Date: 2022-03-31T04:49:07+00:00

Actual source code: hyperbolic.c

      1: #include <petsctao.h>

      3: /*T
      4:    Concepts: TAO^Solving a system of nonlinear equations, nonlinear least squares
      5:    Routines: TaoCreate();
      6:    Routines: TaoSetType();
      7:    Routines: TaoSetSolution();
      8:    Routines: TaoSetObjective();
      9:    Routines: TaoSetGradient();
     10:    Routines: TaoSetConstraintsRoutine();
     11:    Routines: TaoSetJacobianStateRoutine();
     12:    Routines: TaoSetJacobianDesignRoutine();
     13:    Routines: TaoSetStateDesignIS();
     14:    Routines: TaoSetFromOptions();
     15:    Routines: TaoSolve();
     16:    Routines: TaoDestroy();
     17:    Processors: 1
     18: T*/

     20: typedef struct {
     21:   PetscInt n; /*  Number of variables */
     22:   PetscInt m; /*  Number of constraints */
     23:   PetscInt mx; /*  grid points in each direction */
     24:   PetscInt nt; /*  Number of time steps */
     25:   PetscInt ndata; /*  Number of data points per sample */
     26:   IS       s_is;
     27:   IS       d_is;
     28:   VecScatter state_scatter;
     29:   VecScatter design_scatter;
     30:   VecScatter *uxi_scatter,*uyi_scatter,*ux_scatter,*uy_scatter,*ui_scatter;
     31:   VecScatter *yi_scatter;

     33:   Mat       Js,Jd,JsBlockPrec,JsInv,JsBlock;
     34:   PetscBool jformed,c_formed;

     36:   PetscReal alpha; /*  Regularization parameter */
     37:   PetscReal gamma;
     38:   PetscReal ht; /*  Time step */
     39:   PetscReal T; /*  Final time */
     40:   Mat Q,QT;
     41:   Mat L,LT;
     42:   Mat Div,Divwork,Divxy[2];
     43:   Mat Grad,Gradxy[2];
     44:   Mat M;
     45:   Mat *C,*Cwork;
     46:   /* Mat Hs,Hd,Hsd; */
     47:   Vec q;
     48:   Vec ur; /*  reference */

     50:   Vec d;
     51:   Vec dwork;

     53:   Vec y; /*  state variables */
     54:   Vec ywork;
     55:   Vec ytrue;
     56:   Vec *yi,*yiwork,*ziwork;
     57:   Vec *uxi,*uyi,*uxiwork,*uyiwork,*ui,*uiwork;

     59:   Vec u; /*  design variables */
     60:   Vec uwork,vwork;
     61:   Vec utrue;

     63:   Vec js_diag;

     65:   Vec c; /*  constraint vector */
     66:   Vec cwork;

     68:   Vec lwork;

     70:   KSP      solver;
     71:   PC       prec;
     72:   PetscInt block_index;

     74:   PetscInt ksp_its;
     75:   PetscInt ksp_its_initial;
     76: } AppCtx;

     78: PetscErrorCode FormFunction(Tao, Vec, PetscReal*, void*);
     79: PetscErrorCode FormGradient(Tao, Vec, Vec, void*);
     80: PetscErrorCode FormFunctionGradient(Tao, Vec, PetscReal*, Vec, void*);
     81: PetscErrorCode FormJacobianState(Tao, Vec, Mat, Mat, Mat, void*);
     82: PetscErrorCode FormJacobianDesign(Tao, Vec, Mat,void*);
     83: PetscErrorCode FormConstraints(Tao, Vec, Vec, void*);
     84: PetscErrorCode FormHessian(Tao, Vec, Mat, Mat, void*);
     85: PetscErrorCode Gather(Vec x, Vec state, VecScatter s_scat, Vec design, VecScatter d_scat);
     86: PetscErrorCode Scatter(Vec x, Vec state, VecScatter s_scat, Vec design, VecScatter d_scat);
     87: PetscErrorCode HyperbolicInitialize(AppCtx *user);
     88: PetscErrorCode HyperbolicDestroy(AppCtx *user);
     89: PetscErrorCode HyperbolicMonitor(Tao, void*);

     91: PetscErrorCode StateMatMult(Mat,Vec,Vec);
     92: PetscErrorCode StateMatBlockMult(Mat,Vec,Vec);
     93: PetscErrorCode StateMatBlockMultTranspose(Mat,Vec,Vec);
     94: PetscErrorCode StateMatMultTranspose(Mat,Vec,Vec);
     95: PetscErrorCode StateMatGetDiagonal(Mat,Vec);
     96: PetscErrorCode StateMatDuplicate(Mat,MatDuplicateOption,Mat*);
     97: PetscErrorCode StateMatInvMult(Mat,Vec,Vec);
     98: PetscErrorCode StateMatInvTransposeMult(Mat,Vec,Vec);
     99: PetscErrorCode StateMatBlockPrecMult(PC,Vec,Vec);

    101: PetscErrorCode DesignMatMult(Mat,Vec,Vec);
    102: PetscErrorCode DesignMatMultTranspose(Mat,Vec,Vec);

    104: PetscErrorCode Scatter_yi(Vec,Vec*,VecScatter*,PetscInt); /*  y to y1,y2,...,y_nt */
    105: PetscErrorCode Gather_yi(Vec,Vec*,VecScatter*,PetscInt);
    106: PetscErrorCode Scatter_uxi_uyi(Vec,Vec*,VecScatter*,Vec*,VecScatter*,PetscInt); /*  u to ux_1,uy_1,ux_2,uy_2,...,u */
    107: PetscErrorCode Gather_uxi_uyi(Vec,Vec*,VecScatter*,Vec*,VecScatter*,PetscInt);

    109: static  char help[]="";

    111: int main(int argc, char **argv)
    112: {
    113:   PetscErrorCode     ierr;
    114:   Vec                x,x0;
    115:   Tao                tao;
    116:   AppCtx             user;
    117:   IS                 is_allstate,is_alldesign;
    118:   PetscInt           lo,hi,hi2,lo2,ksp_old;
    119:   PetscInt           ntests = 1;
    120:   PetscInt           i;
    121: #if defined(PETSC_USE_LOG)
    122:   PetscLogStage      stages[1];
    123: #endif

    125:   PetscInitialize(&argc, &argv, (char*)0,help);
    126:   user.mx = 32;
    127:   PetscOptionsBegin(PETSC_COMM_WORLD,NULL,"hyperbolic example",NULL);
    128:   PetscOptionsInt("-mx","Number of grid points in each direction","",user.mx,&user.mx,NULL);
    129:   user.nt = 16;
    130:   PetscOptionsInt("-nt","Number of time steps","",user.nt,&user.nt,NULL);
    131:   user.ndata = 64;
    132:   PetscOptionsInt("-ndata","Numbers of data points per sample","",user.ndata,&user.ndata,NULL);
    133:   user.alpha = 10.0;
    134:   PetscOptionsReal("-alpha","Regularization parameter","",user.alpha,&user.alpha,NULL);
    135:   user.T = 1.0/32.0;
    136:   PetscOptionsReal("-Tfinal","Final time","",user.T,&user.T,NULL);
    137:   PetscOptionsInt("-ntests","Number of times to repeat TaoSolve","",ntests,&ntests,NULL);
    138:   PetscOptionsEnd();

    140:   user.m = user.mx*user.mx*user.nt; /*  number of constraints */
    141:   user.n = user.mx*user.mx*3*user.nt; /*  number of variables */
    142:   user.ht = user.T/user.nt; /*  Time step */
    143:   user.gamma = user.T*user.ht / (user.mx*user.mx);

    145:   VecCreate(PETSC_COMM_WORLD,&user.u);
    146:   VecCreate(PETSC_COMM_WORLD,&user.y);
    147:   VecCreate(PETSC_COMM_WORLD,&user.c);
    148:   VecSetSizes(user.u,PETSC_DECIDE,user.n-user.m);
    149:   VecSetSizes(user.y,PETSC_DECIDE,user.m);
    150:   VecSetSizes(user.c,PETSC_DECIDE,user.m);
    151:   VecSetFromOptions(user.u);
    152:   VecSetFromOptions(user.y);
    153:   VecSetFromOptions(user.c);

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

    165:   VecGetOwnershipRange(user.y,&lo,&hi);
    166:   VecGetOwnershipRange(user.u,&lo2,&hi2);

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

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

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

    177:   VecScatterCreate(x,user.s_is,user.y,is_allstate,&user.state_scatter);
    178:   VecScatterCreate(x,user.d_is,user.u,is_alldesign,&user.design_scatter);
    179:   ISDestroy(&is_alldesign);
    180:   ISDestroy(&is_allstate);

    182:   /* Create TAO solver and set desired solution method */
    183:   TaoCreate(PETSC_COMM_WORLD,&tao);
    184:   TaoSetType(tao,TAOLCL);

    186:   /* Set up initial vectors and matrices */
    187:   HyperbolicInitialize(&user);

    189:   Gather(x,user.y,user.state_scatter,user.u,user.design_scatter);
    190:   VecDuplicate(x,&x0);
    191:   VecCopy(x,x0);

    193:   /* Set solution vector with an initial guess */
    194:   TaoSetSolution(tao,x);
    195:   TaoSetObjective(tao, FormFunction, &user);
    196:   TaoSetGradient(tao, NULL, FormGradient, &user);
    197:   TaoSetConstraintsRoutine(tao, user.c, FormConstraints, &user);
    198:   TaoSetJacobianStateRoutine(tao, user.Js, user.Js, user.JsInv, FormJacobianState, &user);
    199:   TaoSetJacobianDesignRoutine(tao, user.Jd, FormJacobianDesign, &user);
    200:   TaoSetFromOptions(tao);
    201:   TaoSetStateDesignIS(tao,user.s_is,user.d_is);

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

    223:   TaoDestroy(&tao);
    224:   VecDestroy(&x);
    225:   VecDestroy(&x0);
    226:   HyperbolicDestroy(&user);
    227:   PetscFinalize();
    228:   return 0;
    229: }
    230: /* ------------------------------------------------------------------- */
    231: /*
    232:    dwork = Qy - d
    233:    lwork = L*(u-ur).^2
    234:    f = 1/2 * (dwork.dork + alpha*y.lwork)
    235: */
    236: PetscErrorCode FormFunction(Tao tao,Vec X,PetscReal *f,void *ptr)
    237: {
    238:   PetscReal      d1=0,d2=0;
    239:   AppCtx         *user = (AppCtx*)ptr;

    241:   Scatter(X,user->y,user->state_scatter,user->u,user->design_scatter);
    242:   MatMult(user->Q,user->y,user->dwork);
    243:   VecAXPY(user->dwork,-1.0,user->d);
    244:   VecDot(user->dwork,user->dwork,&d1);

    246:   VecWAXPY(user->uwork,-1.0,user->ur,user->u);
    247:   VecPointwiseMult(user->uwork,user->uwork,user->uwork);
    248:   MatMult(user->L,user->uwork,user->lwork);
    249:   VecDot(user->y,user->lwork,&d2);
    250:   *f = 0.5 * (d1 + user->alpha*d2);
    251:   return 0;
    252: }

    254: /* ------------------------------------------------------------------- */
    255: /*
    256:     state: g_s = Q' *(Qy - d) + 0.5*alpha*L*(u-ur).^2
    257:     design: g_d = alpha*(L'y).*(u-ur)
    258: */
    259: PetscErrorCode FormGradient(Tao tao,Vec X,Vec G,void *ptr)
    260: {
    261:   AppCtx         *user = (AppCtx*)ptr;

    263:   Scatter(X,user->y,user->state_scatter,user->u,user->design_scatter);
    264:   MatMult(user->Q,user->y,user->dwork);
    265:   VecAXPY(user->dwork,-1.0,user->d);

    267:   MatMult(user->QT,user->dwork,user->ywork);

    269:   MatMult(user->LT,user->y,user->uwork);
    270:   VecWAXPY(user->vwork,-1.0,user->ur,user->u);
    271:   VecPointwiseMult(user->uwork,user->vwork,user->uwork);
    272:   VecScale(user->uwork,user->alpha);

    274:   VecPointwiseMult(user->vwork,user->vwork,user->vwork);
    275:   MatMult(user->L,user->vwork,user->lwork);
    276:   VecAXPY(user->ywork,0.5*user->alpha,user->lwork);

    278:   Gather(G,user->ywork,user->state_scatter,user->uwork,user->design_scatter);
    279:   return 0;
    280: }

    282: PetscErrorCode FormFunctionGradient(Tao tao, Vec X, PetscReal *f, Vec G, void *ptr)
    283: {
    284:   PetscReal      d1,d2;
    285:   AppCtx         *user = (AppCtx*)ptr;

    287:   Scatter(X,user->y,user->state_scatter,user->u,user->design_scatter);
    288:   MatMult(user->Q,user->y,user->dwork);
    289:   VecAXPY(user->dwork,-1.0,user->d);

    291:   MatMult(user->QT,user->dwork,user->ywork);

    293:   VecDot(user->dwork,user->dwork,&d1);

    295:   MatMult(user->LT,user->y,user->uwork);
    296:   VecWAXPY(user->vwork,-1.0,user->ur,user->u);
    297:   VecPointwiseMult(user->uwork,user->vwork,user->uwork);
    298:   VecScale(user->uwork,user->alpha);

    300:   VecPointwiseMult(user->vwork,user->vwork,user->vwork);
    301:   MatMult(user->L,user->vwork,user->lwork);
    302:   VecAXPY(user->ywork,0.5*user->alpha,user->lwork);

    304:   VecDot(user->y,user->lwork,&d2);

    306:   *f = 0.5 * (d1 + user->alpha*d2);
    307:   Gather(G,user->ywork,user->state_scatter,user->uwork,user->design_scatter);
    308:   return 0;
    309: }

    311: /* ------------------------------------------------------------------- */
    312: /* A
    313: MatShell object
    314: */
    315: PetscErrorCode FormJacobianState(Tao tao, Vec X, Mat J, Mat JPre, Mat JInv, void *ptr)
    316: {
    317:   PetscInt       i;
    318:   AppCtx         *user = (AppCtx*)ptr;

    320:   Scatter(X,user->y,user->state_scatter,user->u,user->design_scatter);
    321:   Scatter_yi(user->u,user->ui,user->ui_scatter,user->nt);
    322:   Scatter_uxi_uyi(user->u,user->uxi,user->uxi_scatter,user->uyi,user->uyi_scatter,user->nt);
    323:   for (i=0; i<user->nt; i++) {
    324:     MatCopy(user->Divxy[0],user->C[i],SUBSET_NONZERO_PATTERN);
    325:     MatCopy(user->Divxy[1],user->Cwork[i],SAME_NONZERO_PATTERN);

    327:     MatDiagonalScale(user->C[i],NULL,user->uxi[i]);
    328:     MatDiagonalScale(user->Cwork[i],NULL,user->uyi[i]);
    329:     MatAXPY(user->C[i],1.0,user->Cwork[i],SUBSET_NONZERO_PATTERN);
    330:     MatScale(user->C[i],user->ht);
    331:     MatShift(user->C[i],1.0);
    332:   }
    333:   return 0;
    334: }

    336: /* ------------------------------------------------------------------- */
    337: /* B */
    338: PetscErrorCode FormJacobianDesign(Tao tao, Vec X, Mat J, void *ptr)
    339: {
    340:   AppCtx         *user = (AppCtx*)ptr;

    342:   Scatter(X,user->y,user->state_scatter,user->u,user->design_scatter);
    343:   return 0;
    344: }

    346: PetscErrorCode StateMatMult(Mat J_shell, Vec X, Vec Y)
    347: {
    348:   PetscInt       i;
    349:   AppCtx         *user;

    351:   MatShellGetContext(J_shell,&user);
    352:   Scatter_yi(X,user->yi,user->yi_scatter,user->nt);
    353:   user->block_index = 0;
    354:   MatMult(user->JsBlock,user->yi[0],user->yiwork[0]);

    356:   for (i=1; i<user->nt; i++) {
    357:     user->block_index = i;
    358:     MatMult(user->JsBlock,user->yi[i],user->yiwork[i]);
    359:     MatMult(user->M,user->yi[i-1],user->ziwork[i-1]);
    360:     VecAXPY(user->yiwork[i],-1.0,user->ziwork[i-1]);
    361:   }
    362:   Gather_yi(Y,user->yiwork,user->yi_scatter,user->nt);
    363:   return 0;
    364: }

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

    371:   MatShellGetContext(J_shell,&user);
    372:   Scatter_yi(X,user->yi,user->yi_scatter,user->nt);

    374:   for (i=0; i<user->nt-1; i++) {
    375:     user->block_index = i;
    376:     MatMultTranspose(user->JsBlock,user->yi[i],user->yiwork[i]);
    377:     MatMult(user->M,user->yi[i+1],user->ziwork[i+1]);
    378:     VecAXPY(user->yiwork[i],-1.0,user->ziwork[i+1]);
    379:   }

    381:   i = user->nt-1;
    382:   user->block_index = i;
    383:   MatMultTranspose(user->JsBlock,user->yi[i],user->yiwork[i]);
    384:   Gather_yi(Y,user->yiwork,user->yi_scatter,user->nt);
    385:   return 0;
    386: }

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

    393:   MatShellGetContext(J_shell,&user);
    394:   i = user->block_index;
    395:   VecPointwiseMult(user->uxiwork[i],X,user->uxi[i]);
    396:   VecPointwiseMult(user->uyiwork[i],X,user->uyi[i]);
    397:   Gather(user->uiwork[i],user->uxiwork[i],user->ux_scatter[i],user->uyiwork[i],user->uy_scatter[i]);
    398:   MatMult(user->Div,user->uiwork[i],Y);
    399:   VecAYPX(Y,user->ht,X);
    400:   return 0;
    401: }

    403: PetscErrorCode StateMatBlockMultTranspose(Mat J_shell, Vec X, Vec Y)
    404: {
    405:   PetscInt       i;
    406:   AppCtx         *user;

    408:   MatShellGetContext(J_shell,&user);
    409:   i = user->block_index;
    410:   MatMult(user->Grad,X,user->uiwork[i]);
    411:   Scatter(user->uiwork[i],user->uxiwork[i],user->ux_scatter[i],user->uyiwork[i],user->uy_scatter[i]);
    412:   VecPointwiseMult(user->uxiwork[i],user->uxi[i],user->uxiwork[i]);
    413:   VecPointwiseMult(user->uyiwork[i],user->uyi[i],user->uyiwork[i]);
    414:   VecWAXPY(Y,1.0,user->uxiwork[i],user->uyiwork[i]);
    415:   VecAYPX(Y,user->ht,X);
    416:   return 0;
    417: }

    419: PetscErrorCode DesignMatMult(Mat J_shell, Vec X, Vec Y)
    420: {
    421:   PetscInt       i;
    422:   AppCtx         *user;

    424:   MatShellGetContext(J_shell,&user);
    425:   Scatter_yi(user->y,user->yi,user->yi_scatter,user->nt);
    426:   Scatter_uxi_uyi(X,user->uxiwork,user->uxi_scatter,user->uyiwork,user->uyi_scatter,user->nt);
    427:   for (i=0; i<user->nt; i++) {
    428:     VecPointwiseMult(user->uxiwork[i],user->yi[i],user->uxiwork[i]);
    429:     VecPointwiseMult(user->uyiwork[i],user->yi[i],user->uyiwork[i]);
    430:     Gather(user->uiwork[i],user->uxiwork[i],user->ux_scatter[i],user->uyiwork[i],user->uy_scatter[i]);
    431:     MatMult(user->Div,user->uiwork[i],user->ziwork[i]);
    432:     VecScale(user->ziwork[i],user->ht);
    433:   }
    434:   Gather_yi(Y,user->ziwork,user->yi_scatter,user->nt);
    435:   return 0;
    436: }

    438: PetscErrorCode DesignMatMultTranspose(Mat J_shell, Vec X, Vec Y)
    439: {
    440:   PetscInt       i;
    441:   AppCtx         *user;

    443:   MatShellGetContext(J_shell,&user);
    444:   Scatter_yi(user->y,user->yi,user->yi_scatter,user->nt);
    445:   Scatter_yi(X,user->yiwork,user->yi_scatter,user->nt);
    446:   for (i=0; i<user->nt; i++) {
    447:     MatMult(user->Grad,user->yiwork[i],user->uiwork[i]);
    448:     Scatter(user->uiwork[i],user->uxiwork[i],user->ux_scatter[i],user->uyiwork[i],user->uy_scatter[i]);
    449:     VecPointwiseMult(user->uxiwork[i],user->yi[i],user->uxiwork[i]);
    450:     VecPointwiseMult(user->uyiwork[i],user->yi[i],user->uyiwork[i]);
    451:     Gather(user->uiwork[i],user->uxiwork[i],user->ux_scatter[i],user->uyiwork[i],user->uy_scatter[i]);
    452:     VecScale(user->uiwork[i],user->ht);
    453:   }
    454:   Gather_yi(Y,user->uiwork,user->ui_scatter,user->nt);
    455:   return 0;
    456: }

    458: PetscErrorCode StateMatBlockPrecMult(PC PC_shell, Vec X, Vec Y)
    459: {
    460:   PetscInt       i;
    461:   AppCtx         *user;

    463:   PCShellGetContext(PC_shell,&user);
    464:   i = user->block_index;
    465:   if (user->c_formed) {
    466:     MatSOR(user->C[i],X,1.0,(MatSORType)(SOR_ZERO_INITIAL_GUESS | SOR_LOCAL_SYMMETRIC_SWEEP),0.0,1,1,Y);
    467:   } else SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"Not formed");
    468:   return 0;
    469: }

    471: PetscErrorCode StateMatBlockPrecMultTranspose(PC PC_shell, Vec X, Vec Y)
    472: {
    473:   PetscInt       i;
    474:   AppCtx         *user;

    476:   PCShellGetContext(PC_shell,&user);

    478:   i = user->block_index;
    479:   if (user->c_formed) {
    480:     MatSOR(user->C[i],X,1.0,(MatSORType)(SOR_ZERO_INITIAL_GUESS | SOR_LOCAL_SYMMETRIC_SWEEP),0.0,1,1,Y);
    481:   } else SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"Not formed");
    482:   return 0;
    483: }

    485: PetscErrorCode StateMatInvMult(Mat J_shell, Vec X, Vec Y)
    486: {
    487:   AppCtx         *user;
    488:   PetscInt       its,i;

    490:   MatShellGetContext(J_shell,&user);

    492:   if (Y == user->ytrue) {
    493:     /* First solve is done using true solution to set up problem */
    494:     KSPSetTolerances(user->solver,1e-4,1e-20,PETSC_DEFAULT,PETSC_DEFAULT);
    495:   } else {
    496:     KSPSetTolerances(user->solver,PETSC_DEFAULT,PETSC_DEFAULT,PETSC_DEFAULT,PETSC_DEFAULT);
    497:   }
    498:   Scatter_yi(X,user->yi,user->yi_scatter,user->nt);
    499:   Scatter_yi(Y,user->yiwork,user->yi_scatter,user->nt);
    500:   Scatter_uxi_uyi(user->u,user->uxi,user->uxi_scatter,user->uyi,user->uyi_scatter,user->nt);

    502:   user->block_index = 0;
    503:   KSPSolve(user->solver,user->yi[0],user->yiwork[0]);

    505:   KSPGetIterationNumber(user->solver,&its);
    506:   user->ksp_its = user->ksp_its + its;
    507:   for (i=1; i<user->nt; i++) {
    508:     MatMult(user->M,user->yiwork[i-1],user->ziwork[i-1]);
    509:     VecAXPY(user->yi[i],1.0,user->ziwork[i-1]);
    510:     user->block_index = i;
    511:     KSPSolve(user->solver,user->yi[i],user->yiwork[i]);

    513:     KSPGetIterationNumber(user->solver,&its);
    514:     user->ksp_its = user->ksp_its + its;
    515:   }
    516:   Gather_yi(Y,user->yiwork,user->yi_scatter,user->nt);
    517:   return 0;
    518: }

    520: PetscErrorCode StateMatInvTransposeMult(Mat J_shell, Vec X, Vec Y)
    521: {
    522:   AppCtx         *user;
    523:   PetscInt       its,i;

    525:   MatShellGetContext(J_shell,&user);

    527:   Scatter_yi(X,user->yi,user->yi_scatter,user->nt);
    528:   Scatter_yi(Y,user->yiwork,user->yi_scatter,user->nt);
    529:   Scatter_uxi_uyi(user->u,user->uxi,user->uxi_scatter,user->uyi,user->uyi_scatter,user->nt);

    531:   i = user->nt - 1;
    532:   user->block_index = i;
    533:   KSPSolveTranspose(user->solver,user->yi[i],user->yiwork[i]);

    535:   KSPGetIterationNumber(user->solver,&its);
    536:   user->ksp_its = user->ksp_its + its;

    538:   for (i=user->nt-2; i>=0; i--) {
    539:     MatMult(user->M,user->yiwork[i+1],user->ziwork[i+1]);
    540:     VecAXPY(user->yi[i],1.0,user->ziwork[i+1]);
    541:     user->block_index = i;
    542:     KSPSolveTranspose(user->solver,user->yi[i],user->yiwork[i]);

    544:     KSPGetIterationNumber(user->solver,&its);
    545:     user->ksp_its = user->ksp_its + its;
    546:   }
    547:   Gather_yi(Y,user->yiwork,user->yi_scatter,user->nt);
    548:   return 0;
    549: }

    551: PetscErrorCode StateMatDuplicate(Mat J_shell, MatDuplicateOption opt, Mat *new_shell)
    552: {
    553:   AppCtx         *user;

    555:   MatShellGetContext(J_shell,&user);

    557:   MatCreateShell(PETSC_COMM_WORLD,PETSC_DETERMINE,PETSC_DETERMINE,user->m,user->m,user,new_shell);
    558:   MatShellSetOperation(*new_shell,MATOP_MULT,(void(*)(void))StateMatMult);
    559:   MatShellSetOperation(*new_shell,MATOP_DUPLICATE,(void(*)(void))StateMatDuplicate);
    560:   MatShellSetOperation(*new_shell,MATOP_MULT_TRANSPOSE,(void(*)(void))StateMatMultTranspose);
    561:   MatShellSetOperation(*new_shell,MATOP_GET_DIAGONAL,(void(*)(void))StateMatGetDiagonal);
    562:   return 0;
    563: }

    565: PetscErrorCode StateMatGetDiagonal(Mat J_shell, Vec X)
    566: {
    567:   AppCtx         *user;

    569:   MatShellGetContext(J_shell,&user);
    570:   VecCopy(user->js_diag,X);
    571:   return 0;
    572: }

    574: PetscErrorCode FormConstraints(Tao tao, Vec X, Vec C, void *ptr)
    575: {
    576:   /* con = Ay - q, A = [C(u1)  0     0     ...   0;
    577:                          -M  C(u2)   0     ...   0;
    578:                           0   -M   C(u3)   ...   0;
    579:                                       ...         ;
    580:                           0    ...      -M C(u_nt)]
    581:      C(u) = eye + ht*Div*[diag(u1); diag(u2)]       */
    582:   PetscInt       i;
    583:   AppCtx         *user = (AppCtx*)ptr;

    585:   Scatter(X,user->y,user->state_scatter,user->u,user->design_scatter);
    586:   Scatter_yi(user->y,user->yi,user->yi_scatter,user->nt);
    587:   Scatter_uxi_uyi(user->u,user->uxi,user->uxi_scatter,user->uyi,user->uyi_scatter,user->nt);

    589:   user->block_index = 0;
    590:   MatMult(user->JsBlock,user->yi[0],user->yiwork[0]);

    592:   for (i=1; i<user->nt; i++) {
    593:     user->block_index = i;
    594:     MatMult(user->JsBlock,user->yi[i],user->yiwork[i]);
    595:     MatMult(user->M,user->yi[i-1],user->ziwork[i-1]);
    596:     VecAXPY(user->yiwork[i],-1.0,user->ziwork[i-1]);
    597:   }

    599:   Gather_yi(C,user->yiwork,user->yi_scatter,user->nt);
    600:   VecAXPY(C,-1.0,user->q);

    602:   return 0;
    603: }

    605: PetscErrorCode Scatter(Vec x, Vec state, VecScatter s_scat, Vec design, VecScatter d_scat)
    606: {
    607:   VecScatterBegin(s_scat,x,state,INSERT_VALUES,SCATTER_FORWARD);
    608:   VecScatterEnd(s_scat,x,state,INSERT_VALUES,SCATTER_FORWARD);
    609:   VecScatterBegin(d_scat,x,design,INSERT_VALUES,SCATTER_FORWARD);
    610:   VecScatterEnd(d_scat,x,design,INSERT_VALUES,SCATTER_FORWARD);
    611:   return 0;
    612: }

    614: PetscErrorCode Scatter_uxi_uyi(Vec u, Vec *uxi, VecScatter *scatx, Vec *uyi, VecScatter *scaty, PetscInt nt)
    615: {
    616:   PetscInt       i;

    618:   for (i=0; i<nt; i++) {
    619:     VecScatterBegin(scatx[i],u,uxi[i],INSERT_VALUES,SCATTER_FORWARD);
    620:     VecScatterEnd(scatx[i],u,uxi[i],INSERT_VALUES,SCATTER_FORWARD);
    621:     VecScatterBegin(scaty[i],u,uyi[i],INSERT_VALUES,SCATTER_FORWARD);
    622:     VecScatterEnd(scaty[i],u,uyi[i],INSERT_VALUES,SCATTER_FORWARD);
    623:   }
    624:   return 0;
    625: }

    627: PetscErrorCode Gather(Vec x, Vec state, VecScatter s_scat, Vec design, VecScatter d_scat)
    628: {
    629:   VecScatterBegin(s_scat,state,x,INSERT_VALUES,SCATTER_REVERSE);
    630:   VecScatterEnd(s_scat,state,x,INSERT_VALUES,SCATTER_REVERSE);
    631:   VecScatterBegin(d_scat,design,x,INSERT_VALUES,SCATTER_REVERSE);
    632:   VecScatterEnd(d_scat,design,x,INSERT_VALUES,SCATTER_REVERSE);
    633:   return 0;
    634: }

    636: PetscErrorCode Gather_uxi_uyi(Vec u, Vec *uxi, VecScatter *scatx, Vec *uyi, VecScatter *scaty, PetscInt nt)
    637: {
    638:   PetscInt       i;

    640:   for (i=0; i<nt; i++) {
    641:     VecScatterBegin(scatx[i],uxi[i],u,INSERT_VALUES,SCATTER_REVERSE);
    642:     VecScatterEnd(scatx[i],uxi[i],u,INSERT_VALUES,SCATTER_REVERSE);
    643:     VecScatterBegin(scaty[i],uyi[i],u,INSERT_VALUES,SCATTER_REVERSE);
    644:     VecScatterEnd(scaty[i],uyi[i],u,INSERT_VALUES,SCATTER_REVERSE);
    645:   }
    646:   return 0;
    647: }

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

    653:   for (i=0; i<nt; i++) {
    654:     VecScatterBegin(scat[i],y,yi[i],INSERT_VALUES,SCATTER_FORWARD);
    655:     VecScatterEnd(scat[i],y,yi[i],INSERT_VALUES,SCATTER_FORWARD);
    656:   }
    657:   return 0;
    658: }

    660: PetscErrorCode Gather_yi(Vec y, Vec *yi, VecScatter *scat, PetscInt nt)
    661: {
    662:   PetscInt       i;

    664:   for (i=0; i<nt; i++) {
    665:     VecScatterBegin(scat[i],yi[i],y,INSERT_VALUES,SCATTER_REVERSE);
    666:     VecScatterEnd(scat[i],yi[i],y,INSERT_VALUES,SCATTER_REVERSE);
    667:   }
    668:   return 0;
    669: }

    671: PetscErrorCode HyperbolicInitialize(AppCtx *user)
    672: {
    673:   PetscInt       n,i,j,linear_index,istart,iend,iblock,lo,hi;
    674:   Vec            XX,YY,XXwork,YYwork,yi,uxi,ui,bc;
    675:   PetscReal      h,sum;
    676:   PetscScalar    hinv,neg_hinv,quarter=0.25,one=1.0,half_hinv,neg_half_hinv;
    677:   PetscScalar    vx,vy,zero=0.0;
    678:   IS             is_from_y,is_to_yi,is_from_u,is_to_uxi,is_to_uyi;

    680:   user->jformed = PETSC_FALSE;
    681:   user->c_formed = PETSC_FALSE;

    683:   user->ksp_its = 0;
    684:   user->ksp_its_initial = 0;

    686:   n = user->mx * user->mx;

    688:   h = 1.0/user->mx;
    689:   hinv = user->mx;
    690:   neg_hinv = -hinv;
    691:   half_hinv = hinv / 2.0;
    692:   neg_half_hinv = neg_hinv / 2.0;

    694:   /* Generate Grad matrix */
    695:   MatCreate(PETSC_COMM_WORLD,&user->Grad);
    696:   MatSetSizes(user->Grad,PETSC_DECIDE,PETSC_DECIDE,2*n,n);
    697:   MatSetFromOptions(user->Grad);
    698:   MatMPIAIJSetPreallocation(user->Grad,3,NULL,3,NULL);
    699:   MatSeqAIJSetPreallocation(user->Grad,3,NULL);
    700:   MatGetOwnershipRange(user->Grad,&istart,&iend);

    702:   for (i=istart; i<iend; i++) {
    703:     if (i<n) {
    704:       iblock = i / user->mx;
    705:       j = iblock*user->mx + ((i+user->mx-1) % user->mx);
    706:       MatSetValues(user->Grad,1,&i,1,&j,&half_hinv,INSERT_VALUES);
    707:       j = iblock*user->mx + ((i+1) % user->mx);
    708:       MatSetValues(user->Grad,1,&i,1,&j,&neg_half_hinv,INSERT_VALUES);
    709:     }
    710:     if (i>=n) {
    711:       j = (i - user->mx) % n;
    712:       MatSetValues(user->Grad,1,&i,1,&j,&half_hinv,INSERT_VALUES);
    713:       j = (j + 2*user->mx) % n;
    714:       MatSetValues(user->Grad,1,&i,1,&j,&neg_half_hinv,INSERT_VALUES);
    715:     }
    716:   }

    718:   MatAssemblyBegin(user->Grad,MAT_FINAL_ASSEMBLY);
    719:   MatAssemblyEnd(user->Grad,MAT_FINAL_ASSEMBLY);

    721:   MatCreate(PETSC_COMM_WORLD,&user->Gradxy[0]);
    722:   MatSetSizes(user->Gradxy[0],PETSC_DECIDE,PETSC_DECIDE,n,n);
    723:   MatSetFromOptions(user->Gradxy[0]);
    724:   MatMPIAIJSetPreallocation(user->Gradxy[0],3,NULL,3,NULL);
    725:   MatSeqAIJSetPreallocation(user->Gradxy[0],3,NULL);
    726:   MatGetOwnershipRange(user->Gradxy[0],&istart,&iend);

    728:   for (i=istart; i<iend; i++) {
    729:     iblock = i / user->mx;
    730:     j = iblock*user->mx + ((i+user->mx-1) % user->mx);
    731:     MatSetValues(user->Gradxy[0],1,&i,1,&j,&half_hinv,INSERT_VALUES);
    732:     j = iblock*user->mx + ((i+1) % user->mx);
    733:     MatSetValues(user->Gradxy[0],1,&i,1,&j,&neg_half_hinv,INSERT_VALUES);
    734:     MatSetValues(user->Gradxy[0],1,&i,1,&i,&zero,INSERT_VALUES);
    735:   }
    736:   MatAssemblyBegin(user->Gradxy[0],MAT_FINAL_ASSEMBLY);
    737:   MatAssemblyEnd(user->Gradxy[0],MAT_FINAL_ASSEMBLY);

    739:   MatCreate(PETSC_COMM_WORLD,&user->Gradxy[1]);
    740:   MatSetSizes(user->Gradxy[1],PETSC_DECIDE,PETSC_DECIDE,n,n);
    741:   MatSetFromOptions(user->Gradxy[1]);
    742:   MatMPIAIJSetPreallocation(user->Gradxy[1],3,NULL,3,NULL);
    743:   MatSeqAIJSetPreallocation(user->Gradxy[1],3,NULL);
    744:   MatGetOwnershipRange(user->Gradxy[1],&istart,&iend);

    746:   for (i=istart; i<iend; i++) {
    747:     j = (i + n - user->mx) % n;
    748:     MatSetValues(user->Gradxy[1],1,&i,1,&j,&half_hinv,INSERT_VALUES);
    749:     j = (j + 2*user->mx) % n;
    750:     MatSetValues(user->Gradxy[1],1,&i,1,&j,&neg_half_hinv,INSERT_VALUES);
    751:     MatSetValues(user->Gradxy[1],1,&i,1,&i,&zero,INSERT_VALUES);
    752:   }
    753:   MatAssemblyBegin(user->Gradxy[1],MAT_FINAL_ASSEMBLY);
    754:   MatAssemblyEnd(user->Gradxy[1],MAT_FINAL_ASSEMBLY);

    756:   /* Generate Div matrix */
    757:   MatTranspose(user->Grad,MAT_INITIAL_MATRIX,&user->Div);
    758:   MatTranspose(user->Gradxy[0],MAT_INITIAL_MATRIX,&user->Divxy[0]);
    759:   MatTranspose(user->Gradxy[1],MAT_INITIAL_MATRIX,&user->Divxy[1]);

    761:   /* Off-diagonal averaging matrix */
    762:   MatCreate(PETSC_COMM_WORLD,&user->M);
    763:   MatSetSizes(user->M,PETSC_DECIDE,PETSC_DECIDE,n,n);
    764:   MatSetFromOptions(user->M);
    765:   MatMPIAIJSetPreallocation(user->M,4,NULL,4,NULL);
    766:   MatSeqAIJSetPreallocation(user->M,4,NULL);
    767:   MatGetOwnershipRange(user->M,&istart,&iend);

    769:   for (i=istart; i<iend; i++) {
    770:     /* kron(Id,Av) */
    771:     iblock = i / user->mx;
    772:     j = iblock*user->mx + ((i+user->mx-1) % user->mx);
    773:     MatSetValues(user->M,1,&i,1,&j,&quarter,INSERT_VALUES);
    774:     j = iblock*user->mx + ((i+1) % user->mx);
    775:     MatSetValues(user->M,1,&i,1,&j,&quarter,INSERT_VALUES);

    777:     /* kron(Av,Id) */
    778:     j = (i + user->mx) % n;
    779:     MatSetValues(user->M,1,&i,1,&j,&quarter,INSERT_VALUES);
    780:     j = (i + n - user->mx) % n;
    781:     MatSetValues(user->M,1,&i,1,&j,&quarter,INSERT_VALUES);
    782:   }
    783:   MatAssemblyBegin(user->M,MAT_FINAL_ASSEMBLY);
    784:   MatAssemblyEnd(user->M,MAT_FINAL_ASSEMBLY);

    786:   /* Generate 2D grid */
    787:   VecCreate(PETSC_COMM_WORLD,&XX);
    788:   VecCreate(PETSC_COMM_WORLD,&user->q);
    789:   VecSetSizes(XX,PETSC_DECIDE,n);
    790:   VecSetSizes(user->q,PETSC_DECIDE,n*user->nt);
    791:   VecSetFromOptions(XX);
    792:   VecSetFromOptions(user->q);

    794:   VecDuplicate(XX,&YY);
    795:   VecDuplicate(XX,&XXwork);
    796:   VecDuplicate(XX,&YYwork);
    797:   VecDuplicate(XX,&user->d);
    798:   VecDuplicate(XX,&user->dwork);

    800:   VecGetOwnershipRange(XX,&istart,&iend);
    801:   for (linear_index=istart; linear_index<iend; linear_index++) {
    802:     i = linear_index % user->mx;
    803:     j = (linear_index-i)/user->mx;
    804:     vx = h*(i+0.5);
    805:     vy = h*(j+0.5);
    806:     VecSetValues(XX,1,&linear_index,&vx,INSERT_VALUES);
    807:     VecSetValues(YY,1,&linear_index,&vy,INSERT_VALUES);
    808:   }

    810:   VecAssemblyBegin(XX);
    811:   VecAssemblyEnd(XX);
    812:   VecAssemblyBegin(YY);
    813:   VecAssemblyEnd(YY);

    815:   /* Compute final density function yT
    816:      yT = 1.0 + exp(-30*((x-0.25)^2+(y-0.25)^2)) + exp(-30*((x-0.75)^2+(y-0.75)^2))
    817:      yT = yT / (h^2*sum(yT)) */
    818:   VecCopy(XX,XXwork);
    819:   VecCopy(YY,YYwork);

    821:   VecShift(XXwork,-0.25);
    822:   VecShift(YYwork,-0.25);

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

    827:   VecCopy(XXwork,user->dwork);
    828:   VecAXPY(user->dwork,1.0,YYwork);
    829:   VecScale(user->dwork,-30.0);
    830:   VecExp(user->dwork);
    831:   VecCopy(user->dwork,user->d);

    833:   VecCopy(XX,XXwork);
    834:   VecCopy(YY,YYwork);

    836:   VecShift(XXwork,-0.75);
    837:   VecShift(YYwork,-0.75);

    839:   VecPointwiseMult(XXwork,XXwork,XXwork);
    840:   VecPointwiseMult(YYwork,YYwork,YYwork);

    842:   VecCopy(XXwork,user->dwork);
    843:   VecAXPY(user->dwork,1.0,YYwork);
    844:   VecScale(user->dwork,-30.0);
    845:   VecExp(user->dwork);

    847:   VecAXPY(user->d,1.0,user->dwork);
    848:   VecShift(user->d,1.0);
    849:   VecSum(user->d,&sum);
    850:   VecScale(user->d,1.0/(h*h*sum));

    852:   /* Initial conditions of forward problem */
    853:   VecDuplicate(XX,&bc);
    854:   VecCopy(XX,XXwork);
    855:   VecCopy(YY,YYwork);

    857:   VecShift(XXwork,-0.5);
    858:   VecShift(YYwork,-0.5);

    860:   VecPointwiseMult(XXwork,XXwork,XXwork);
    861:   VecPointwiseMult(YYwork,YYwork,YYwork);

    863:   VecWAXPY(bc,1.0,XXwork,YYwork);
    864:   VecScale(bc,-50.0);
    865:   VecExp(bc);
    866:   VecShift(bc,1.0);
    867:   VecSum(bc,&sum);
    868:   VecScale(bc,1.0/(h*h*sum));

    870:   /* Create scatter from y to y_1,y_2,...,y_nt */
    871:   /*  TODO: Reorder for better parallelism. (This will require reordering Q and L as well.) */
    872:   PetscMalloc1(user->nt*user->mx*user->mx,&user->yi_scatter);
    873:   VecCreate(PETSC_COMM_WORLD,&yi);
    874:   VecSetSizes(yi,PETSC_DECIDE,user->mx*user->mx);
    875:   VecSetFromOptions(yi);
    876:   VecDuplicateVecs(yi,user->nt,&user->yi);
    877:   VecDuplicateVecs(yi,user->nt,&user->yiwork);
    878:   VecDuplicateVecs(yi,user->nt,&user->ziwork);
    879:   for (i=0; i<user->nt; i++) {
    880:     VecGetOwnershipRange(user->yi[i],&lo,&hi);
    881:     ISCreateStride(PETSC_COMM_SELF,hi-lo,lo,1,&is_to_yi);
    882:     ISCreateStride(PETSC_COMM_SELF,hi-lo,lo+i*user->mx*user->mx,1,&is_from_y);
    883:     VecScatterCreate(user->y,is_from_y,user->yi[i],is_to_yi,&user->yi_scatter[i]);
    884:     ISDestroy(&is_to_yi);
    885:     ISDestroy(&is_from_y);
    886:   }

    888:   /* Create scatter from u to ux_1,uy_1,ux_2,uy_2,...,ux_nt,uy_nt */
    889:   /*  TODO: reorder for better parallelism */
    890:   PetscMalloc1(user->nt*user->mx*user->mx,&user->uxi_scatter);
    891:   PetscMalloc1(user->nt*user->mx*user->mx,&user->uyi_scatter);
    892:   PetscMalloc1(user->nt*user->mx*user->mx,&user->ux_scatter);
    893:   PetscMalloc1(user->nt*user->mx*user->mx,&user->uy_scatter);
    894:   PetscMalloc1(2*user->nt*user->mx*user->mx,&user->ui_scatter);
    895:   VecCreate(PETSC_COMM_WORLD,&uxi);
    896:   VecCreate(PETSC_COMM_WORLD,&ui);
    897:   VecSetSizes(uxi,PETSC_DECIDE,user->mx*user->mx);
    898:   VecSetSizes(ui,PETSC_DECIDE,2*user->mx*user->mx);
    899:   VecSetFromOptions(uxi);
    900:   VecSetFromOptions(ui);
    901:   VecDuplicateVecs(uxi,user->nt,&user->uxi);
    902:   VecDuplicateVecs(uxi,user->nt,&user->uyi);
    903:   VecDuplicateVecs(uxi,user->nt,&user->uxiwork);
    904:   VecDuplicateVecs(uxi,user->nt,&user->uyiwork);
    905:   VecDuplicateVecs(ui,user->nt,&user->ui);
    906:   VecDuplicateVecs(ui,user->nt,&user->uiwork);
    907:   for (i=0; i<user->nt; i++) {
    908:     VecGetOwnershipRange(user->uxi[i],&lo,&hi);
    909:     ISCreateStride(PETSC_COMM_SELF,hi-lo,lo,1,&is_to_uxi);
    910:     ISCreateStride(PETSC_COMM_SELF,hi-lo,lo+2*i*user->mx*user->mx,1,&is_from_u);
    911:     VecScatterCreate(user->u,is_from_u,user->uxi[i],is_to_uxi,&user->uxi_scatter[i]);

    913:     ISDestroy(&is_to_uxi);
    914:     ISDestroy(&is_from_u);

    916:     VecGetOwnershipRange(user->uyi[i],&lo,&hi);
    917:     ISCreateStride(PETSC_COMM_SELF,hi-lo,lo,1,&is_to_uyi);
    918:     ISCreateStride(PETSC_COMM_SELF,hi-lo,lo+(2*i+1)*user->mx*user->mx,1,&is_from_u);
    919:     VecScatterCreate(user->u,is_from_u,user->uyi[i],is_to_uyi,&user->uyi_scatter[i]);

    921:     ISDestroy(&is_to_uyi);
    922:     ISDestroy(&is_from_u);

    924:     VecGetOwnershipRange(user->uxi[i],&lo,&hi);
    925:     ISCreateStride(PETSC_COMM_SELF,hi-lo,lo,1,&is_to_uxi);
    926:     ISCreateStride(PETSC_COMM_SELF,hi-lo,lo,1,&is_from_u);
    927:     VecScatterCreate(user->ui[i],is_from_u,user->uxi[i],is_to_uxi,&user->ux_scatter[i]);

    929:     ISDestroy(&is_to_uxi);
    930:     ISDestroy(&is_from_u);

    932:     VecGetOwnershipRange(user->uyi[i],&lo,&hi);
    933:     ISCreateStride(PETSC_COMM_SELF,hi-lo,lo,1,&is_to_uyi);
    934:     ISCreateStride(PETSC_COMM_SELF,hi-lo,lo+user->mx*user->mx,1,&is_from_u);
    935:     VecScatterCreate(user->ui[i],is_from_u,user->uyi[i],is_to_uyi,&user->uy_scatter[i]);

    937:     ISDestroy(&is_to_uyi);
    938:     ISDestroy(&is_from_u);

    940:     VecGetOwnershipRange(user->ui[i],&lo,&hi);
    941:     ISCreateStride(PETSC_COMM_SELF,hi-lo,lo,1,&is_to_uxi);
    942:     ISCreateStride(PETSC_COMM_SELF,hi-lo,lo+2*i*user->mx*user->mx,1,&is_from_u);
    943:     VecScatterCreate(user->u,is_from_u,user->ui[i],is_to_uxi,&user->ui_scatter[i]);

    945:     ISDestroy(&is_to_uxi);
    946:     ISDestroy(&is_from_u);
    947:   }

    949:   /* RHS of forward problem */
    950:   MatMult(user->M,bc,user->yiwork[0]);
    951:   for (i=1; i<user->nt; i++) {
    952:     VecSet(user->yiwork[i],0.0);
    953:   }
    954:   Gather_yi(user->q,user->yiwork,user->yi_scatter,user->nt);

    956:   /* Compute true velocity field utrue */
    957:   VecDuplicate(user->u,&user->utrue);
    958:   for (i=0; i<user->nt; i++) {
    959:     VecCopy(YY,user->uxi[i]);
    960:     VecScale(user->uxi[i],150.0*i*user->ht);
    961:     VecCopy(XX,user->uyi[i]);
    962:     VecShift(user->uyi[i],-10.0);
    963:     VecScale(user->uyi[i],15.0*i*user->ht);
    964:   }
    965:   Gather_uxi_uyi(user->utrue,user->uxi,user->uxi_scatter,user->uyi,user->uyi_scatter,user->nt);

    967:   /* Initial guess and reference model */
    968:   VecDuplicate(user->utrue,&user->ur);
    969:   for (i=0; i<user->nt; i++) {
    970:     VecCopy(XX,user->uxi[i]);
    971:     VecShift(user->uxi[i],i*user->ht);
    972:     VecCopy(YY,user->uyi[i]);
    973:     VecShift(user->uyi[i],-i*user->ht);
    974:   }
    975:   Gather_uxi_uyi(user->ur,user->uxi,user->uxi_scatter,user->uyi,user->uyi_scatter,user->nt);

    977:   /* Generate regularization matrix L */
    978:   MatCreate(PETSC_COMM_WORLD,&user->LT);
    979:   MatSetSizes(user->LT,PETSC_DECIDE,PETSC_DECIDE,2*n*user->nt,n*user->nt);
    980:   MatSetFromOptions(user->LT);
    981:   MatMPIAIJSetPreallocation(user->LT,1,NULL,1,NULL);
    982:   MatSeqAIJSetPreallocation(user->LT,1,NULL);
    983:   MatGetOwnershipRange(user->LT,&istart,&iend);

    985:   for (i=istart; i<iend; i++) {
    986:     iblock = (i+n) / (2*n);
    987:     j = i - iblock*n;
    988:     MatSetValues(user->LT,1,&i,1,&j,&user->gamma,INSERT_VALUES);
    989:   }

    991:   MatAssemblyBegin(user->LT,MAT_FINAL_ASSEMBLY);
    992:   MatAssemblyEnd(user->LT,MAT_FINAL_ASSEMBLY);

    994:   MatTranspose(user->LT,MAT_INITIAL_MATRIX,&user->L);

    996:   /* Build work vectors and matrices */
    997:   VecCreate(PETSC_COMM_WORLD,&user->lwork);
    998:   VecSetType(user->lwork,VECMPI);
    999:   VecSetSizes(user->lwork,PETSC_DECIDE,user->m);
    1000:   VecSetFromOptions(user->lwork);

    1002:   MatDuplicate(user->Div,MAT_SHARE_NONZERO_PATTERN,&user->Divwork);

    1004:   VecDuplicate(user->y,&user->ywork);
    1005:   VecDuplicate(user->u,&user->uwork);
    1006:   VecDuplicate(user->u,&user->vwork);
    1007:   VecDuplicate(user->u,&user->js_diag);
    1008:   VecDuplicate(user->c,&user->cwork);

    1010:   /* Create matrix-free shell user->Js for computing A*x */
    1011:   MatCreateShell(PETSC_COMM_WORLD,PETSC_DETERMINE,PETSC_DETERMINE,user->m,user->m,user,&user->Js);
    1012:   MatShellSetOperation(user->Js,MATOP_MULT,(void(*)(void))StateMatMult);
    1013:   MatShellSetOperation(user->Js,MATOP_DUPLICATE,(void(*)(void))StateMatDuplicate);
    1014:   MatShellSetOperation(user->Js,MATOP_MULT_TRANSPOSE,(void(*)(void))StateMatMultTranspose);
    1015:   MatShellSetOperation(user->Js,MATOP_GET_DIAGONAL,(void(*)(void))StateMatGetDiagonal);

    1017:   /* Diagonal blocks of user->Js */
    1018:   MatCreateShell(PETSC_COMM_WORLD,PETSC_DETERMINE,PETSC_DETERMINE,n,n,user,&user->JsBlock);
    1019:   MatShellSetOperation(user->JsBlock,MATOP_MULT,(void(*)(void))StateMatBlockMult);
    1020:   MatShellSetOperation(user->JsBlock,MATOP_MULT_TRANSPOSE,(void(*)(void))StateMatBlockMultTranspose);

    1022:   /* Create a matrix-free shell user->JsBlockPrec for computing (U+D)\D*(L+D)\x, where JsBlock = L+D+U,
    1023:      D is diagonal, L is strictly lower triangular, and U is strictly upper triangular.
    1024:      This is an SOR preconditioner for user->JsBlock. */
    1025:   MatCreateShell(PETSC_COMM_WORLD,PETSC_DETERMINE,PETSC_DETERMINE,n,n,user,&user->JsBlockPrec);
    1026:   MatShellSetOperation(user->JsBlockPrec,MATOP_MULT,(void(*)(void))StateMatBlockPrecMult);
    1027:   MatShellSetOperation(user->JsBlockPrec,MATOP_MULT_TRANSPOSE,(void(*)(void))StateMatBlockPrecMultTranspose);

    1029:   /* Create a matrix-free shell user->Jd for computing B*x */
    1030:   MatCreateShell(PETSC_COMM_WORLD,PETSC_DETERMINE,PETSC_DETERMINE,user->m,user->n-user->m,user,&user->Jd);
    1031:   MatShellSetOperation(user->Jd,MATOP_MULT,(void(*)(void))DesignMatMult);
    1032:   MatShellSetOperation(user->Jd,MATOP_MULT_TRANSPOSE,(void(*)(void))DesignMatMultTranspose);

    1034:   /* User-defined routines for computing user->Js\x and user->Js^T\x*/
    1035:   MatCreateShell(PETSC_COMM_WORLD,PETSC_DETERMINE,PETSC_DETERMINE,user->m,user->m,user,&user->JsInv);
    1036:   MatShellSetOperation(user->JsInv,MATOP_MULT,(void(*)(void))StateMatInvMult);
    1037:   MatShellSetOperation(user->JsInv,MATOP_MULT_TRANSPOSE,(void(*)(void))StateMatInvTransposeMult);

    1039:   /* Build matrices for SOR preconditioner */
    1040:   Scatter_uxi_uyi(user->u,user->uxi,user->uxi_scatter,user->uyi,user->uyi_scatter,user->nt);
    1041:   PetscMalloc1(5*n,&user->C);
    1042:   PetscMalloc1(2*n,&user->Cwork);
    1043:   for (i=0; i<user->nt; i++) {
    1044:     MatDuplicate(user->Divxy[0],MAT_COPY_VALUES,&user->C[i]);
    1045:     MatDuplicate(user->Divxy[1],MAT_COPY_VALUES,&user->Cwork[i]);

    1047:     MatDiagonalScale(user->C[i],NULL,user->uxi[i]);
    1048:     MatDiagonalScale(user->Cwork[i],NULL,user->uyi[i]);
    1049:     MatAXPY(user->C[i],1.0,user->Cwork[i],DIFFERENT_NONZERO_PATTERN);
    1050:     MatScale(user->C[i],user->ht);
    1051:     MatShift(user->C[i],1.0);
    1052:   }

    1054:   /* Solver options and tolerances */
    1055:   KSPCreate(PETSC_COMM_WORLD,&user->solver);
    1056:   KSPSetType(user->solver,KSPGMRES);
    1057:   KSPSetOperators(user->solver,user->JsBlock,user->JsBlockPrec);
    1058:   KSPSetTolerances(user->solver,1e-4,1e-20,1e3,500);
    1059:   /* KSPSetTolerances(user->solver,1e-8,1e-16,1e3,500); */
    1060:   KSPGetPC(user->solver,&user->prec);
    1061:   PCSetType(user->prec,PCSHELL);

    1063:   PCShellSetApply(user->prec,StateMatBlockPrecMult);
    1064:   PCShellSetApplyTranspose(user->prec,StateMatBlockPrecMultTranspose);
    1065:   PCShellSetContext(user->prec,user);

    1067:   /* Compute true state function yt given ut */
    1068:   VecCreate(PETSC_COMM_WORLD,&user->ytrue);
    1069:   VecSetSizes(user->ytrue,PETSC_DECIDE,n*user->nt);
    1070:   VecSetFromOptions(user->ytrue);
    1071:   user->c_formed = PETSC_TRUE;
    1072:   VecCopy(user->utrue,user->u); /*  Set u=utrue temporarily for StateMatInv */
    1073:   VecSet(user->ytrue,0.0); /*  Initial guess */
    1074:   StateMatInvMult(user->Js,user->q,user->ytrue);
    1075:   VecCopy(user->ur,user->u); /*  Reset u=ur */

    1077:   /* Initial guess y0 for state given u0 */
    1078:   StateMatInvMult(user->Js,user->q,user->y);

    1080:   /* Data discretization */
    1081:   MatCreate(PETSC_COMM_WORLD,&user->Q);
    1082:   MatSetSizes(user->Q,PETSC_DECIDE,PETSC_DECIDE,user->mx*user->mx,user->m);
    1083:   MatSetFromOptions(user->Q);
    1084:   MatMPIAIJSetPreallocation(user->Q,0,NULL,1,NULL);
    1085:   MatSeqAIJSetPreallocation(user->Q,1,NULL);

    1087:   MatGetOwnershipRange(user->Q,&istart,&iend);

    1089:   for (i=istart; i<iend; i++) {
    1090:     j = i + user->m - user->mx*user->mx;
    1091:     MatSetValues(user->Q,1,&i,1,&j,&one,INSERT_VALUES);
    1092:   }

    1094:   MatAssemblyBegin(user->Q,MAT_FINAL_ASSEMBLY);
    1095:   MatAssemblyEnd(user->Q,MAT_FINAL_ASSEMBLY);

    1097:   MatTranspose(user->Q,MAT_INITIAL_MATRIX,&user->QT);

    1099:   VecDestroy(&XX);
    1100:   VecDestroy(&YY);
    1101:   VecDestroy(&XXwork);
    1102:   VecDestroy(&YYwork);
    1103:   VecDestroy(&yi);
    1104:   VecDestroy(&uxi);
    1105:   VecDestroy(&ui);
    1106:   VecDestroy(&bc);

    1108:   /* Now that initial conditions have been set, let the user pass tolerance options to the KSP solver */
    1109:   KSPSetFromOptions(user->solver);
    1110:   return 0;
    1111: }

    1113: PetscErrorCode HyperbolicDestroy(AppCtx *user)
    1114: {
    1115:   PetscInt       i;

    1117:   MatDestroy(&user->Q);
    1118:   MatDestroy(&user->QT);
    1119:   MatDestroy(&user->Div);
    1120:   MatDestroy(&user->Divwork);
    1121:   MatDestroy(&user->Grad);
    1122:   MatDestroy(&user->L);
    1123:   MatDestroy(&user->LT);
    1124:   KSPDestroy(&user->solver);
    1125:   MatDestroy(&user->Js);
    1126:   MatDestroy(&user->Jd);
    1127:   MatDestroy(&user->JsBlockPrec);
    1128:   MatDestroy(&user->JsInv);
    1129:   MatDestroy(&user->JsBlock);
    1130:   MatDestroy(&user->Divxy[0]);
    1131:   MatDestroy(&user->Divxy[1]);
    1132:   MatDestroy(&user->Gradxy[0]);
    1133:   MatDestroy(&user->Gradxy[1]);
    1134:   MatDestroy(&user->M);
    1135:   for (i=0; i<user->nt; i++) {
    1136:     MatDestroy(&user->C[i]);
    1137:     MatDestroy(&user->Cwork[i]);
    1138:   }
    1139:   PetscFree(user->C);
    1140:   PetscFree(user->Cwork);
    1141:   VecDestroy(&user->u);
    1142:   VecDestroy(&user->uwork);
    1143:   VecDestroy(&user->vwork);
    1144:   VecDestroy(&user->utrue);
    1145:   VecDestroy(&user->y);
    1146:   VecDestroy(&user->ywork);
    1147:   VecDestroy(&user->ytrue);
    1148:   VecDestroyVecs(user->nt,&user->yi);
    1149:   VecDestroyVecs(user->nt,&user->yiwork);
    1150:   VecDestroyVecs(user->nt,&user->ziwork);
    1151:   VecDestroyVecs(user->nt,&user->uxi);
    1152:   VecDestroyVecs(user->nt,&user->uyi);
    1153:   VecDestroyVecs(user->nt,&user->uxiwork);
    1154:   VecDestroyVecs(user->nt,&user->uyiwork);
    1155:   VecDestroyVecs(user->nt,&user->ui);
    1156:   VecDestroyVecs(user->nt,&user->uiwork);
    1157:   VecDestroy(&user->c);
    1158:   VecDestroy(&user->cwork);
    1159:   VecDestroy(&user->ur);
    1160:   VecDestroy(&user->q);
    1161:   VecDestroy(&user->d);
    1162:   VecDestroy(&user->dwork);
    1163:   VecDestroy(&user->lwork);
    1164:   VecDestroy(&user->js_diag);
    1165:   ISDestroy(&user->s_is);
    1166:   ISDestroy(&user->d_is);
    1167:   VecScatterDestroy(&user->state_scatter);
    1168:   VecScatterDestroy(&user->design_scatter);
    1169:   for (i=0; i<user->nt; i++) {
    1170:     VecScatterDestroy(&user->uxi_scatter[i]);
    1171:     VecScatterDestroy(&user->uyi_scatter[i]);
    1172:     VecScatterDestroy(&user->ux_scatter[i]);
    1173:     VecScatterDestroy(&user->uy_scatter[i]);
    1174:     VecScatterDestroy(&user->ui_scatter[i]);
    1175:     VecScatterDestroy(&user->yi_scatter[i]);
    1176:   }
    1177:   PetscFree(user->uxi_scatter);
    1178:   PetscFree(user->uyi_scatter);
    1179:   PetscFree(user->ux_scatter);
    1180:   PetscFree(user->uy_scatter);
    1181:   PetscFree(user->ui_scatter);
    1182:   PetscFree(user->yi_scatter);
    1183:   return 0;
    1184: }

    1186: PetscErrorCode HyperbolicMonitor(Tao tao, void *ptr)
    1187: {
    1188:   Vec            X;
    1189:   PetscReal      unorm,ynorm;
    1190:   AppCtx         *user = (AppCtx*)ptr;

    1192:   TaoGetSolution(tao,&X);
    1193:   Scatter(X,user->ywork,user->state_scatter,user->uwork,user->design_scatter);
    1194:   VecAXPY(user->ywork,-1.0,user->ytrue);
    1195:   VecAXPY(user->uwork,-1.0,user->utrue);
    1196:   VecNorm(user->uwork,NORM_2,&unorm);
    1197:   VecNorm(user->ywork,NORM_2,&ynorm);
    1198:   PetscPrintf(MPI_COMM_WORLD, "||u-ut||=%g ||y-yt||=%g\n",(double)unorm,(double)ynorm);
    1199:   return 0;
    1200: }

    1202: /*TEST

    1204:    build:
    1205:       requires: !complex

    1207:    test:
    1208:       requires: !single
    1209:       args: -tao_cmonitor -tao_max_funcs 10 -tao_type lcl -tao_gatol 1.e-5

    1211:    test:
    1212:       suffix: guess_pod
    1213:       requires: !single
    1214:       args: -tao_cmonitor -tao_max_funcs 10 -tao_type lcl -ksp_guess_type pod -tao_gatol 1.e-5

    1216: TEST*/