Date: 2022-08-02T03:15:04+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:   Notes:
      5:   This code demonstrates how to solve an ODE-constrained optimization problem with TAO, TSAdjoint and TS.
      6:   The nonlinear problem is written in an ODE equivalent form.
      7:   The objective is to minimize the difference between observation and model prediction by finding optimal values for initial conditions.
      8:   The gradient is computed with the discrete adjoint of an implicit method or an explicit method, see ex20adj.c for details.
      9: */

     11: #include <petsctao.h>
     12: #include <petscts.h>

     14: typedef struct _n_User *User;
     15: struct _n_User {
     16:   TS        ts;
     17:   PetscReal mu;
     18:   PetscReal next_output;

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

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

     37: static PetscErrorCode RHSFunction(TS ts,PetscReal t,Vec U,Vec F,void *ctx)
     38: {
     39:   User              user = (User)ctx;
     40:   PetscScalar       *f;
     41:   const PetscScalar *u;

     44:   VecGetArrayRead(U,&u);
     45:   VecGetArray(F,&f);
     46:   f[0] = u[1];
     47:   f[1] = user->mu*((1.-u[0]*u[0])*u[1]-u[0]);
     48:   VecRestoreArrayRead(U,&u);
     49:   VecRestoreArray(F,&f);
     50:   return 0;
     51: }

     53: static PetscErrorCode RHSJacobian(TS ts,PetscReal t,Vec U,Mat A,Mat B,void *ctx)
     54: {
     55:   User              user = (User)ctx;
     56:   PetscReal         mu   = user->mu;
     57:   PetscInt          rowcol[] = {0,1};
     58:   PetscScalar       J[2][2];
     59:   const PetscScalar *u;

     62:   VecGetArrayRead(U,&u);
     63:   J[0][0] = 0;
     64:   J[1][0] = -mu*(2.0*u[1]*u[0]+1.);
     65:   J[0][1] = 1.0;
     66:   J[1][1] = mu*(1.0-u[0]*u[0]);
     67:   MatSetValues(A,2,rowcol,2,rowcol,&J[0][0],INSERT_VALUES);
     68:   MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
     69:   MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
     70:   if (A != B) {
     71:     MatAssemblyBegin(B,MAT_FINAL_ASSEMBLY);
     72:     MatAssemblyEnd(B,MAT_FINAL_ASSEMBLY);
     73:   }
     74:   VecRestoreArrayRead(U,&u);
     75:   return 0;
     76: }

     78: static PetscErrorCode RHSHessianProductUU(TS ts,PetscReal t,Vec U,Vec *Vl,Vec Vr,Vec *VHV,void *ctx)
     79: {
     80:   const PetscScalar *vl,*vr,*u;
     81:   PetscScalar       *vhv;
     82:   PetscScalar       dJdU[2][2][2]={{{0}}};
     83:   PetscInt          i,j,k;
     84:   User              user = (User)ctx;

     87:   VecGetArrayRead(U,&u);
     88:   VecGetArrayRead(Vl[0],&vl);
     89:   VecGetArrayRead(Vr,&vr);
     90:   VecGetArray(VHV[0],&vhv);

     92:   dJdU[1][0][0] = -2.*user->mu*u[1];
     93:   dJdU[1][1][0] = -2.*user->mu*u[0];
     94:   dJdU[1][0][1] = -2.*user->mu*u[0];
     95:   for (j=0;j<2;j++) {
     96:     vhv[j] = 0;
     97:     for (k=0;k<2;k++)
     98:       for (i=0;i<2;i++)
     99:         vhv[j] += vl[i]*dJdU[i][j][k]*vr[k];
    100:   }
    101:   VecRestoreArrayRead(U,&u);
    102:   VecRestoreArrayRead(Vl[0],&vl);
    103:   VecRestoreArrayRead(Vr,&vr);
    104:   VecRestoreArray(VHV[0],&vhv);
    105:   return 0;
    106: }

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

    110: static PetscErrorCode IFunction(TS ts,PetscReal t,Vec U,Vec Udot,Vec F,void *ctx)
    111: {
    112:   User              user = (User)ctx;
    113:   const PetscScalar *u,*udot;
    114:   PetscScalar       *f;

    117:   VecGetArrayRead(U,&u);
    118:   VecGetArrayRead(Udot,&udot);
    119:   VecGetArray(F,&f);
    120:   f[0] = udot[0] - u[1];
    121:   f[1] = udot[1] - user->mu*((1.0-u[0]*u[0])*u[1] - u[0]) ;
    122:   VecRestoreArrayRead(U,&u);
    123:   VecRestoreArrayRead(Udot,&udot);
    124:   VecRestoreArray(F,&f);
    125:   return 0;
    126: }

    128: static PetscErrorCode IJacobian(TS ts,PetscReal t,Vec U,Vec Udot,PetscReal a,Mat A,Mat B,void *ctx)
    129: {
    130:   User              user = (User)ctx;
    131:   PetscInt          rowcol[] = {0,1};
    132:   PetscScalar       J[2][2];
    133:   const PetscScalar *u;

    136:   VecGetArrayRead(U,&u);
    137:   J[0][0] = a;     J[0][1] =  -1.0;
    138:   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]);
    139:   MatSetValues(B,2,rowcol,2,rowcol,&J[0][0],INSERT_VALUES);
    140:   VecRestoreArrayRead(U,&u);
    141:   MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
    142:   MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
    143:   if (A != B) {
    144:     MatAssemblyBegin(B,MAT_FINAL_ASSEMBLY);
    145:     MatAssemblyEnd(B,MAT_FINAL_ASSEMBLY);
    146:   }
    147:   return 0;
    148: }

    150: /* Monitor timesteps and use interpolation to output at integer multiples of 0.1 */
    151: static PetscErrorCode Monitor(TS ts,PetscInt step,PetscReal t,Vec U,void *ctx)
    152: {
    153:   const PetscScalar *u;
    154:   PetscReal         tfinal, dt;
    155:   User              user = (User)ctx;
    156:   Vec               interpolatedU;

    159:   TSGetTimeStep(ts,&dt);
    160:   TSGetMaxTime(ts,&tfinal);

    162:   while (user->next_output <= t && user->next_output <= tfinal) {
    163:     VecDuplicate(U,&interpolatedU);
    164:     TSInterpolate(ts,user->next_output,interpolatedU);
    165:     VecGetArrayRead(interpolatedU,&u);
    166:     PetscCall(PetscPrintf(PETSC_COMM_WORLD,"[%g] %D TS %g (dt = %g) X %g %g\n",
    167:                         (double)user->next_output,step,(double)t,(double)dt,(double)PetscRealPart(u[0]),
    168:                         (double)PetscRealPart(u[1])));
    169:     VecRestoreArrayRead(interpolatedU,&u);
    170:     VecDestroy(&interpolatedU);
    171:     user->next_output += 0.1;
    172:   }
    173:   return 0;
    174: }

    176: static PetscErrorCode IHessianProductUU(TS ts,PetscReal t,Vec U,Vec *Vl,Vec Vr,Vec *VHV,void *ctx)
    177: {
    178:   const PetscScalar *vl,*vr,*u;
    179:   PetscScalar       *vhv;
    180:   PetscScalar       dJdU[2][2][2]={{{0}}};
    181:   PetscInt          i,j,k;
    182:   User              user = (User)ctx;

    185:   VecGetArrayRead(U,&u);
    186:   VecGetArrayRead(Vl[0],&vl);
    187:   VecGetArrayRead(Vr,&vr);
    188:   VecGetArray(VHV[0],&vhv);
    189:   dJdU[1][0][0] = 2.*user->mu*u[1];
    190:   dJdU[1][1][0] = 2.*user->mu*u[0];
    191:   dJdU[1][0][1] = 2.*user->mu*u[0];
    192:   for (j=0;j<2;j++) {
    193:     vhv[j] = 0;
    194:     for (k=0;k<2;k++)
    195:       for (i=0;i<2;i++)
    196:         vhv[j] += vl[i]*dJdU[i][j][k]*vr[k];
    197:   }
    198:   VecRestoreArrayRead(U,&u);
    199:   VecRestoreArrayRead(Vl[0],&vl);
    200:   VecRestoreArrayRead(Vr,&vr);
    201:   VecRestoreArray(VHV[0],&vhv);
    202:   return 0;
    203: }

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

    207: static PetscErrorCode FormFunctionGradient(Tao tao,Vec IC,PetscReal *f,Vec G,void *ctx)
    208: {
    209:   User              user_ptr = (User)ctx;
    210:   TS                ts = user_ptr->ts;
    211:   const PetscScalar *x_ptr;
    212:   PetscScalar       *y_ptr;

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

    217:   TSSetTime(ts,0.0);
    218:   TSSetStepNumber(ts,0);
    219:   TSSetTimeStep(ts,0.001); /* can be overwritten by command line options */
    220:   TSSetFromOptions(ts);
    221:   TSSolve(ts,user_ptr->U);

    223:   VecGetArrayRead(user_ptr->U,&x_ptr);
    224:   VecGetArray(user_ptr->Lambda[0],&y_ptr);
    225:   *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]);
    226:   y_ptr[0] = 2.*(x_ptr[0]-user_ptr->ob[0]);
    227:   y_ptr[1] = 2.*(x_ptr[1]-user_ptr->ob[1]);
    228:   VecRestoreArray(user_ptr->Lambda[0],&y_ptr);
    229:   VecRestoreArrayRead(user_ptr->U,&x_ptr);

    231:   TSSetCostGradients(ts,1,user_ptr->Lambda,NULL);
    232:   TSAdjointSolve(ts);
    233:   VecCopy(user_ptr->Lambda[0],G);
    234:   return 0;
    235: }

    237: static PetscErrorCode FormHessian(Tao tao,Vec U,Mat H,Mat Hpre,void *ctx)
    238: {
    239:   User           user_ptr = (User)ctx;
    240:   PetscScalar    harr[2];
    241:   PetscScalar    *x_ptr;
    242:   const PetscInt rows[2] = {0,1};
    243:   PetscInt       col;

    246:   VecCopy(U,user_ptr->U);
    247:   VecGetArray(user_ptr->Dir,&x_ptr);
    248:   x_ptr[0] = 1.;
    249:   x_ptr[1] = 0.;
    250:   VecRestoreArray(user_ptr->Dir,&x_ptr);
    251:   Adjoint2(user_ptr->U,harr,user_ptr);
    252:   col      = 0;
    253:   MatSetValues(H,2,rows,1,&col,harr,INSERT_VALUES);

    255:   VecCopy(U,user_ptr->U);
    256:   VecGetArray(user_ptr->Dir,&x_ptr);
    257:   x_ptr[0] = 0.;
    258:   x_ptr[1] = 1.;
    259:   VecRestoreArray(user_ptr->Dir,&x_ptr);
    260:   Adjoint2(user_ptr->U,harr,user_ptr);
    261:   col      = 1;
    262:   MatSetValues(H,2,rows,1,&col,harr,INSERT_VALUES);

    264:   MatAssemblyBegin(H,MAT_FINAL_ASSEMBLY);
    265:   MatAssemblyEnd(H,MAT_FINAL_ASSEMBLY);
    266:   if (H != Hpre) {
    267:     MatAssemblyBegin(Hpre,MAT_FINAL_ASSEMBLY);
    268:     MatAssemblyEnd(Hpre,MAT_FINAL_ASSEMBLY);
    269:   }
    270:   return 0;
    271: }

    273: static PetscErrorCode MatrixFreeHessian(Tao tao,Vec U,Mat H,Mat Hpre,void *ctx)
    274: {
    275:   User           user_ptr = (User)ctx;

    278:   VecCopy(U,user_ptr->U);
    279:   return 0;
    280: }

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

    284: /*
    285:   Compute the Hessian-vector product for the cost function using Second-order adjoint
    286: */
    287: PetscErrorCode Adjoint2(Vec U,PetscScalar arr[],User ctx)
    288: {
    289:   TS             ts = ctx->ts;
    290:   PetscScalar    *x_ptr,*y_ptr;
    291:   Mat            tlmsen;

    294:   TSAdjointReset(ts);

    296:   TSSetTime(ts,0.0);
    297:   TSSetStepNumber(ts,0);
    298:   TSSetTimeStep(ts,0.001);
    299:   TSSetFromOptions(ts);
    300:   TSSetCostHessianProducts(ts,1,ctx->Lambda2,NULL,ctx->Dir);
    301:   TSAdjointSetForward(ts,NULL);
    302:   TSSolve(ts,U);

    304:   /* Set terminal conditions for first- and second-order adjonts */
    305:   VecGetArray(U,&x_ptr);
    306:   VecGetArray(ctx->Lambda[0],&y_ptr);
    307:   y_ptr[0] = 2.*(x_ptr[0]-ctx->ob[0]);
    308:   y_ptr[1] = 2.*(x_ptr[1]-ctx->ob[1]);
    309:   VecRestoreArray(ctx->Lambda[0],&y_ptr);
    310:   VecRestoreArray(U,&x_ptr);
    311:   TSForwardGetSensitivities(ts,NULL,&tlmsen);
    312:   MatDenseGetColumn(tlmsen,0,&x_ptr);
    313:   VecGetArray(ctx->Lambda2[0],&y_ptr);
    314:   y_ptr[0] = 2.*x_ptr[0];
    315:   y_ptr[1] = 2.*x_ptr[1];
    316:   VecRestoreArray(ctx->Lambda2[0],&y_ptr);
    317:   MatDenseRestoreColumn(tlmsen,&x_ptr);

    319:   TSSetCostGradients(ts,1,ctx->Lambda,NULL);
    320:   if (ctx->implicitform) {
    321:     TSSetIHessianProduct(ts,ctx->Ihp1,IHessianProductUU,NULL,NULL,NULL,NULL,NULL,NULL,ctx);
    322:   } else {
    323:     TSSetRHSHessianProduct(ts,ctx->Ihp1,RHSHessianProductUU,NULL,NULL,NULL,NULL,NULL,NULL,ctx);
    324:   }
    325:   TSAdjointSolve(ts);

    327:   VecGetArray(ctx->Lambda2[0],&x_ptr);
    328:   arr[0] = x_ptr[0];
    329:   arr[1] = x_ptr[1];
    330:   VecRestoreArray(ctx->Lambda2[0],&x_ptr);

    332:   TSAdjointReset(ts);
    333:   TSAdjointResetForward(ts);
    334:   return 0;
    335: }

    337: PetscErrorCode FiniteDiff(Vec U,PetscScalar arr[],User ctx)
    338: {
    339:   Vec               Up,G,Gp;
    340:   const PetscScalar eps = PetscRealConstant(1e-7);
    341:   PetscScalar       *u;
    342:   Tao               tao = NULL;
    343:   PetscReal         f;

    346:   VecDuplicate(U,&Up);
    347:   VecDuplicate(U,&G);
    348:   VecDuplicate(U,&Gp);

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

    352:   VecCopy(U,Up);
    353:   VecGetArray(Up,&u);
    354:   u[0] += eps;
    355:   VecRestoreArray(Up,&u);
    356:   FormFunctionGradient(tao,Up,&f,Gp,ctx);
    357:   VecAXPY(Gp,-1,G);
    358:   VecScale(Gp,1./eps);
    359:   VecGetArray(Gp,&u);
    360:   arr[0] = u[0];
    361:   arr[1] = u[1];
    362:   VecRestoreArray(Gp,&u);

    364:   VecCopy(U,Up);
    365:   VecGetArray(Up,&u);
    366:   u[1] += eps;
    367:   VecRestoreArray(Up,&u);
    368:   FormFunctionGradient(tao,Up,&f,Gp,ctx);
    369:   VecAXPY(Gp,-1,G);
    370:   VecScale(Gp,1./eps);
    371:   VecGetArray(Gp,&u);
    372:   arr[2] = u[0];
    373:   arr[3] = u[1];
    374:   VecRestoreArray(Gp,&u);

    376:   VecDestroy(&G);
    377:   VecDestroy(&Gp);
    378:   VecDestroy(&Up);
    379:   return 0;
    380: }

    382: static PetscErrorCode HessianProductMat(Mat mat,Vec svec,Vec y)
    383: {
    384:   User           user_ptr;
    385:   PetscScalar    *y_ptr;

    388:   MatShellGetContext(mat,&user_ptr);
    389:   VecCopy(svec,user_ptr->Dir);
    390:   VecGetArray(y,&y_ptr);
    391:   Adjoint2(user_ptr->U,y_ptr,user_ptr);
    392:   VecRestoreArray(y,&y_ptr);
    393:   return 0;
    394: }

    396: int main(int argc,char **argv)
    397: {
    398:   PetscBool      monitor = PETSC_FALSE,mf = PETSC_TRUE;
    399:   PetscInt       mode = 0;
    400:   PetscMPIInt    size;
    401:   struct _n_User user;
    402:   Vec            x; /* working vector for TAO */
    403:   PetscScalar    *x_ptr,arr[4];
    404:   PetscScalar    ic1 = 2.2,ic2 = -0.7; /* initial guess for TAO */
    405:   Tao            tao;
    406:   KSP            ksp;
    407:   PC             pc;

    409:   /* Initialize program */
    410:   PetscInitialize(&argc,&argv,NULL,help);
    411:   MPI_Comm_size(PETSC_COMM_WORLD,&size);

    414:   /* Set runtime options */
    415:   user.next_output  = 0.0;
    416:   user.mu           = 1.0e3;
    417:   user.steps        = 0;
    418:   user.ftime        = 0.5;
    419:   user.implicitform = PETSC_TRUE;

    421:   PetscOptionsGetBool(NULL,NULL,"-monitor",&monitor,NULL);
    422:   PetscOptionsGetReal(NULL,NULL,"-mu",&user.mu,NULL);
    423:   PetscOptionsGetInt(NULL,NULL,"-mode",&mode,NULL);
    424:   PetscOptionsGetReal(NULL,NULL,"-ic1",&ic1,NULL);
    425:   PetscOptionsGetReal(NULL,NULL,"-ic2",&ic2,NULL);
    426:   PetscOptionsGetBool(NULL,NULL,"-my_tao_mf",&mf,NULL); /* matrix-free hessian for optimization */
    427:   PetscOptionsGetBool(NULL,NULL,"-implicitform",&user.implicitform,NULL);

    429:   /* Create necessary matrix and vectors, solve same ODE on every process */
    430:   MatCreate(PETSC_COMM_WORLD,&user.A);
    431:   MatSetSizes(user.A,PETSC_DECIDE,PETSC_DECIDE,2,2);
    432:   MatSetFromOptions(user.A);
    433:   MatSetUp(user.A);
    434:   MatCreateVecs(user.A,&user.U,NULL);
    435:   MatCreateVecs(user.A,&user.Dir,NULL);
    436:   MatCreateVecs(user.A,&user.Lambda[0],NULL);
    437:   MatCreateVecs(user.A,&user.Lambda2[0],NULL);
    438:   MatCreateVecs(user.A,&user.Ihp1[0],NULL);

    440:   /* Create timestepping solver context */
    441:   TSCreate(PETSC_COMM_WORLD,&user.ts);
    442:   TSSetEquationType(user.ts,TS_EQ_ODE_EXPLICIT); /* less Jacobian evaluations when adjoint BEuler is used, otherwise no effect */
    443:   if (user.implicitform) {
    444:     TSSetIFunction(user.ts,NULL,IFunction,&user);
    445:     TSSetIJacobian(user.ts,user.A,user.A,IJacobian,&user);
    446:     TSSetType(user.ts,TSCN);
    447:   } else {
    448:     TSSetRHSFunction(user.ts,NULL,RHSFunction,&user);
    449:     TSSetRHSJacobian(user.ts,user.A,user.A,RHSJacobian,&user);
    450:     TSSetType(user.ts,TSRK);
    451:   }
    452:   TSSetMaxTime(user.ts,user.ftime);
    453:   TSSetExactFinalTime(user.ts,TS_EXACTFINALTIME_MATCHSTEP);

    455:   if (monitor) {
    456:     TSMonitorSet(user.ts,Monitor,&user,NULL);
    457:   }

    459:   /* Set ODE initial conditions */
    460:   VecGetArray(user.U,&x_ptr);
    461:   x_ptr[0] = 2.0;
    462:   x_ptr[1] = -2.0/3.0 + 10.0/(81.0*user.mu) - 292.0/(2187.0*user.mu*user.mu);
    463:   VecRestoreArray(user.U,&x_ptr);

    465:   /* Set runtime options */
    466:   TSSetFromOptions(user.ts);

    468:   /* Obtain the observation */
    469:   TSSolve(user.ts,user.U);
    470:   VecGetArray(user.U,&x_ptr);
    471:   user.ob[0] = x_ptr[0];
    472:   user.ob[1] = x_ptr[1];
    473:   VecRestoreArray(user.U,&x_ptr);

    475:   VecDuplicate(user.U,&x);
    476:   VecGetArray(x,&x_ptr);
    477:   x_ptr[0] = ic1;
    478:   x_ptr[1] = ic2;
    479:   VecRestoreArray(x,&x_ptr);

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

    484:   /* Compare finite difference and second-order adjoint. */
    485:   switch (mode) {
    486:     case 2 :
    487:       FiniteDiff(x,arr,&user);
    488:       PetscPrintf(PETSC_COMM_WORLD,"\n Finite difference approximation of the Hessian\n");
    489:       PetscPrintf(PETSC_COMM_WORLD,"%g %g\n%g %g\n",(double)arr[0],(double)arr[1],(double)arr[2],(double)arr[3]);
    490:       break;
    491:     case 1 : /* Compute the Hessian column by column */
    492:       VecCopy(x,user.U);
    493:       VecGetArray(user.Dir,&x_ptr);
    494:       x_ptr[0] = 1.;
    495:       x_ptr[1] = 0.;
    496:       VecRestoreArray(user.Dir,&x_ptr);
    497:       Adjoint2(user.U,arr,&user);
    498:       PetscPrintf(PETSC_COMM_WORLD,"\nFirst column of the Hessian\n");
    499:       PetscPrintf(PETSC_COMM_WORLD,"%g\n%g\n",(double)arr[0],(double)arr[1]);
    500:       VecCopy(x,user.U);
    501:       VecGetArray(user.Dir,&x_ptr);
    502:       x_ptr[0] = 0.;
    503:       x_ptr[1] = 1.;
    504:       VecRestoreArray(user.Dir,&x_ptr);
    505:       Adjoint2(user.U,arr,&user);
    506:       PetscPrintf(PETSC_COMM_WORLD,"\nSecond column of the Hessian\n");
    507:       PetscPrintf(PETSC_COMM_WORLD,"%g\n%g\n",(double)arr[0],(double)arr[1]);
    508:       break;
    509:     case 0 :
    510:       /* Create optimization context and set up */
    511:       TaoCreate(PETSC_COMM_WORLD,&tao);
    512:       TaoSetType(tao,TAOBLMVM);
    513:       TaoSetObjectiveAndGradient(tao,NULL,FormFunctionGradient,(void *)&user);

    515:       if (mf) {
    516:         MatCreateShell(PETSC_COMM_SELF,2,2,2,2,(void*)&user,&user.H);
    517:         MatShellSetOperation(user.H,MATOP_MULT,(void(*)(void))HessianProductMat);
    518:         MatSetOption(user.H,MAT_SYMMETRIC,PETSC_TRUE);
    519:         TaoSetHessian(tao,user.H,user.H,MatrixFreeHessian,(void *)&user);
    520:       } else { /* Create Hessian matrix */
    521:         MatCreate(PETSC_COMM_WORLD,&user.H);
    522:         MatSetSizes(user.H,PETSC_DECIDE,PETSC_DECIDE,2,2);
    523:         MatSetUp(user.H);
    524:         TaoSetHessian(tao,user.H,user.H,FormHessian,(void *)&user);
    525:       }

    527:       /* Not use any preconditioner */
    528:       TaoGetKSP(tao,&ksp);
    529:       if (ksp) {
    530:         KSPGetPC(ksp,&pc);
    531:         PCSetType(pc,PCNONE);
    532:       }

    534:       /* Set initial solution guess */
    535:       TaoSetSolution(tao,x);
    536:       TaoSetFromOptions(tao);
    537:       TaoSolve(tao);
    538:       TaoDestroy(&tao);
    539:       MatDestroy(&user.H);
    540:       break;
    541:     default:
    542:       break;
    543:   }

    545:   /* Free work space.  All PETSc objects should be destroyed when they are no longer needed. */
    546:   MatDestroy(&user.A);
    547:   VecDestroy(&user.U);
    548:   VecDestroy(&user.Lambda[0]);
    549:   VecDestroy(&user.Lambda2[0]);
    550:   VecDestroy(&user.Ihp1[0]);
    551:   VecDestroy(&user.Dir);
    552:   TSDestroy(&user.ts);
    553:   VecDestroy(&x);
    554:   PetscFinalize();
    555:   return 0;
    556: }

    558: /*TEST
    559:     build:
    560:       requires: !complex !single

    562:     test:
    563:       args:  -ts_type cn -viewer_binary_skip_info -tao_monitor -tao_view -mu 1000 -ts_dt 0.03125
    564:       output_file: output/ex20opt_ic_1.out

    566:     test:
    567:       suffix: 2
    568:       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
    569:       output_file: output/ex20opt_ic_2.out

    571:     test:
    572:       suffix: 3
    573:       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
    574:       output_file: output/ex20opt_ic_3.out

    576:     test:
    577:       suffix: 4
    578:       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
    579: TEST*/