Actual source code: parabolic.c
petsc-3.12.4 2020-02-04
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      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*/

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

     33:   VecScatter state_scatter;
     34:   VecScatter design_scatter;
     35:   VecScatter *yi_scatter;
     36:   VecScatter *di_scatter;

     38:   Mat       Js,Jd,JsBlockPrec,JsInv,JsBlock;
     39:   PetscBool jformed,dsg_formed;

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

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

     55:   Vec d;
     56:   Vec dwork;
     57:   Vec *di;

     59:   Vec y; /*  state variables */
     60:   Vec ywork;

     62:   Vec ytrue;
     63:   Vec *yi,*yiwork;

     65:   Vec u; /*  design variables */
     66:   Vec uwork;

     68:   Vec utrue;
     69:   Vec js_diag;
     70:   Vec c; /*  constraint vector */
     71:   Vec cwork;

     73:   Vec lwork;
     74:   Vec S;
     75:   Vec Rwork,Swork,Twork;
     76:   Vec Av_u;

     78:   KSP solver;
     79:   PC prec;

     81:   PetscInt ksp_its;
     82:   PetscInt ksp_its_initial;
     83: } AppCtx;

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

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

    108: PetscErrorCode DesignMatMult(Mat,Vec,Vec);
    109: PetscErrorCode DesignMatMultTranspose(Mat,Vec,Vec);

    111: PetscErrorCode Gather_i(Vec,Vec*,VecScatter*,PetscInt);
    112: PetscErrorCode Scatter_i(Vec,Vec*,VecScatter*,PetscInt);

    114: static  char help[]="";

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    233:   TaoDestroy(&tao);
    234:   VecDestroy(&x);
    235:   VecDestroy(&x0);
    236:   ParabolicDestroy(&user);

    238:   PetscFinalize();
    239:   return ierr;
    240: }
    241: /* ------------------------------------------------------------------- */
    242: /*
    243:    dwork = Qy - d
    244:    lwork = L*(u-ur)
    245:    f = 1/2 * (dwork.dork + alpha*lwork.lwork)
    246: */
    247: PetscErrorCode FormFunction(Tao tao,Vec X,PetscReal *f,void *ptr)
    248: {
    250:   PetscReal      d1=0,d2=0;
    251:   PetscInt       i,j;
    252:   AppCtx         *user = (AppCtx*)ptr;

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

    265:   VecWAXPY(user->uwork,-1.0,user->ur,user->u);
    266:   MatMult(user->L,user->uwork,user->lwork);
    267:   VecDot(user->lwork,user->lwork,&d2);

    269:   *f = 0.5 * (d1 + user->alpha*d2);
    270:   return(0);
    271: }

    273: /* ------------------------------------------------------------------- */
    274: /*
    275:     state: g_s = Q' *(Qy - d)
    276:     design: g_d = alpha*L'*L*(u-ur)
    277: */
    278: PetscErrorCode FormGradient(Tao tao,Vec X,Vec G,void *ptr)
    279: {
    281:   PetscInt       i,j;
    282:   AppCtx         *user = (AppCtx*)ptr;

    285:   Scatter(X,user->y,user->state_scatter,user->u,user->design_scatter);
    286:   Scatter_i(user->y,user->yi,user->yi_scatter,user->nt);
    287:   for (j=0; j<user->ns; j++){
    288:     i = user->sample_times[j];
    289:     MatMult(user->Qblock,user->yi[i],user->di[j]);
    290:   }
    291:   Gather_i(user->dwork,user->di,user->di_scatter,user->ns);
    292:   VecAXPY(user->dwork,-1.0,user->d);
    293:   Scatter_i(user->dwork,user->di,user->di_scatter,user->ns);
    294:   VecSet(user->ywork,0.0);
    295:   Scatter_i(user->ywork,user->yiwork,user->yi_scatter,user->nt);
    296:   for (j=0; j<user->ns; j++){
    297:     i = user->sample_times[j];
    298:     MatMult(user->QblockT,user->di[j],user->yiwork[i]);
    299:   }
    300:   Gather_i(user->ywork,user->yiwork,user->yi_scatter,user->nt);

    302:   VecWAXPY(user->uwork,-1.0,user->ur,user->u);
    303:   MatMult(user->L,user->uwork,user->lwork);
    304:   MatMult(user->LT,user->lwork,user->uwork);
    305:   VecScale(user->uwork, user->alpha);
    306:   Gather(G,user->ywork,user->state_scatter,user->uwork,user->design_scatter);
    307:   return(0);
    308: }

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

    318:   Scatter(X,user->y,user->state_scatter,user->u,user->design_scatter);
    319:   Scatter_i(user->y,user->yi,user->yi_scatter,user->nt);
    320:   for (j=0; j<user->ns; j++){
    321:     i = user->sample_times[j];
    322:     MatMult(user->Qblock,user->yi[i],user->di[j]);
    323:   }
    324:   Gather_i(user->dwork,user->di,user->di_scatter,user->ns);
    325:   VecAXPY(user->dwork,-1.0,user->d);
    326:   VecDot(user->dwork,user->dwork,&d1);
    327:   Scatter_i(user->dwork,user->di,user->di_scatter,user->ns);
    328:   VecSet(user->ywork,0.0);
    329:   Scatter_i(user->ywork,user->yiwork,user->yi_scatter,user->nt);
    330:   for (j=0; j<user->ns; j++){
    331:     i = user->sample_times[j];
    332:     MatMult(user->QblockT,user->di[j],user->yiwork[i]);
    333:   }
    334:   Gather_i(user->ywork,user->yiwork,user->yi_scatter,user->nt);

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

    343:   Gather(G,user->ywork,user->state_scatter,user->uwork,user->design_scatter);
    344:   return(0);
    345: }

    347: /* ------------------------------------------------------------------- */
    348: /* A
    349: MatShell object
    350: */
    351: PetscErrorCode FormJacobianState(Tao tao, Vec X, Mat J, Mat JPre, Mat JInv, void *ptr)
    352: {
    354:   AppCtx         *user = (AppCtx*)ptr;

    357:   Scatter(X,user->y,user->state_scatter,user->u,user->design_scatter);
    358:   VecSet(user->uwork,0);
    359:   VecAXPY(user->uwork,-1.0,user->u);
    360:   VecExp(user->uwork);
    361:   MatMult(user->Av,user->uwork,user->Av_u);
    362:   VecCopy(user->Av_u,user->Swork);
    363:   VecReciprocal(user->Swork);
    364:   MatCopy(user->Div,user->Divwork,SAME_NONZERO_PATTERN);
    365:   MatDiagonalScale(user->Divwork,NULL,user->Swork);
    366:   if (user->dsg_formed) {
    367:     MatMatMultNumeric(user->Divwork,user->Grad,user->DSG);
    368:   } else {
    369:     MatMatMultSymbolic(user->Divwork,user->Grad,PETSC_DECIDE,&user->DSG);
    370:     MatMatMultNumeric(user->Divwork,user->Grad,user->DSG);
    371:     user->dsg_formed = PETSC_TRUE;
    372:   }

    374:   /* B = speye(nx^3) + ht*DSG; */
    375:   MatScale(user->DSG,user->ht);
    376:   MatShift(user->DSG,1.0);
    377:   return(0);
    378: }

    380: /* ------------------------------------------------------------------- */
    381: /* B */
    382: PetscErrorCode FormJacobianDesign(Tao tao, Vec X, Mat J, void *ptr)
    383: {
    385:   AppCtx         *user = (AppCtx*)ptr;

    388:   Scatter(X,user->y,user->state_scatter,user->u,user->design_scatter);
    389:   return(0);
    390: }

    392: PetscErrorCode StateMatMult(Mat J_shell, Vec X, Vec Y)
    393: {
    395:   PetscInt       i;
    396:   AppCtx         *user;

    399:   MatShellGetContext(J_shell,(void**)&user);
    400:   Scatter_i(X,user->yi,user->yi_scatter,user->nt);
    401:   MatMult(user->JsBlock,user->yi[0],user->yiwork[0]);
    402:   for (i=1; i<user->nt; i++){
    403:     MatMult(user->JsBlock,user->yi[i],user->yiwork[i]);
    404:     VecAXPY(user->yiwork[i],-1.0,user->yi[i-1]);
    405:   }
    406:   Gather_i(Y,user->yiwork,user->yi_scatter,user->nt);
    407:   return(0);
    408: }

    410: PetscErrorCode StateMatMultTranspose(Mat J_shell, Vec X, Vec Y)
    411: {
    413:   PetscInt       i;
    414:   AppCtx         *user;

    417:   MatShellGetContext(J_shell,(void**)&user);
    418:   Scatter_i(X,user->yi,user->yi_scatter,user->nt);
    419:   for (i=0; i<user->nt-1; i++){
    420:     MatMult(user->JsBlock,user->yi[i],user->yiwork[i]);
    421:     VecAXPY(user->yiwork[i],-1.0,user->yi[i+1]);
    422:   }
    423:   i = user->nt-1;
    424:   MatMult(user->JsBlock,user->yi[i],user->yiwork[i]);
    425:   Gather_i(Y,user->yiwork,user->yi_scatter,user->nt);
    426:   return(0);
    427: }

    429: PetscErrorCode StateMatBlockMult(Mat J_shell, Vec X, Vec Y)
    430: {
    432:   AppCtx         *user;

    435:   MatShellGetContext(J_shell,(void**)&user);
    436:   MatMult(user->Grad,X,user->Swork);
    437:   VecPointwiseDivide(user->Swork,user->Swork,user->Av_u);
    438:   MatMult(user->Div,user->Swork,Y);
    439:   VecAYPX(Y,user->ht,X);
    440:   return(0);
    441: }

    443: PetscErrorCode DesignMatMult(Mat J_shell, Vec X, Vec Y)
    444: {
    446:   PetscInt       i;
    447:   AppCtx         *user;

    450:   MatShellGetContext(J_shell,(void**)&user);

    452:   /* sdiag(1./v) */
    453:   VecSet(user->uwork,0);
    454:   VecAXPY(user->uwork,-1.0,user->u);
    455:   VecExp(user->uwork);

    457:   /* sdiag(1./((Av*(1./v)).^2)) */
    458:   MatMult(user->Av,user->uwork,user->Swork);
    459:   VecPointwiseMult(user->Swork,user->Swork,user->Swork);
    460:   VecReciprocal(user->Swork);

    462:   /* (Av * (sdiag(1./v) * b)) */
    463:   VecPointwiseMult(user->uwork,user->uwork,X);
    464:   MatMult(user->Av,user->uwork,user->Twork);

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

    469:   Scatter_i(user->y,user->yi,user->yi_scatter,user->nt);
    470:   for (i=0; i<user->nt; i++){
    471:     /* (sdiag(Grad*y(:,i)) */
    472:     MatMult(user->Grad,user->yi[i],user->Twork);

    474:     /* ht * Div * (sdiag(Grad*y(:,i)) * (sdiag(1./((Av*(1./v)).^2)) * (Av * (sdiag(1./v) * b)))) */
    475:     VecPointwiseMult(user->Twork,user->Twork,user->Swork);
    476:     MatMult(user->Div,user->Twork,user->yiwork[i]);
    477:     VecScale(user->yiwork[i],user->ht);
    478:   }
    479:   Gather_i(Y,user->yiwork,user->yi_scatter,user->nt);

    481:   return(0);
    482: }

    484: PetscErrorCode DesignMatMultTranspose(Mat J_shell, Vec X, Vec Y)
    485: {
    487:   PetscInt       i;
    488:   AppCtx         *user;

    491:   MatShellGetContext(J_shell,(void**)&user);

    493:   /* sdiag(1./((Av*(1./v)).^2)) */
    494:   VecSet(user->uwork,0);
    495:   VecAXPY(user->uwork,-1.0,user->u);
    496:   VecExp(user->uwork);
    497:   MatMult(user->Av,user->uwork,user->Rwork);
    498:   VecPointwiseMult(user->Rwork,user->Rwork,user->Rwork);
    499:   VecReciprocal(user->Rwork);

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

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

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

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

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

    520:     /* sdiag(1./v) * (Av' * (sdiag(1./((Av*(1./v)).^2)) * (sdiag(Grad*y(:,i)) * (Div' * b(:,i))))) */
    521:     VecPointwiseMult(user->yiwork[i],user->uwork,user->yiwork[i]);
    522:     VecAXPY(Y,user->ht,user->yiwork[i]);
    523:   }
    524:   return(0);
    525: }

    527: PetscErrorCode StateMatBlockPrecMult(PC PC_shell, Vec X, Vec Y)
    528: {
    530:   AppCtx         *user;

    533:   PCShellGetContext(PC_shell,(void**)&user);

    535:   if (user->dsg_formed) {
    536:     MatSOR(user->DSG,X,1.0,(MatSORType)(SOR_ZERO_INITIAL_GUESS | SOR_LOCAL_SYMMETRIC_SWEEP),0.0,1,1,Y);
    537:   } else SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"DSG not formed");
    538:   return(0);
    539: }

    541: PetscErrorCode StateMatInvMult(Mat J_shell, Vec X, Vec Y)
    542: {
    544:   AppCtx         *user;
    545:   PetscInt       its,i;

    548:   MatShellGetContext(J_shell,(void**)&user);

    550:   if (Y == user->ytrue) {
    551:     /* First solve is done with true solution to set up problem */
    552:     KSPSetTolerances(user->solver,1e-8,PETSC_DEFAULT,PETSC_DEFAULT,PETSC_DEFAULT);
    553:   } else {
    554:     KSPSetTolerances(user->solver,PETSC_DEFAULT,PETSC_DEFAULT,PETSC_DEFAULT,PETSC_DEFAULT);
    555:   }

    557:   Scatter_i(X,user->yi,user->yi_scatter,user->nt);
    558:   KSPSolve(user->solver,user->yi[0],user->yiwork[0]);
    559:   KSPGetIterationNumber(user->solver,&its);
    560:   user->ksp_its = user->ksp_its + its;

    562:   for (i=1; i<user->nt; i++){
    563:     VecAXPY(user->yi[i],1.0,user->yiwork[i-1]);
    564:     KSPSolve(user->solver,user->yi[i],user->yiwork[i]);
    565:     KSPGetIterationNumber(user->solver,&its);
    566:     user->ksp_its = user->ksp_its + its;
    567:   }
    568:   Gather_i(Y,user->yiwork,user->yi_scatter,user->nt);
    569:   return(0);
    570: }

    572: PetscErrorCode StateMatInvTransposeMult(Mat J_shell, Vec X, Vec Y)
    573: {
    575:   AppCtx         *user;
    576:   PetscInt       its,i;

    579:   MatShellGetContext(J_shell,(void**)&user);

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

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

    586:   KSPGetIterationNumber(user->solver,&its);
    587:   user->ksp_its = user->ksp_its + its;

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

    593:     KSPGetIterationNumber(user->solver,&its);
    594:     user->ksp_its = user->ksp_its + its;

    596:   }

    598:   Gather_i(Y,user->yiwork,user->yi_scatter,user->nt);
    599:   return(0);
    600: }

    602: PetscErrorCode StateMatDuplicate(Mat J_shell, MatDuplicateOption opt, Mat *new_shell)
    603: {
    605:   AppCtx         *user;

    608:   MatShellGetContext(J_shell,(void**)&user);

    610:   MatCreateShell(PETSC_COMM_WORLD,PETSC_DETERMINE,PETSC_DETERMINE,user->m,user->m,user,new_shell);
    611:   MatShellSetOperation(*new_shell,MATOP_MULT,(void(*)(void))StateMatMult);
    612:   MatShellSetOperation(*new_shell,MATOP_DUPLICATE,(void(*)(void))StateMatDuplicate);
    613:   MatShellSetOperation(*new_shell,MATOP_MULT_TRANSPOSE,(void(*)(void))StateMatMultTranspose);
    614:   MatShellSetOperation(*new_shell,MATOP_GET_DIAGONAL,(void(*)(void))StateMatGetDiagonal);
    615:   return(0);
    616: }

    618: PetscErrorCode StateMatGetDiagonal(Mat J_shell, Vec X)
    619: {
    621:   AppCtx         *user;

    624:   MatShellGetContext(J_shell,(void**)&user);
    625:    VecCopy(user->js_diag,X);
    626:   return(0);

    628: }

    630: PetscErrorCode FormConstraints(Tao tao, Vec X, Vec C, void *ptr)
    631: {
    632:   /* con = Ay - q, A = [B  0  0 ... 0;
    633:                        -I  B  0 ... 0;
    634:                         0 -I  B ... 0;
    635:                              ...     ;
    636:                         0    ... -I B]
    637:      B = ht * Div * Sigma * Grad + eye */
    639:   PetscInt       i;
    640:   AppCtx         *user = (AppCtx*)ptr;

    643:   Scatter(X,user->y,user->state_scatter,user->u,user->design_scatter);
    644:   Scatter_i(user->y,user->yi,user->yi_scatter,user->nt);
    645:   MatMult(user->JsBlock,user->yi[0],user->yiwork[0]);
    646:   for (i=1; i<user->nt; i++){
    647:     MatMult(user->JsBlock,user->yi[i],user->yiwork[i]);
    648:     VecAXPY(user->yiwork[i],-1.0,user->yi[i-1]);
    649:   }
    650:   Gather_i(C,user->yiwork,user->yi_scatter,user->nt);
    651:   VecAXPY(C,-1.0,user->q);
    652:   return(0);
    653: }

    656: PetscErrorCode Scatter(Vec x, Vec state, VecScatter s_scat, Vec design, VecScatter d_scat)
    657: {

    661:   VecScatterBegin(s_scat,x,state,INSERT_VALUES,SCATTER_FORWARD);
    662:   VecScatterEnd(s_scat,x,state,INSERT_VALUES,SCATTER_FORWARD);
    663:   VecScatterBegin(d_scat,x,design,INSERT_VALUES,SCATTER_FORWARD);
    664:   VecScatterEnd(d_scat,x,design,INSERT_VALUES,SCATTER_FORWARD);
    665:   return(0);
    666: }

    668: PetscErrorCode Scatter_i(Vec y, Vec *yi, VecScatter *scat, PetscInt nt)
    669: {
    671:   PetscInt       i;

    674:   for (i=0; i<nt; i++){
    675:     VecScatterBegin(scat[i],y,yi[i],INSERT_VALUES,SCATTER_FORWARD);
    676:     VecScatterEnd(scat[i],y,yi[i],INSERT_VALUES,SCATTER_FORWARD);
    677:   }
    678:   return(0);
    679: }

    682: PetscErrorCode Gather(Vec x, Vec state, VecScatter s_scat, Vec design, VecScatter d_scat)
    683: {

    687:   VecScatterBegin(s_scat,state,x,INSERT_VALUES,SCATTER_REVERSE);
    688:   VecScatterEnd(s_scat,state,x,INSERT_VALUES,SCATTER_REVERSE);
    689:   VecScatterBegin(d_scat,design,x,INSERT_VALUES,SCATTER_REVERSE);
    690:   VecScatterEnd(d_scat,design,x,INSERT_VALUES,SCATTER_REVERSE);
    691:   return(0);
    692: }

    694: PetscErrorCode Gather_i(Vec y, Vec *yi, VecScatter *scat, PetscInt nt)
    695: {
    697:   PetscInt       i;

    700:   for (i=0; i<nt; i++){
    701:     VecScatterBegin(scat[i],yi[i],y,INSERT_VALUES,SCATTER_REVERSE);
    702:     VecScatterEnd(scat[i],yi[i],y,INSERT_VALUES,SCATTER_REVERSE);
    703:   }
    704:   return(0);
    705: }

    707: PetscErrorCode ParabolicInitialize(AppCtx *user)
    708: {
    710:   PetscInt       m,n,i,j,k,linear_index,istart,iend,iblock,lo,hi,lo2,hi2;
    711:   Vec            XX,YY,ZZ,XXwork,YYwork,ZZwork,UTwork,yi,di,bc;
    712:   PetscReal      *x, *y, *z;
    713:   PetscReal      h,stime;
    714:   PetscScalar    hinv,neg_hinv,half = 0.5,sqrt_beta;
    715:   PetscInt       im,indx1,indx2,indy1,indy2,indz1,indz2,nx,ny,nz;
    716:   PetscReal      xri,yri,zri,xim,yim,zim,dx1,dx2,dy1,dy2,dz1,dz2,Dx,Dy,Dz;
    717:   PetscScalar    v,vx,vy,vz;
    718:   IS             is_from_y,is_to_yi,is_from_d,is_to_di;
    719:   /* Data locations */
    720:   PetscScalar xr[64] = {0.4970,     0.8498,     0.7814,     0.6268,     0.7782,     0.6402,     0.3617,     0.3160,
    721:                         0.3610,     0.5298,     0.6987,     0.3331,     0.7962,     0.5596,     0.3866,     0.6774,
    722:                         0.5407,     0.4518,     0.6702,     0.6061,     0.7580,     0.8997,     0.5198,     0.8326,
    723:                         0.2138,     0.9198,     0.3000,     0.2833,     0.8288,     0.7076,     0.1820,     0.0728,
    724:                         0.8447,     0.2367,     0.3239,     0.6413,     0.3114,     0.4731,     0.1192,     0.9273,
    725:                         0.5724,     0.4331,     0.5136,     0.3547,     0.4413,     0.2602,     0.5698,     0.7278,
    726:                         0.5261,     0.6230,     0.2454,     0.3948,     0.7479,     0.6582,     0.4660,     0.5594,
    727:                         0.7574,     0.1143,     0.5900,     0.1065,     0.4260,     0.3294,     0.8276,     0.0756};

    729:   PetscScalar yr[64] = {0.7345,     0.9120,     0.9288,     0.7528,     0.4463,     0.4985,     0.2497,     0.6256,
    730:                         0.3425,     0.9026,     0.6983,     0.4230,     0.7140,     0.2970,     0.4474,     0.8792,
    731:                         0.6604,     0.2485,     0.7968,     0.6127,     0.1796,     0.2437,     0.5938,     0.6137,
    732:                         0.3867,     0.5658,     0.4575,     0.1009,     0.0863,     0.3361,     0.0738,     0.3985,
    733:                         0.6602,     0.1437,     0.0934,     0.5983,     0.5950,     0.0763,     0.0768,     0.2288,
    734:                         0.5761,     0.1129,     0.3841,     0.6150,     0.6904,     0.6686,     0.1361,     0.4601,
    735:                         0.4491,     0.3716,     0.1969,     0.6537,     0.6743,     0.6991,     0.4811,     0.5480,
    736:                         0.1684,     0.4569,     0.6889,     0.8437,     0.3015,     0.2854,     0.8199,     0.2658};

    738:   PetscScalar zr[64] = {0.7668,     0.8573,     0.2654,     0.2719,     0.1060,     0.1311,     0.6232,     0.2295,
    739:                         0.8009,     0.2147,     0.2119,     0.9325,     0.4473,     0.3600,     0.3374,     0.3819,
    740:                         0.4066,     0.5801,     0.1673,     0.0959,     0.4638,     0.8236,     0.8800,     0.2939,
    741:                         0.2028,     0.8262,     0.2706,     0.6276,     0.9085,     0.6443,     0.8241,     0.0712,
    742:                         0.1824,     0.7789,     0.4389,     0.8415,     0.7055,     0.6639,     0.3653,     0.2078,
    743:                         0.1987,     0.2297,     0.4321,     0.8115,     0.4915,     0.7764,     0.4657,     0.4627,
    744:                         0.4569,     0.4232,     0.8514,     0.0674,     0.3227,     0.1055,     0.6690,     0.6313,
    745:                         0.9226,     0.5461,     0.4126,     0.2364,     0.6096,     0.7042,     0.3914,     0.0711};

    748:   PetscMalloc1(user->mx,&x);
    749:   PetscMalloc1(user->mx,&y);
    750:   PetscMalloc1(user->mx,&z);
    751:   user->jformed = PETSC_FALSE;
    752:   user->dsg_formed = PETSC_FALSE;

    754:   n = user->mx * user->mx * user->mx;
    755:   m = 3 * user->mx * user->mx * (user->mx-1);
    756:   sqrt_beta = PetscSqrtScalar(user->beta);

    758:   user->ksp_its = 0;
    759:   user->ksp_its_initial = 0;

    761:   stime = (PetscReal)user->nt/user->ns;
    762:   PetscMalloc1(user->ns,&user->sample_times);
    763:   for (i=0; i<user->ns; i++){
    764:     user->sample_times[i] = (PetscInt)(stime*((PetscReal)i+1.0)-0.5);
    765:   }

    767:   VecCreate(PETSC_COMM_WORLD,&XX);
    768:   VecCreate(PETSC_COMM_WORLD,&user->q);
    769:   VecSetSizes(XX,PETSC_DECIDE,n);
    770:   VecSetSizes(user->q,PETSC_DECIDE,n*user->nt);
    771:   VecSetFromOptions(XX);
    772:   VecSetFromOptions(user->q);

    774:   VecDuplicate(XX,&YY);
    775:   VecDuplicate(XX,&ZZ);
    776:   VecDuplicate(XX,&XXwork);
    777:   VecDuplicate(XX,&YYwork);
    778:   VecDuplicate(XX,&ZZwork);
    779:   VecDuplicate(XX,&UTwork);
    780:   VecDuplicate(XX,&user->utrue);
    781:   VecDuplicate(XX,&bc);

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

    788:   VecGetOwnershipRange(XX,&istart,&iend);
    789:   for (linear_index=istart; linear_index<iend; linear_index++){
    790:     i = linear_index % user->mx;
    791:     j = ((linear_index-i)/user->mx) % user->mx;
    792:     k = ((linear_index-i)/user->mx-j) / user->mx;
    793:     vx = h*(i+0.5);
    794:     vy = h*(j+0.5);
    795:     vz = h*(k+0.5);
    796:     VecSetValues(XX,1,&linear_index,&vx,INSERT_VALUES);
    797:     VecSetValues(YY,1,&linear_index,&vy,INSERT_VALUES);
    798:     VecSetValues(ZZ,1,&linear_index,&vz,INSERT_VALUES);
    799:     if ((vx<0.6) && (vx>0.4) && (vy<0.6) && (vy>0.4) && (vy<0.6) && (vz<0.6) && (vz>0.4)){
    800:       v = 1000.0;
    801:       VecSetValues(bc,1,&linear_index,&v,INSERT_VALUES);
    802:     }
    803:   }

    805:   VecAssemblyBegin(XX);
    806:   VecAssemblyEnd(XX);
    807:   VecAssemblyBegin(YY);
    808:   VecAssemblyEnd(YY);
    809:   VecAssemblyBegin(ZZ);
    810:   VecAssemblyEnd(ZZ);
    811:   VecAssemblyBegin(bc);
    812:   VecAssemblyEnd(bc);

    814:   /* Compute true parameter function
    815:      ut = 0.5 + exp(-10*((x-0.5)^2+(y-0.5)^2+(z-0.5)^2)) */
    816:   VecCopy(XX,XXwork);
    817:   VecCopy(YY,YYwork);
    818:   VecCopy(ZZ,ZZwork);

    820:   VecShift(XXwork,-0.5);
    821:   VecShift(YYwork,-0.5);
    822:   VecShift(ZZwork,-0.5);

    824:   VecPointwiseMult(XXwork,XXwork,XXwork);
    825:   VecPointwiseMult(YYwork,YYwork,YYwork);
    826:   VecPointwiseMult(ZZwork,ZZwork,ZZwork);

    828:   VecCopy(XXwork,user->utrue);
    829:   VecAXPY(user->utrue,1.0,YYwork);
    830:   VecAXPY(user->utrue,1.0,ZZwork);
    831:   VecScale(user->utrue,-10.0);
    832:   VecExp(user->utrue);
    833:   VecShift(user->utrue,0.5);

    835:   VecDestroy(&XX);
    836:   VecDestroy(&YY);
    837:   VecDestroy(&ZZ);
    838:   VecDestroy(&XXwork);
    839:   VecDestroy(&YYwork);
    840:   VecDestroy(&ZZwork);
    841:   VecDestroy(&UTwork);

    843:   /* Initial guess and reference model */
    844:   VecDuplicate(user->utrue,&user->ur);
    845:   VecSet(user->ur,0.0);

    847:   /* Generate Grad matrix */
    848:   MatCreate(PETSC_COMM_WORLD,&user->Grad);
    849:   MatSetSizes(user->Grad,PETSC_DECIDE,PETSC_DECIDE,m,n);
    850:   MatSetFromOptions(user->Grad);
    851:   MatMPIAIJSetPreallocation(user->Grad,2,NULL,2,NULL);
    852:   MatSeqAIJSetPreallocation(user->Grad,2,NULL);
    853:   MatGetOwnershipRange(user->Grad,&istart,&iend);

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

    878:   MatAssemblyBegin(user->Grad,MAT_FINAL_ASSEMBLY);
    879:   MatAssemblyEnd(user->Grad,MAT_FINAL_ASSEMBLY);

    882:   /* Generate arithmetic averaging matrix Av */
    883:   MatCreate(PETSC_COMM_WORLD,&user->Av);
    884:   MatSetSizes(user->Av,PETSC_DECIDE,PETSC_DECIDE,m,n);
    885:   MatSetFromOptions(user->Av);
    886:   MatMPIAIJSetPreallocation(user->Av,2,NULL,2,NULL);
    887:   MatSeqAIJSetPreallocation(user->Av,2,NULL);
    888:   MatGetOwnershipRange(user->Av,&istart,&iend);

    890:   for (i=istart; i<iend; i++){
    891:     if (i<m/3){
    892:       iblock = i / (user->mx-1);
    893:       j = iblock*user->mx + (i % (user->mx-1));
    894:       MatSetValues(user->Av,1,&i,1,&j,&half,INSERT_VALUES);
    895:       j = j+1;
    896:       MatSetValues(user->Av,1,&i,1,&j,&half,INSERT_VALUES);
    897:     }
    898:     if (i>=m/3 && i<2*m/3){
    899:       iblock = (i-m/3) / (user->mx*(user->mx-1));
    900:       j = iblock*user->mx*user->mx + ((i-m/3) % (user->mx*(user->mx-1)));
    901:       MatSetValues(user->Av,1,&i,1,&j,&half,INSERT_VALUES);
    902:       j = j + user->mx;
    903:       MatSetValues(user->Av,1,&i,1,&j,&half,INSERT_VALUES);
    904:     }
    905:     if (i>=2*m/3){
    906:       j = i-2*m/3;
    907:       MatSetValues(user->Av,1,&i,1,&j,&half,INSERT_VALUES);
    908:       j = j + user->mx*user->mx;
    909:       MatSetValues(user->Av,1,&i,1,&j,&half,INSERT_VALUES);
    910:     }
    911:   }

    913:   MatAssemblyBegin(user->Av,MAT_FINAL_ASSEMBLY);
    914:   MatAssemblyEnd(user->Av,MAT_FINAL_ASSEMBLY);

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

    919:   MatCreate(PETSC_COMM_WORLD,&user->L);
    920:   MatSetSizes(user->L,PETSC_DECIDE,PETSC_DECIDE,m+n,n);
    921:   MatSetFromOptions(user->L);
    922:   MatMPIAIJSetPreallocation(user->L,2,NULL,2,NULL);
    923:   MatSeqAIJSetPreallocation(user->L,2,NULL);
    924:   MatGetOwnershipRange(user->L,&istart,&iend);

    926:   for (i=istart; i<iend; i++){
    927:     if (i<m/3){
    928:       iblock = i / (user->mx-1);
    929:       j = iblock*user->mx + (i % (user->mx-1));
    930:       MatSetValues(user->L,1,&i,1,&j,&neg_hinv,INSERT_VALUES);
    931:       j = j+1;
    932:       MatSetValues(user->L,1,&i,1,&j,&hinv,INSERT_VALUES);
    933:     }
    934:     if (i>=m/3 && i<2*m/3){
    935:       iblock = (i-m/3) / (user->mx*(user->mx-1));
    936:       j = iblock*user->mx*user->mx + ((i-m/3) % (user->mx*(user->mx-1)));
    937:       MatSetValues(user->L,1,&i,1,&j,&neg_hinv,INSERT_VALUES);
    938:       j = j + user->mx;
    939:       MatSetValues(user->L,1,&i,1,&j,&hinv,INSERT_VALUES);
    940:     }
    941:     if (i>=2*m/3 && i<m){
    942:       j = i-2*m/3;
    943:       MatSetValues(user->L,1,&i,1,&j,&neg_hinv,INSERT_VALUES);
    944:       j = j + user->mx*user->mx;
    945:       MatSetValues(user->L,1,&i,1,&j,&hinv,INSERT_VALUES);
    946:     }
    947:     if (i>=m){
    948:       j = i - m;
    949:       MatSetValues(user->L,1,&i,1,&j,&sqrt_beta,INSERT_VALUES);
    950:     }
    951:   }

    953:   MatAssemblyBegin(user->L,MAT_FINAL_ASSEMBLY);
    954:   MatAssemblyEnd(user->L,MAT_FINAL_ASSEMBLY);

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

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

    961:   /* Build work vectors and matrices */
    962:   VecCreate(PETSC_COMM_WORLD,&user->S);
    963:   VecSetSizes(user->S, PETSC_DECIDE, user->mx*user->mx*(user->mx-1)*3);
    964:   VecSetFromOptions(user->S);

    966:   VecCreate(PETSC_COMM_WORLD,&user->lwork);
    967:   VecSetSizes(user->lwork,PETSC_DECIDE,m+user->mx*user->mx*user->mx);
    968:   VecSetFromOptions(user->lwork);

    970:   MatDuplicate(user->Div,MAT_SHARE_NONZERO_PATTERN,&user->Divwork);
    971:   MatDuplicate(user->Av,MAT_SHARE_NONZERO_PATTERN,&user->Avwork);

    973:   VecCreate(PETSC_COMM_WORLD,&user->d);
    974:   VecSetSizes(user->d,PETSC_DECIDE,user->ndata*user->nt);
    975:   VecSetFromOptions(user->d);

    977:   VecDuplicate(user->S,&user->Swork);
    978:   VecDuplicate(user->S,&user->Av_u);
    979:   VecDuplicate(user->S,&user->Twork);
    980:   VecDuplicate(user->S,&user->Rwork);
    981:   VecDuplicate(user->y,&user->ywork);
    982:   VecDuplicate(user->u,&user->uwork);
    983:   VecDuplicate(user->u,&user->js_diag);
    984:   VecDuplicate(user->c,&user->cwork);
    985:   VecDuplicate(user->d,&user->dwork);

    987:   /* Create matrix-free shell user->Js for computing A*x */
    988:   MatCreateShell(PETSC_COMM_WORLD,PETSC_DETERMINE,PETSC_DETERMINE,user->m*user->nt,user->m*user->nt,user,&user->Js);
    989:   MatShellSetOperation(user->Js,MATOP_MULT,(void(*)(void))StateMatMult);
    990:   MatShellSetOperation(user->Js,MATOP_DUPLICATE,(void(*)(void))StateMatDuplicate);
    991:   MatShellSetOperation(user->Js,MATOP_MULT_TRANSPOSE,(void(*)(void))StateMatMultTranspose);
    992:   MatShellSetOperation(user->Js,MATOP_GET_DIAGONAL,(void(*)(void))StateMatGetDiagonal);

    994:   /* Diagonal blocks of user->Js */
    995:   MatCreateShell(PETSC_COMM_WORLD,PETSC_DETERMINE,PETSC_DETERMINE,user->m,user->m,user,&user->JsBlock);
    996:   MatShellSetOperation(user->JsBlock,MATOP_MULT,(void(*)(void))StateMatBlockMult);
    997:   /* Blocks are symmetric */
    998:   MatShellSetOperation(user->JsBlock,MATOP_MULT_TRANSPOSE,(void(*)(void))StateMatBlockMult);

    1000:   /* Create a matrix-free shell user->JsBlockPrec for computing (U+D)\D*(L+D)\x, where JsBlock = L+D+U,
    1001:      D is diagonal, L is strictly lower triangular, and U is strictly upper triangular.
    1002:      This is an SSOR preconditioner for user->JsBlock. */
    1003:   MatCreateShell(PETSC_COMM_WORLD,PETSC_DETERMINE,PETSC_DETERMINE,user->m,user->m,user,&user->JsBlockPrec);
    1004:   MatShellSetOperation(user->JsBlockPrec,MATOP_MULT,(void(*)(void))StateMatBlockPrecMult);
    1005:   /* JsBlockPrec is symmetric */
    1006:   MatShellSetOperation(user->JsBlockPrec,MATOP_MULT_TRANSPOSE,(void(*)(void))StateMatBlockPrecMult);
    1007:   MatSetOption(user->JsBlockPrec,MAT_SYMMETRY_ETERNAL,PETSC_TRUE);

    1009:   /* Create a matrix-free shell user->Jd for computing B*x */
    1010:   MatCreateShell(PETSC_COMM_WORLD,PETSC_DETERMINE,PETSC_DETERMINE,user->m*user->nt,user->m,user,&user->Jd);
    1011:   MatShellSetOperation(user->Jd,MATOP_MULT,(void(*)(void))DesignMatMult);
    1012:   MatShellSetOperation(user->Jd,MATOP_MULT_TRANSPOSE,(void(*)(void))DesignMatMultTranspose);

    1014:   /* User-defined routines for computing user->Js\x and user->Js^T\x*/
    1015:   MatCreateShell(PETSC_COMM_WORLD,PETSC_DETERMINE,PETSC_DETERMINE,user->m*user->nt,user->m*user->nt,user,&user->JsInv);
    1016:   MatShellSetOperation(user->JsInv,MATOP_MULT,(void(*)(void))StateMatInvMult);
    1017:   MatShellSetOperation(user->JsInv,MATOP_MULT_TRANSPOSE,(void(*)(void))StateMatInvTransposeMult);

    1019:   /* Solver options and tolerances */
    1020:   KSPCreate(PETSC_COMM_WORLD,&user->solver);
    1021:   KSPSetType(user->solver,KSPCG);
    1022:   KSPSetOperators(user->solver,user->JsBlock,user->JsBlockPrec);
    1023:   KSPSetInitialGuessNonzero(user->solver,PETSC_FALSE);
    1024:   KSPSetTolerances(user->solver,1e-4,1e-20,1e3,500);
    1025:   KSPSetFromOptions(user->solver);
    1026:   KSPGetPC(user->solver,&user->prec);
    1027:   PCSetType(user->prec,PCSHELL);

    1029:   PCShellSetApply(user->prec,StateMatBlockPrecMult);
    1030:   PCShellSetApplyTranspose(user->prec,StateMatBlockPrecMult);
    1031:   PCShellSetContext(user->prec,user);

    1033:   /* Create scatter from y to y_1,y_2,...,y_nt */
    1034:   PetscMalloc1(user->nt*user->m,&user->yi_scatter);
    1035:   VecCreate(PETSC_COMM_WORLD,&yi);
    1036:   VecSetSizes(yi,PETSC_DECIDE,user->mx*user->mx*user->mx);
    1037:   VecSetFromOptions(yi);
    1038:   VecDuplicateVecs(yi,user->nt,&user->yi);
    1039:   VecDuplicateVecs(yi,user->nt,&user->yiwork);

    1041:   VecGetOwnershipRange(user->y,&lo2,&hi2);
    1042:   istart = 0;
    1043:   for (i=0; i<user->nt; i++){
    1044:     VecGetOwnershipRange(user->yi[i],&lo,&hi);
    1045:     ISCreateStride(PETSC_COMM_SELF,hi-lo,lo,1,&is_to_yi);
    1046:     ISCreateStride(PETSC_COMM_SELF,hi-lo,lo2+istart,1,&is_from_y);
    1047:     VecScatterCreate(user->y,is_from_y,user->yi[i],is_to_yi,&user->yi_scatter[i]);
    1048:     istart = istart + hi-lo;
    1049:     ISDestroy(&is_to_yi);
    1050:     ISDestroy(&is_from_y);
    1051:   }
    1052:   VecDestroy(&yi);

    1054:   /* Create scatter from d to d_1,d_2,...,d_ns */
    1055:   PetscMalloc1(user->ns*user->ndata,&user->di_scatter);
    1056:   VecCreate(PETSC_COMM_WORLD,&di);
    1057:   VecSetSizes(di,PETSC_DECIDE,user->ndata);
    1058:   VecSetFromOptions(di);
    1059:   VecDuplicateVecs(di,user->ns,&user->di);
    1060:   VecGetOwnershipRange(user->d,&lo2,&hi2);
    1061:   istart = 0;
    1062:   for (i=0; i<user->ns; i++){
    1063:     VecGetOwnershipRange(user->di[i],&lo,&hi);
    1064:     ISCreateStride(PETSC_COMM_SELF,hi-lo,lo,1,&is_to_di);
    1065:     ISCreateStride(PETSC_COMM_SELF,hi-lo,lo2+istart,1,&is_from_d);
    1066:     VecScatterCreate(user->d,is_from_d,user->di[i],is_to_di,&user->di_scatter[i]);
    1067:     istart = istart + hi-lo;
    1068:     ISDestroy(&is_to_di);
    1069:     ISDestroy(&is_from_d);
    1070:   }
    1071:   VecDestroy(&di);

    1073:   /* Assemble RHS of forward problem */
    1074:   VecCopy(bc,user->yiwork[0]);
    1075:   for (i=1; i<user->nt; i++){
    1076:     VecSet(user->yiwork[i],0.0);
    1077:   }
    1078:   Gather_i(user->q,user->yiwork,user->yi_scatter,user->nt);
    1079:   VecDestroy(&bc);

    1081:   /* Compute true state function yt given ut */
    1082:   VecCreate(PETSC_COMM_WORLD,&user->ytrue);
    1083:   VecSetSizes(user->ytrue,PETSC_DECIDE,n*user->nt);
    1084:   VecSetFromOptions(user->ytrue);

    1086:   /* First compute Av_u = Av*exp(-u) */
    1087:   VecSet(user->uwork,0);
    1088:   VecAXPY(user->uwork,-1.0,user->utrue); /*  Note: user->utrue */
    1089:   VecExp(user->uwork);
    1090:   MatMult(user->Av,user->uwork,user->Av_u);

    1092:   MatMatMultSymbolic(user->Div,user->Grad,PETSC_DEFAULT,&user->DSG);
    1093:   user->dsg_formed = PETSC_TRUE;

    1095:   /* Next form DSG = Div*S*Grad */
    1096:   MatCopy(user->Div,user->Divwork,SAME_NONZERO_PATTERN);
    1097:   MatDiagonalScale(user->Divwork,NULL,user->Av_u);
    1098:   if (user->dsg_formed) {
    1099:     MatMatMultNumeric(user->Divwork,user->Grad,user->DSG);
    1100:   } else {
    1101:     MatMatMultSymbolic(user->Div,user->Grad,PETSC_DEFAULT,&user->DSG);
    1102:     MatMatMultNumeric(user->Div,user->Grad,user->DSG);
    1103:     user->dsg_formed = PETSC_TRUE;
    1104:   }
    1105:   /* B = speye(nx^3) + ht*DSG; */
    1106:   MatScale(user->DSG,user->ht);
    1107:   MatShift(user->DSG,1.0);

    1109:   /* Now solve for ytrue */
    1110:   StateMatInvMult(user->Js,user->q,user->ytrue);

    1112:   /* Initial guess y0 for state given u0 */

    1114:   /* First compute Av_u = Av*exp(-u) */
    1115:   VecSet(user->uwork,0);
    1116:   VecAXPY(user->uwork,-1.0,user->u); /*  Note: user->u */
    1117:   VecExp(user->uwork);
    1118:   MatMult(user->Av,user->uwork,user->Av_u);

    1120:   /* Next form DSG = Div*S*Grad */
    1121:   MatCopy(user->Div,user->Divwork,SAME_NONZERO_PATTERN);
    1122:   MatDiagonalScale(user->Divwork,NULL,user->Av_u);
    1123:   if (user->dsg_formed) {
    1124:     MatMatMultNumeric(user->Divwork,user->Grad,user->DSG);
    1125:   } else {
    1126:     MatMatMultSymbolic(user->Div,user->Grad,PETSC_DEFAULT,&user->DSG);
    1127:     MatMatMultNumeric(user->Div,user->Grad,user->DSG);
    1128:     user->dsg_formed = PETSC_TRUE;
    1129:   }
    1130:   /* B = speye(nx^3) + ht*DSG; */
    1131:   MatScale(user->DSG,user->ht);
    1132:   MatShift(user->DSG,1.0);

    1134:   /* Now solve for y */
    1135:   StateMatInvMult(user->Js,user->q,user->y);

    1137:   /* Construct projection matrix Q, a block diagonal matrix consisting of nt copies of Qblock along the diagonal */
    1138:   MatCreate(PETSC_COMM_WORLD,&user->Qblock);
    1139:   MatSetSizes(user->Qblock,PETSC_DECIDE,PETSC_DECIDE,user->ndata,n);
    1140:   MatSetFromOptions(user->Qblock);
    1141:   MatMPIAIJSetPreallocation(user->Qblock,8,NULL,8,NULL);
    1142:   MatSeqAIJSetPreallocation(user->Qblock,8,NULL);

    1144:   for (i=0; i<user->mx; i++){
    1145:     x[i] = h*(i+0.5);
    1146:     y[i] = h*(i+0.5);
    1147:     z[i] = h*(i+0.5);
    1148:   }

    1150:   MatGetOwnershipRange(user->Qblock,&istart,&iend);
    1151:   nx = user->mx; ny = user->mx; nz = user->mx;
    1152:   for (i=istart; i<iend; i++){
    1153:     xri = xr[i];
    1154:     im = 0;
    1155:     xim = x[im];
    1156:     while (xri>xim && im<nx){
    1157:       im = im+1;
    1158:       xim = x[im];
    1159:     }
    1160:     indx1 = im-1;
    1161:     indx2 = im;
    1162:     dx1 = xri - x[indx1];
    1163:     dx2 = x[indx2] - xri;

    1165:     yri = yr[i];
    1166:     im = 0;
    1167:     yim = y[im];
    1168:     while (yri>yim && im<ny){
    1169:       im = im+1;
    1170:       yim = y[im];
    1171:     }
    1172:     indy1 = im-1;
    1173:     indy2 = im;
    1174:     dy1 = yri - y[indy1];
    1175:     dy2 = y[indy2] - yri;

    1177:     zri = zr[i];
    1178:     im = 0;
    1179:     zim = z[im];
    1180:     while (zri>zim && im<nz){
    1181:       im = im+1;
    1182:       zim = z[im];
    1183:     }
    1184:     indz1 = im-1;
    1185:     indz2 = im;
    1186:     dz1 = zri - z[indz1];
    1187:     dz2 = z[indz2] - zri;

    1189:     Dx = x[indx2] - x[indx1];
    1190:     Dy = y[indy2] - y[indy1];
    1191:     Dz = z[indz2] - z[indz1];

    1193:     j = indx1 + indy1*nx + indz1*nx*ny;
    1194:     v = (1-dx1/Dx)*(1-dy1/Dy)*(1-dz1/Dz);
    1195:     MatSetValues(user->Qblock,1,&i,1,&j,&v,INSERT_VALUES);

    1197:     j = indx1 + indy1*nx + indz2*nx*ny;
    1198:     v = (1-dx1/Dx)*(1-dy1/Dy)*(1-dz2/Dz);
    1199:     MatSetValues(user->Qblock,1,&i,1,&j,&v,INSERT_VALUES);

    1201:     j = indx1 + indy2*nx + indz1*nx*ny;
    1202:     v = (1-dx1/Dx)*(1-dy2/Dy)*(1-dz1/Dz);
    1203:     MatSetValues(user->Qblock,1,&i,1,&j,&v,INSERT_VALUES);

    1205:     j = indx1 + indy2*nx + indz2*nx*ny;
    1206:     v = (1-dx1/Dx)*(1-dy2/Dy)*(1-dz2/Dz);
    1207:     MatSetValues(user->Qblock,1,&i,1,&j,&v,INSERT_VALUES);

    1209:     j = indx2 + indy1*nx + indz1*nx*ny;
    1210:     v = (1-dx2/Dx)*(1-dy1/Dy)*(1-dz1/Dz);
    1211:     MatSetValues(user->Qblock,1,&i,1,&j,&v,INSERT_VALUES);

    1213:     j = indx2 + indy1*nx + indz2*nx*ny;
    1214:     v = (1-dx2/Dx)*(1-dy1/Dy)*(1-dz2/Dz);
    1215:     MatSetValues(user->Qblock,1,&i,1,&j,&v,INSERT_VALUES);

    1217:     j = indx2 + indy2*nx + indz1*nx*ny;
    1218:     v = (1-dx2/Dx)*(1-dy2/Dy)*(1-dz1/Dz);
    1219:     MatSetValues(user->Qblock,1,&i,1,&j,&v,INSERT_VALUES);

    1221:     j = indx2 + indy2*nx + indz2*nx*ny;
    1222:     v = (1-dx2/Dx)*(1-dy2/Dy)*(1-dz2/Dz);
    1223:     MatSetValues(user->Qblock,1,&i,1,&j,&v,INSERT_VALUES);
    1224:   }
    1225:   MatAssemblyBegin(user->Qblock,MAT_FINAL_ASSEMBLY);
    1226:   MatAssemblyEnd(user->Qblock,MAT_FINAL_ASSEMBLY);

    1228:   MatTranspose(user->Qblock,MAT_INITIAL_MATRIX,&user->QblockT);
    1229:   MatTranspose(user->L,MAT_INITIAL_MATRIX,&user->LT);

    1231:   /* Add noise to the measurement data */
    1232:   VecSet(user->ywork,1.0);
    1233:   VecAYPX(user->ywork,user->noise,user->ytrue);
    1234:   Scatter_i(user->ywork,user->yiwork,user->yi_scatter,user->nt);
    1235:   for (j=0; j<user->ns; j++){
    1236:     i = user->sample_times[j];
    1237:     MatMult(user->Qblock,user->yiwork[i],user->di[j]);
    1238:   }
    1239:   Gather_i(user->d,user->di,user->di_scatter,user->ns);

    1241:   /* Now that initial conditions have been set, let the user pass tolerance options to the KSP solver */
    1242:   KSPSetFromOptions(user->solver);
    1243:   PetscFree(x);
    1244:   PetscFree(y);
    1245:   PetscFree(z);
    1246:   return(0);
    1247: }

    1249: PetscErrorCode ParabolicDestroy(AppCtx *user)
    1250: {
    1252:   PetscInt       i;

    1255:   MatDestroy(&user->Qblock);
    1256:   MatDestroy(&user->QblockT);
    1257:   MatDestroy(&user->Div);
    1258:   MatDestroy(&user->Divwork);
    1259:   MatDestroy(&user->Grad);
    1260:   MatDestroy(&user->Av);
    1261:   MatDestroy(&user->Avwork);
    1262:   MatDestroy(&user->AvT);
    1263:   MatDestroy(&user->DSG);
    1264:   MatDestroy(&user->L);
    1265:   MatDestroy(&user->LT);
    1266:   KSPDestroy(&user->solver);
    1267:   MatDestroy(&user->Js);
    1268:   MatDestroy(&user->Jd);
    1269:   MatDestroy(&user->JsInv);
    1270:   MatDestroy(&user->JsBlock);
    1271:   MatDestroy(&user->JsBlockPrec);
    1272:   VecDestroy(&user->u);
    1273:   VecDestroy(&user->uwork);
    1274:   VecDestroy(&user->utrue);
    1275:   VecDestroy(&user->y);
    1276:   VecDestroy(&user->ywork);
    1277:   VecDestroy(&user->ytrue);
    1278:   VecDestroyVecs(user->nt,&user->yi);
    1279:   VecDestroyVecs(user->nt,&user->yiwork);
    1280:   VecDestroyVecs(user->ns,&user->di);
    1281:   PetscFree(user->yi);
    1282:   PetscFree(user->yiwork);
    1283:   PetscFree(user->di);
    1284:   VecDestroy(&user->c);
    1285:   VecDestroy(&user->cwork);
    1286:   VecDestroy(&user->ur);
    1287:   VecDestroy(&user->q);
    1288:   VecDestroy(&user->d);
    1289:   VecDestroy(&user->dwork);
    1290:   VecDestroy(&user->lwork);
    1291:   VecDestroy(&user->S);
    1292:   VecDestroy(&user->Swork);
    1293:   VecDestroy(&user->Av_u);
    1294:   VecDestroy(&user->Twork);
    1295:   VecDestroy(&user->Rwork);
    1296:   VecDestroy(&user->js_diag);
    1297:   ISDestroy(&user->s_is);
    1298:   ISDestroy(&user->d_is);
    1299:   VecScatterDestroy(&user->state_scatter);
    1300:   VecScatterDestroy(&user->design_scatter);
    1301:   for (i=0; i<user->nt; i++){
    1302:     VecScatterDestroy(&user->yi_scatter[i]);
    1303:   }
    1304:   for (i=0; i<user->ns; i++){
    1305:     VecScatterDestroy(&user->di_scatter[i]);
    1306:   }
    1307:   PetscFree(user->yi_scatter);
    1308:   PetscFree(user->di_scatter);
    1309:   PetscFree(user->sample_times);
    1310:   return(0);
    1311: }

    1313: PetscErrorCode ParabolicMonitor(Tao tao, void *ptr)
    1314: {
    1316:   Vec            X;
    1317:   PetscReal      unorm,ynorm;
    1318:   AppCtx         *user = (AppCtx*)ptr;

    1321:   TaoGetSolutionVector(tao,&X);
    1322:   Scatter(X,user->ywork,user->state_scatter,user->uwork,user->design_scatter);
    1323:   VecAXPY(user->ywork,-1.0,user->ytrue);
    1324:   VecAXPY(user->uwork,-1.0,user->utrue);
    1325:   VecNorm(user->uwork,NORM_2,&unorm);
    1326:   VecNorm(user->ywork,NORM_2,&ynorm);
    1327:   PetscPrintf(MPI_COMM_WORLD, "||u-ut||=%g ||y-yt||=%g\n",(double)unorm,(double)ynorm);
    1328:   return(0);
    1329: }

    1332: /*TEST

    1334:    build:
    1335:       requires: !complex

    1337:    test:
    1338:       args: -tao_cmonitor -tao_type lcl -ns 1 -tao_gatol 1.e-4 -ksp_max_it 30
    1339:       requires: !single

    1341: TEST*/