user->d,user->di,user->di_scatter,user->ns);

    1240:   /* Now that initial conditions have been set, let the user pass tolerance options to the KSP solver */
    1241:   KSPSetFromOptions(user->solver);
    1242:   PetscFree(x);
    1243:   PetscFree(y);
    1244:   PetscFree(z);
    1245:   return(0);
    1246: }

    1248: PetscErrorCode ParabolicDestroy(AppCtx *user)
    1249: {
    1251:   PetscInt       i;

    1254:   MatDestroy(&user->Qblock);
    1255:   MatDestroy(&user->QblockT);
    1256:   MatDestroy(&user->Div);
    1257:   MatDestroy(&user->Divwork);
    1258:   MatDestroy(&user->Grad);
    1259:   MatDestroy(&user->Av);
    1260:   MatDestroy(&user->Avwork);
    1261:   MatDestroy(&user->AvT);
    1262:   MatDestroy(&user->DSG);
    1263:   MatDestroy(&user->L);
    1264:   MatDestroy(&user->LT);
    1265:   KSPDestroy(&user->solver);
    1266:   MatDestroy(&user->Js);
    1267:   MatDestroy(&user->Jd);
    1268:   MatDestroy(&user->JsInv);
    1269:   MatDestroy(&user->JsBlock);
    1270:   MatDestroy(&user->JsBlockPrec);
    1271:   VecDestroy(&user->u);
    1272:   VecDestroy(&user->uwork);
    1273:   VecDestroy(&user->utrue);
    1274:   VecDestroy(&user->y);
    1275:   VecDestroy(&user->ywork);
    1276:   VecDestroy(&user->ytrue);
    1277:   VecDestroyVecs(user->nt,&user->yi);
    1278:   VecDestroyVecs(user->nt,&user->yiwork);
    1279:   VecDestroyVecs(user->ns,&user->di);
    1280:   PetscFree(user->yi);
    1281:   PetscFree(user->yiwork);
    1282:   PetscFree(user->di);
    1283:   VecDestroy(&user->c);
    1284:   VecDestroy(&user->cwork);
    1285:   VecDestroy(&user->ur);
    1286:   VecDestroy(&user->q);
    1287:   VecDestroy(&user->d);
    1288:   VecDestroy(&user->dwork);
    1289:   VecDestroy(&user->lwork);
    1290:   VecDestroy(&user->S);
    1291:   VecDestroy(&user->Swork);
    1292:   VecDestroy(&user->Av_u);
    1293:   VecDestroy(&user->Twork);
    1294:   VecDestroy(&user->Rwork);
    1295:   VecDestroy(&user->js_diag);
    1296:   ISDestroy(&user->s_is);
    1297:   ISDestroy(&user->d_is);
    1298:   VecScatterDestroy(&user->state_scatter);
    1299:   VecScatterDestroy(&user->design_scatter);
    1300:   for (i=0; i<user->nt; i++) {
    1301:     VecScatterDestroy(&user->yi_scatter[i]);
    1302:   }
    1303:   for (i=0; i<user->ns; i++) {
    1304:     VecScatterDestroy(&user->di_scatter[i]);
    1305:   }
    1306:   PetscFree(user->yi_scatter);
    1307:   PetscFree(user->di_scatter);
    1308:   PetscFree(user->sample_times);
    1309:   return(0);
    1310: }

    1312: PetscErrorCode ParabolicMonitor(Tao tao, void *ptr)
    1313: {
    1315:   Vec            X;
    1316:   PetscReal      unorm,ynorm;
    1317:   AppCtx         *user = (AppCtx*)ptr;

    1320:   TaoGetSolutionVector(tao,&X);
    1321:   Scatter(X,user->ywork,user->state_scatter,user->uwork,user->design_scatter);
    1322:   VecAXPY(user->ywork,-1.0,user->ytrue);
    1323:   VecAXPY(user->uwork,-1.0,user->utrue);
    1324:   VecNorm(user->uwork,NORM_2,&unorm);
    1325:   VecNorm(user->ywork,NORM_2,&ynorm);
    1326:   PetscPrintf(MPI_COMM_WORLD, "||u-ut||=%g ||y-yt||=%g\n",(double)unorm,(double)ynorm);
    1327:   return(0);
    1328: }

    1330: /*TEST

    1332:    build:
    1333:       requires: !complex

    1335:    test:
    1336:       args: -tao_cmonitor -tao_type lcl -ns 1 -tao_gatol 1.e-4 -ksp_max_it 30
    1337:       requires: !single

    1339: TEST*/