Date: 2022-03-04T17:39:38+00:00

Actual source code: ex20opt_ic.c

      1: static char help[] = "Solves a ODE-constrained optimization problem -- finding the optimal initial conditions for the van der Pol equation.\n";

      3: /*
      4:   Concepts: TS^time-dependent nonlinear problems
      5:   Concepts: TS^van der Pol equation DAE equivalent
      6:   Concepts: TS^Optimization using adjoint sensitivity analysis
      7:   Processors: 1
      8: */
      9: /*
     10:   Notes:
     11:   This code demonstrates how to solve an ODE-constrained optimization problem with TAO, TSAdjoint and TS.
     12:   The nonlinear problem is written in an ODE equivalent form.
     13:   The objective is to minimize the difference between observation and model prediction by finding optimal values for initial conditions.
     14:   The gradient is computed with the discrete adjoint of an implicit method or an explicit method, see ex20adj.c for details.
     15: */

     17: #include <petsctao.h>
     18: #include <petscts.h>

     20: typedef struct _n_User *User;
     21: struct _n_User {
     22:   TS        ts;
     23:   PetscReal mu;
     24:   PetscReal next_output;

     26:   /* Sensitivity analysis support */
     27:   PetscInt  steps;
     28:   PetscReal ftime;
     29:   Mat       A;                       /* Jacobian matrix for ODE */
     30:   Mat       Jacp;                    /* JacobianP matrix for ODE*/
     31:   Mat       H;                       /* Hessian matrix for optimization */
     32:   Vec       U,Lambda[1],Mup[1];      /* first-order adjoint variables */
     33:   Vec       Lambda2[2];              /* second-order adjoint variables */
     34:   Vec       Ihp1[1];                 /* working space for Hessian evaluations */
     35:   Vec       Dir;                     /* direction vector */
     36:   PetscReal ob[2];                   /* observation used by the cost function */
     37:   PetscBool implicitform;            /* implicit ODE? */
     38: };
     39: PetscErrorCode Adjoint2(Vec,PetscScalar[],User);

     41: /* ----------------------- Explicit form of the ODE  -------------------- */

     43: static PetscErrorCode RHSFunction(TS ts,PetscReal t,Vec U,Vec F,void *ctx)
     44: {
     45:   PetscErrorCode    ierr;
     46:   User              user = (User)ctx;
     47:   PetscScalar       *f;
     48:   const PetscScalar *u;

     51:   VecGetArrayRead(U,&u);
     52:   VecGetArray(F,&f);
     53:   f[0] = u[1];
     54:   f[1] = user->mu*((1.-u[0]*u[0])*u[1]-u[0]);
     55:   VecRestoreArrayRead(U,&u);
     56:   VecRestoreArray(F,&f);
     57:   return(0);
     58: }

     60: static PetscErrorCode RHSJacobian(TS ts,PetscReal t,Vec U,Mat A,Mat B,void *ctx)
     61: {
     62:   PetscErrorCode    ierr;
     63:   User              user = (User)ctx;
     64:   PetscReal         mu   = user->mu;
     65:   PetscInt          rowcol[] = {0,1};
     66:   PetscScalar       J[2][2];
     67:   const PetscScalar *u;

     70:   VecGetArrayRead(U,&u);
     71:   J[0][0] = 0;
     72:   J[1][0] = -mu*(2.0*u[1]*u[0]+1.);
     73:   J[0][1] = 1.0;
     74:   J[1][1] = mu*(1.0-u[0]*u[0]);
     75:   MatSetValues(A,2,rowcol,2,rowcol,&J[0][0],INSERT_VALUES);
     76:   MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
     77:   MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
     78:   if (A != B) {
     79:     MatAssemblyBegin(B,MAT_FINAL_ASSEMBLY);
     80:     MatAssemblyEnd(B,MAT_FINAL_ASSEMBLY);
     81:   }
     82:   VecRestoreArrayRead(U,&u);
     83:   return(0);
     84: }

     86: static PetscErrorCode RHSHessianProductUU(TS ts,PetscReal t,Vec U,Vec *Vl,Vec Vr,Vec *VHV,void *ctx)
     87: {
     88:   const PetscScalar *vl,*vr,*u;
     89:   PetscScalar       *vhv;
     90:   PetscScalar       dJdU[2][2][2]={{{0}}};
     91:   PetscInt          i,j,k;
     92:   User              user = (User)ctx;
     93:   PetscErrorCode    ierr;

     96:   VecGetArrayRead(U,&u);
     97:   VecGetArrayRead(Vl[0],&vl);
     98:   VecGetArrayRead(Vr,&vr);
     99:   VecGetArray(VHV[0],&vhv);

    101:   dJdU[1][0][0] = -2.*user->mu*u[1];
    102:   dJdU[1][1][0] = -2.*user->mu*u[0];
    103:   dJdU[1][0][1] = -2.*user->mu*u[0];
    104:   for (j=0;j<2;j++) {
    105:     vhv[j] = 0;
    106:     for (k=0;k<2;k++)
    107:       for (i=0;i<2;i++)
    108:         vhv[j] += vl[i]*dJdU[i][j][k]*vr[k];
    109:   }
    110:   VecRestoreArrayRead(U,&u);
    111:   VecRestoreArrayRead(Vl[0],&vl);
    112:   VecRestoreArrayRead(Vr,&vr);
    113:   VecRestoreArray(VHV[0],&vhv);
    114:   return(0);
    115: }

    117: /* ----------------------- Implicit form of the ODE  -------------------- */

    119: static PetscErrorCode IFunction(TS ts,PetscReal t,Vec U,Vec Udot,Vec F,void *ctx)
    120: {
    121:   PetscErrorCode    ierr;
    122:   User              user = (User)ctx;
    123:   const PetscScalar *u,*udot;
    124:   PetscScalar       *f;

    127:   VecGetArrayRead(U,&u);
    128:   VecGetArrayRead(Udot,&udot);
    129:   VecGetArray(F,&f);
    130:   f[0] = udot[0] - u[1];
    131:   f[1] = udot[1] - user->mu*((1.0-u[0]*u[0])*u[1] - u[0]) ;
    132:   VecRestoreArrayRead(U,&u);
    133:   VecRestoreArrayRead(Udot,&udot);
    134:   VecRestoreArray(F,&f);
    135:   return(0);
    136: }

    138: static PetscErrorCode IJacobian(TS ts,PetscReal t,Vec U,Vec Udot,PetscReal a,Mat A,Mat B,void *ctx)
    139: {
    140:   PetscErrorCode    ierr;
    141:   User              user = (User)ctx;
    142:   PetscInt          rowcol[] = {0,1};
    143:   PetscScalar       J[2][2];
    144:   const PetscScalar *u;

    147:   VecGetArrayRead(U,&u);
    148:   J[0][0] = a;     J[0][1] =  -1.0;
    149:   J[1][0] = user->mu*(1.0 + 2.0*u[0]*u[1]);   J[1][1] = a - user->mu*(1.0-u[0]*u[0]);
    150:   MatSetValues(B,2,rowcol,2,rowcol,&J[0][0],INSERT_VALUES);
    151:   VecRestoreArrayRead(U,&u);
    152:   MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
    153:   MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
    154:   if (A != B) {
    155:     MatAssemblyBegin(B,MAT_FINAL_ASSEMBLY);
    156:     MatAssemblyEnd(B,MAT_FINAL_ASSEMBLY);
    157:   }
    158:   return(0);
    159: }

    161: /* Monitor timesteps and use interpolation to output at integer multiples of 0.1 */
    162: static PetscErrorCode Monitor(TS ts,PetscInt step,PetscReal t,Vec U,void *ctx)
    163: {
    164:   PetscErrorCode    ierr;
    165:   const PetscScalar *u;
    166:   PetscReal         tfinal, dt;
    167:   User              user = (User)ctx;
    168:   Vec               interpolatedU;

    171:   TSGetTimeStep(ts,&dt);
    172:   TSGetMaxTime(ts,&tfinal);

    174:   while (user->next_output <= t && user->next_output <= tfinal) {
    175:     VecDuplicate(U,&interpolatedU);
    176:     TSInterpolate(ts,user->next_output,interpolatedU);
    177:     VecGetArrayRead(interpolatedU,&u);
    178:     PetscPrintf(PETSC_COMM_WORLD,"[%g] %D TS %g (dt = %g) X %g %g\n",
    179:                        (double)user->next_output,step,(double)t,(double)dt,(double)PetscRealPart(u[0]),
    180:                        (double)PetscRealPart(u[1]));
    181:     VecRestoreArrayRead(interpolatedU,&u);
    182:     VecDestroy(&interpolatedU);
    183:     user->next_output += 0.1;
    184:   }
    185:   return(0);
    186: }

    188: static PetscErrorCode IHessianProductUU(TS ts,PetscReal t,Vec U,Vec *Vl,Vec Vr,Vec *VHV,void *ctx)
    189: {
    190:   const PetscScalar *vl,*vr,*u;
    191:   PetscScalar       *vhv;
    192:   PetscScalar       dJdU[2][2][2]={{{0}}};
    193:   PetscInt          i,j,k;
    194:   User              user = (User)ctx;
    195:   PetscErrorCode    ierr;

    198:   VecGetArrayRead(U,&u);
    199:   VecGetArrayRead(Vl[0],&vl);
    200:   VecGetArrayRead(Vr,&vr);
    201:   VecGetArray(VHV[0],&vhv);
    202:   dJdU[1][0][0] = 2.*user->mu*u[1];
    203:   dJdU[1][1][0] = 2.*user->mu*u[0];
    204:   dJdU[1][0][1] = 2.*user->mu*u[0];
    205:   for (j=0;j<2;j++) {
    206:     vhv[j] = 0;
    207:     for (k=0;k<2;k++)
    208:       for (i=0;i<2;i++)
    209:         vhv[j] += vl[i]*dJdU[i][j][k]*vr[k];
    210:   }
    211:   VecRestoreArrayRead(U,&u);
    212:   VecRestoreArrayRead(Vl[0],&vl);
    213:   VecRestoreArrayRead(Vr,&vr);
    214:   VecRestoreArray(VHV[0],&vhv);
    215:   return(0);
    216: }

    218: /* ------------------ User-defined routines for TAO -------------------------- */

    220: static PetscErrorCode FormFunctionGradient(Tao tao,Vec IC,PetscReal *f,Vec G,void *ctx)
    221: {
    222:   User              user_ptr = (User)ctx;
    223:   TS                ts = user_ptr->ts;
    224:   const PetscScalar *x_ptr;
    225:   PetscScalar       *y_ptr;

    229:   VecCopy(IC,user_ptr->U); /* set up the initial condition */

    231:   TSSetTime(ts,0.0);
    232:   TSSetStepNumber(ts,0);
    233:   TSSetTimeStep(ts,0.001); /* can be overwritten by command line options */
    234:   TSSetFromOptions(ts);
    235:   TSSolve(ts,user_ptr->U);

    237:   VecGetArrayRead(user_ptr->U,&x_ptr);
    238:   VecGetArray(user_ptr->Lambda[0],&y_ptr);
    239:   *f       = (x_ptr[0]-user_ptr->ob[0])*(x_ptr[0]-user_ptr->ob[0])+(x_ptr[1]-user_ptr->ob[1])*(x_ptr[1]-user_ptr->ob[1]);
    240:   y_ptr[0] = 2.*(x_ptr[0]-user_ptr->ob[0]);
    241:   y_ptr[1] = 2.*(x_ptr[1]-user_ptr->ob[1]);
    242:   VecRestoreArray(user_ptr->Lambda[0],&y_ptr);
    243:   VecRestoreArrayRead(user_ptr->U,&x_ptr);

    245:   TSSetCostGradients(ts,1,user_ptr->Lambda,NULL);
    246:   TSAdjointSolve(ts);
    247:   VecCopy(user_ptr->Lambda[0],G);
    248:   return(0);
    249: }

    251: static PetscErrorCode FormHessian(Tao tao,Vec U,Mat H,Mat Hpre,void *ctx)
    252: {
    253:   User           user_ptr = (User)ctx;
    254:   PetscScalar    harr[2];
    255:   PetscScalar    *x_ptr;
    256:   const PetscInt rows[2] = {0,1};
    257:   PetscInt       col;

    261:   VecCopy(U,user_ptr->U);
    262:   VecGetArray(user_ptr->Dir,&x_ptr);
    263:   x_ptr[0] = 1.;
    264:   x_ptr[1] = 0.;
    265:   VecRestoreArray(user_ptr->Dir,&x_ptr);
    266:   Adjoint2(user_ptr->U,harr,user_ptr);
    267:   col      = 0;
    268:   MatSetValues(H,2,rows,1,&col,harr,INSERT_VALUES);

    270:   VecCopy(U,user_ptr->U);
    271:   VecGetArray(user_ptr->Dir,&x_ptr);
    272:   x_ptr[0] = 0.;
    273:   x_ptr[1] = 1.;
    274:   VecRestoreArray(user_ptr->Dir,&x_ptr);
    275:   Adjoint2(user_ptr->U,harr,user_ptr);
    276:   col      = 1;
    277:   MatSetValues(H,2,rows,1,&col,harr,INSERT_VALUES);

    279:   MatAssemblyBegin(H,MAT_FINAL_ASSEMBLY);
    280:   MatAssemblyEnd(H,MAT_FINAL_ASSEMBLY);
    281:   if (H != Hpre) {
    282:     MatAssemblyBegin(Hpre,MAT_FINAL_ASSEMBLY);
    283:     MatAssemblyEnd(Hpre,MAT_FINAL_ASSEMBLY);
    284:   }
    285:   return(0);
    286: }

    288: static PetscErrorCode MatrixFreeHessian(Tao tao,Vec U,Mat H,Mat Hpre,void *ctx)
    289: {
    290:   User           user_ptr = (User)ctx;

    294:   VecCopy(U,user_ptr->U);
    295:   return(0);
    296: }

    298: /* ------------ Routines calculating second-order derivatives -------------- */

    300: /*
    301:   Compute the Hessian-vector product for the cost function using Second-order adjoint
    302: */
    303: PetscErrorCode Adjoint2(Vec U,PetscScalar arr[],User ctx)
    304: {
    305:   TS             ts = ctx->ts;
    306:   PetscScalar    *x_ptr,*y_ptr;
    307:   Mat            tlmsen;

    311:   TSAdjointReset(ts);

    313:   TSSetTime(ts,0.0);
    314:   TSSetStepNumber(ts,0);
    315:   TSSetTimeStep(ts,0.001);
    316:   TSSetFromOptions(ts);
    317:   TSSetCostHessianProducts(ts,1,ctx->Lambda2,NULL,ctx->Dir);
    318:   TSAdjointSetForward(ts,NULL);
    319:   TSSolve(ts,U);

    321:   /* Set terminal conditions for first- and second-order adjonts */
    322:   VecGetArray(U,&x_ptr);
    323:   VecGetArray(ctx->Lambda[0],&y_ptr);
    324:   y_ptr[0] = 2.*(x_ptr[0]-ctx->ob[0]);
    325:   y_ptr[1] = 2.*(x_ptr[1]-ctx->ob[1]);
    326:   VecRestoreArray(ctx->Lambda[0],&y_ptr);
    327:   VecRestoreArray(U,&x_ptr);
    328:   TSForwardGetSensitivities(ts,NULL,&tlmsen);
    329:   MatDenseGetColumn(tlmsen,0,&x_ptr);
    330:   VecGetArray(ctx->Lambda2[0],&y_ptr);
    331:   y_ptr[0] = 2.*x_ptr[0];
    332:   y_ptr[1] = 2.*x_ptr[1];
    333:   VecRestoreArray(ctx->Lambda2[0],&y_ptr);
    334:   MatDenseRestoreColumn(tlmsen,&x_ptr);

    336:   TSSetCostGradients(ts,1,ctx->Lambda,NULL);
    337:   if (ctx->implicitform) {
    338:     TSSetIHessianProduct(ts,ctx->Ihp1,IHessianProductUU,NULL,NULL,NULL,NULL,NULL,NULL,ctx);
    339:   } else {
    340:     TSSetRHSHessianProduct(ts,ctx->Ihp1,RHSHessianProductUU,NULL,NULL,NULL,NULL,NULL,NULL,ctx);
    341:   }
    342:   TSAdjointSolve(ts);

    344:   VecGetArray(ctx->Lambda2[0],&x_ptr);
    345:   arr[0] = x_ptr[0];
    346:   arr[1] = x_ptr[1];
    347:   VecRestoreArray(ctx->Lambda2[0],&x_ptr);

    349:   TSAdjointReset(ts);
    350:   TSAdjointResetForward(ts);
    351:   return(0);
    352: }

    354: PetscErrorCode FiniteDiff(Vec U,PetscScalar arr[],User ctx)
    355: {
    356:   Vec               Up,G,Gp;
    357:   const PetscScalar eps = PetscRealConstant(1e-7);
    358:   PetscScalar       *u;
    359:   Tao               tao = NULL;
    360:   PetscReal         f;
    361:   PetscErrorCode    ierr;

    364:   VecDuplicate(U,&Up);
    365:   VecDuplicate(U,&G);
    366:   VecDuplicate(U,&Gp);

    368:   FormFunctionGradient(tao,U,&f,G,ctx);

    370:   VecCopy(U,Up);
    371:   VecGetArray(Up,&u);
    372:   u[0] += eps;
    373:   VecRestoreArray(Up,&u);
    374:   FormFunctionGradient(tao,Up,&f,Gp,ctx);
    375:   VecAXPY(Gp,-1,G);
    376:   VecScale(Gp,1./eps);
    377:   VecGetArray(Gp,&u);
    378:   arr[0] = u[0];
    379:   arr[1] = u[1];
    380:   VecRestoreArray(Gp,&u);

    382:   VecCopy(U,Up);
    383:   VecGetArray(Up,&u);
    384:   u[1] += eps;
    385:   VecRestoreArray(Up,&u);
    386:   FormFunctionGradient(tao,Up,&f,Gp,ctx);
    387:   VecAXPY(Gp,-1,G);
    388:   VecScale(Gp,1./eps);
    389:   VecGetArray(Gp,&u);
    390:   arr[2] = u[0];
    391:   arr[3] = u[1];
    392:   VecRestoreArray(Gp,&u);

    394:   VecDestroy(&G);
    395:   VecDestroy(&Gp);
    396:   VecDestroy(&Up);
    397:   return(0);
    398: }

    400: static PetscErrorCode HessianProductMat(Mat mat,Vec svec,Vec y)
    401: {
    402:   User           user_ptr;
    403:   PetscScalar    *y_ptr;

    407:   MatShellGetContext(mat,&user_ptr);
    408:   VecCopy(svec,user_ptr->Dir);
    409:   VecGetArray(y,&y_ptr);
    410:   Adjoint2(user_ptr->U,y_ptr,user_ptr);
    411:   VecRestoreArray(y,&y_ptr);
    412:   return(0);
    413: }

    415: int main(int argc,char **argv)
    416: {
    417:   PetscBool      monitor = PETSC_FALSE,mf = PETSC_TRUE;
    418:   PetscInt       mode = 0;
    419:   PetscMPIInt    size;
    420:   struct _n_User user;
    421:   Vec            x; /* working vector for TAO */
    422:   PetscScalar    *x_ptr,arr[4];
    423:   PetscScalar    ic1 = 2.2,ic2 = -0.7; /* initial guess for TAO */
    424:   Tao            tao;
    425:   KSP            ksp;
    426:   PC             pc;

    429:   /* Initialize program */
    430:   PetscInitialize(&argc,&argv,NULL,help);if (ierr) return ierr;
    431:   MPI_Comm_size(PETSC_COMM_WORLD,&size);
    432:   if (size != 1) SETERRQ(PETSC_COMM_WORLD,PETSC_ERR_WRONG_MPI_SIZE,"This is a uniprocessor example only!");

    434:   /* Set runtime options */
    435:   user.next_output  = 0.0;
    436:   user.mu           = 1.0e3;
    437:   user.steps        = 0;
    438:   user.ftime        = 0.5;
    439:   user.implicitform = PETSC_TRUE;

    441:   PetscOptionsGetBool(NULL,NULL,"-monitor",&monitor,NULL);
    442:   PetscOptionsGetReal(NULL,NULL,"-mu",&user.mu,NULL);
    443:   PetscOptionsGetInt(NULL,NULL,"-mode",&mode,NULL);
    444:   PetscOptionsGetReal(NULL,NULL,"-ic1",&ic1,NULL);
    445:   PetscOptionsGetReal(NULL,NULL,"-ic2",&ic2,NULL);
    446:   PetscOptionsGetBool(NULL,NULL,"-my_tao_mf",&mf,NULL); /* matrix-free hessian for optimization */
    447:   PetscOptionsGetBool(NULL,NULL,"-implicitform",&user.implicitform,NULL);

    449:   /* Create necessary matrix and vectors, solve same ODE on every process */
    450:   MatCreate(PETSC_COMM_WORLD,&user.A);
    451:   MatSetSizes(user.A,PETSC_DECIDE,PETSC_DECIDE,2,2);
    452:   MatSetFromOptions(user.A);
    453:   MatSetUp(user.A);
    454:   MatCreateVecs(user.A,&user.U,NULL);
    455:   MatCreateVecs(user.A,&user.Dir,NULL);
    456:   MatCreateVecs(user.A,&user.Lambda[0],NULL);
    457:   MatCreateVecs(user.A,&user.Lambda2[0],NULL);
    458:   MatCreateVecs(user.A,&user.Ihp1[0],NULL);

    460:   /* Create timestepping solver context */
    461:   TSCreate(PETSC_COMM_WORLD,&user.ts);
    462:   TSSetEquationType(user.ts,TS_EQ_ODE_EXPLICIT); /* less Jacobian evaluations when adjoint BEuler is used, otherwise no effect */
    463:   if (user.implicitform) {
    464:     TSSetIFunction(user.ts,NULL,IFunction,&user);
    465:     TSSetIJacobian(user.ts,user.A,user.A,IJacobian,&user);
    466:     TSSetType(user.ts,TSCN);
    467:   } else {
    468:     TSSetRHSFunction(user.ts,NULL,RHSFunction,&user);
    469:     TSSetRHSJacobian(user.ts,user.A,user.A,RHSJacobian,&user);
    470:     TSSetType(user.ts,TSRK);
    471:   }
    472:   TSSetMaxTime(user.ts,user.ftime);
    473:   TSSetExactFinalTime(user.ts,TS_EXACTFINALTIME_MATCHSTEP);

    475:   if (monitor) {
    476:     TSMonitorSet(user.ts,Monitor,&user,NULL);
    477:   }

    479:   /* Set ODE initial conditions */
    480:   VecGetArray(user.U,&x_ptr);
    481:   x_ptr[0] = 2.0;
    482:   x_ptr[1] = -2.0/3.0 + 10.0/(81.0*user.mu) - 292.0/(2187.0*user.mu*user.mu);
    483:   VecRestoreArray(user.U,&x_ptr);

    485:   /* Set runtime options */
    486:   TSSetFromOptions(user.ts);

    488:   /* Obtain the observation */
    489:   TSSolve(user.ts,user.U);
    490:   VecGetArray(user.U,&x_ptr);
    491:   user.ob[0] = x_ptr[0];
    492:   user.ob[1] = x_ptr[1];
    493:   VecRestoreArray(user.U,&x_ptr);

    495:   VecDuplicate(user.U,&x);
    496:   VecGetArray(x,&x_ptr);
    497:   x_ptr[0] = ic1;
    498:   x_ptr[1] = ic2;
    499:   VecRestoreArray(x,&x_ptr);

    501:   /* Save trajectory of solution so that TSAdjointSolve() may be used */
    502:   TSSetSaveTrajectory(user.ts);

    504:   /* Compare finite difference and second-order adjoint. */
    505:   switch (mode) {
    506:     case 2 :
    507:       FiniteDiff(x,arr,&user);
    508:       PetscPrintf(PETSC_COMM_WORLD,"\n Finite difference approximation of the Hessian\n");
    509:       PetscPrintf(PETSC_COMM_WORLD,"%g %g\n%g %g\n",(double)arr[0],(double)arr[1],(double)arr[2],(double)arr[3]);
    510:       break;
    511:     case 1 : /* Compute the Hessian column by column */
    512:       VecCopy(x,user.U);
    513:       VecGetArray(user.Dir,&x_ptr);
    514:       x_ptr[0] = 1.;
    515:       x_ptr[1] = 0.;
    516:       VecRestoreArray(user.Dir,&x_ptr);
    517:       Adjoint2(user.U,arr,&user);
    518:       PetscPrintf(PETSC_COMM_WORLD,"\nFirst column of the Hessian\n");
    519:       PetscPrintf(PETSC_COMM_WORLD,"%g\n%g\n",(double)arr[0],(double)arr[1]);
    520:       VecCopy(x,user.U);
    521:       VecGetArray(user.Dir,&x_ptr);
    522:       x_ptr[0] = 0.;
    523:       x_ptr[1] = 1.;
    524:       VecRestoreArray(user.Dir,&x_ptr);
    525:       Adjoint2(user.U,arr,&user);
    526:       PetscPrintf(PETSC_COMM_WORLD,"\nSecond column of the Hessian\n");
    527:       PetscPrintf(PETSC_COMM_WORLD,"%g\n%g\n",(double)arr[0],(double)arr[1]);
    528:       break;
    529:     case 0 :
    530:       /* Create optimization context and set up */
    531:       TaoCreate(PETSC_COMM_WORLD,&tao);
    532:       TaoSetType(tao,TAOBLMVM);
    533:       TaoSetObjectiveAndGradientRoutine(tao,FormFunctionGradient,(void *)&user);

    535:       if (mf) {
    536:         MatCreateShell(PETSC_COMM_SELF,2,2,2,2,(void*)&user,&user.H);
    537:         MatShellSetOperation(user.H,MATOP_MULT,(void(*)(void))HessianProductMat);
    538:         MatSetOption(user.H,MAT_SYMMETRIC,PETSC_TRUE);
    539:         TaoSetHessianRoutine(tao,user.H,user.H,MatrixFreeHessian,(void *)&user);
    540:       } else { /* Create Hessian matrix */
    541:         MatCreate(PETSC_COMM_WORLD,&user.H);
    542:         MatSetSizes(user.H,PETSC_DECIDE,PETSC_DECIDE,2,2);
    543:         MatSetUp(user.H);
    544:         TaoSetHessianRoutine(tao,user.H,user.H,FormHessian,(void *)&user);
    545:       }

    547:       /* Not use any preconditioner */
    548:       TaoGetKSP(tao,&ksp);
    549:       if (ksp) {
    550:         KSPGetPC(ksp,&pc);
    551:         PCSetType(pc,PCNONE);
    552:       }

    554:       /* Set initial solution guess */
    555:       TaoSetInitialVector(tao,x);
    556:       TaoSetFromOptions(tao);
    557:       TaoSolve(tao);
    558:       TaoDestroy(&tao);
    559:       MatDestroy(&user.H);
    560:       break;
    561:     default:
    562:       break;
    563:   }

    565:   /* Free work space.  All PETSc objects should be destroyed when they are no longer needed. */
    566:   MatDestroy(&user.A);
    567:   VecDestroy(&user.U);
    568:   VecDestroy(&user.Lambda[0]);
    569:   VecDestroy(&user.Lambda2[0]);
    570:   VecDestroy(&user.Ihp1[0]);
    571:   VecDestroy(&user.Dir);
    572:   TSDestroy(&user.ts);
    573:   VecDestroy(&x);
    574:   PetscFinalize();
    575:   return(ierr);
    576: }

    578: /*TEST
    579:     build:
    580:       requires: !complex !single

    582:     test:
    583:       args:  -ts_type cn -viewer_binary_skip_info -tao_monitor -tao_view -mu 1000 -ts_dt 0.03125
    584:       output_file: output/ex20opt_ic_1.out

    586:     test:
    587:       suffix: 2
    588:       args:  -ts_type beuler -viewer_binary_skip_info -tao_monitor -tao_view -mu 100 -ts_dt 0.01 -tao_type bntr -tao_bnk_pc_type none
    589:       output_file: output/ex20opt_ic_2.out

    591:     test:
    592:       suffix: 3
    593:       args:  -ts_type cn -viewer_binary_skip_info -tao_monitor -tao_view -mu 100 -ts_dt 0.01 -tao_type bntr -tao_bnk_pc_type none
    594:       output_file: output/ex20opt_ic_3.out

    596:     test:
    597:       suffix: 4
    598:       args: -implicitform 0 -ts_dt 0.01 -ts_max_steps 2 -ts_rhs_jacobian_test_mult_transpose -mat_shell_test_mult_transpose_view -ts_rhs_jacobian_test_mult -mat_shell_test_mult_view -mode 1 -my_tao_mf
    599: TEST*/