Actual source code: nls.c
petsc-3.7.1 2016-05-15
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      1: #include <petsctaolinesearch.h>
      2: #include <../src/tao/matrix/lmvmmat.h>
      3: #include <../src/tao/unconstrained/impls/nls/nls.h>

      5: #include <petscksp.h>
      6: #include <petscpc.h>
      7: #include <petsc/private/kspimpl.h>
      8: #include <petsc/private/pcimpl.h>

     10: #define NLS_KSP_CG      0
     11: #define NLS_KSP_NASH    1
     12: #define NLS_KSP_STCG    2
     13: #define NLS_KSP_GLTR    3
     14: #define NLS_KSP_PETSC   4
     15: #define NLS_KSP_TYPES   5

     17: #define NLS_PC_NONE     0
     18: #define NLS_PC_AHESS    1
     19: #define NLS_PC_BFGS     2
     20: #define NLS_PC_PETSC    3
     21: #define NLS_PC_TYPES    4

     23: #define BFGS_SCALE_AHESS        0
     24: #define BFGS_SCALE_PHESS        1
     25: #define BFGS_SCALE_BFGS         2
     26: #define BFGS_SCALE_TYPES        3

     28: #define NLS_INIT_CONSTANT         0
     29: #define NLS_INIT_DIRECTION        1
     30: #define NLS_INIT_INTERPOLATION    2
     31: #define NLS_INIT_TYPES            3

     33: #define NLS_UPDATE_STEP           0
     34: #define NLS_UPDATE_REDUCTION      1
     35: #define NLS_UPDATE_INTERPOLATION  2
     36: #define NLS_UPDATE_TYPES          3

     38: static const char *NLS_KSP[64] = {"cg", "nash", "stcg", "gltr", "petsc"};

     40: static const char *NLS_PC[64] = {"none", "ahess", "bfgs", "petsc"};

     42: static const char *BFGS_SCALE[64] = {"ahess", "phess", "bfgs"};

     44: static const char *NLS_INIT[64] = {"constant", "direction", "interpolation"};

     46: static const char *NLS_UPDATE[64] = {"step", "reduction", "interpolation"};

     48: static PetscErrorCode MatLMVMSolveShell(PC pc, Vec b, Vec x);
     49: /* Routine for BFGS preconditioner

     52:  Implements Newton's Method with a line search approach for solving
     53:  unconstrained minimization problems.  A More'-Thuente line search
     54:  is used to guarantee that the bfgs preconditioner remains positive
     55:  definite.

     57:  The method can shift the Hessian matrix.  The shifting procedure is
     58:  adapted from the PATH algorithm for solving complementarity
     59:  problems.

     61:  The linear system solve should be done with a conjugate gradient
     62:  method, although any method can be used. */

     64: #define NLS_NEWTON              0
     65: #define NLS_BFGS                1
     66: #define NLS_SCALED_GRADIENT     2
     67: #define NLS_GRADIENT            3

     71: static PetscErrorCode TaoSolve_NLS(Tao tao)
     72: {
     73:   PetscErrorCode               ierr;
     74:   TAO_NLS                      *nlsP = (TAO_NLS *)tao->data;
     75:   PC                           pc;
     76:   KSPConvergedReason           ksp_reason;
     77:   TaoLineSearchConvergedReason ls_reason;
     78:   TaoConvergedReason           reason;

     80:   PetscReal                    fmin, ftrial, f_full, prered, actred, kappa, sigma;
     81:   PetscReal                    tau, tau_1, tau_2, tau_max, tau_min, max_radius;
     82:   PetscReal                    f, fold, gdx, gnorm, pert;
     83:   PetscReal                    step = 1.0;
     84:   PetscReal                    delta;
     85:   PetscReal                    norm_d = 0.0, e_min;

     87:   PetscInt                     stepType;
     88:   PetscInt                     bfgsUpdates = 0;
     89:   PetscInt                     n,N,kspits;
     90:   PetscInt                     needH = 1;

     92:   PetscInt                     i_max = 5;
     93:   PetscInt                     j_max = 1;
     94:   PetscInt                     i, j;

     97:   if (tao->XL || tao->XU || tao->ops->computebounds) {
     98:     PetscPrintf(((PetscObject)tao)->comm,"WARNING: Variable bounds have been set but will be ignored by nls algorithm\n");
     99:   }

    101:   /* Initialized variables */
    102:   pert = nlsP->sval;

    104:   /* Number of times ksp stopped because of these reasons */
    105:   nlsP->ksp_atol = 0;
    106:   nlsP->ksp_rtol = 0;
    107:   nlsP->ksp_dtol = 0;
    108:   nlsP->ksp_ctol = 0;
    109:   nlsP->ksp_negc = 0;
    110:   nlsP->ksp_iter = 0;
    111:   nlsP->ksp_othr = 0;

    113:   /* Modify the linear solver to a trust region method if desired */
    114:   switch(nlsP->ksp_type) {
    115:   case NLS_KSP_CG:
    116:     KSPSetType(tao->ksp, KSPCG);
    117:     KSPSetFromOptions(tao->ksp);
    118:     break;

    120:   case NLS_KSP_NASH:
    121:     KSPSetType(tao->ksp, KSPNASH);
    122:     KSPSetFromOptions(tao->ksp);
    123:     break;

    125:   case NLS_KSP_STCG:
    126:     KSPSetType(tao->ksp, KSPSTCG);
    127:     KSPSetFromOptions(tao->ksp);
    128:     break;

    130:   case NLS_KSP_GLTR:
    131:     KSPSetType(tao->ksp, KSPGLTR);
    132:     KSPSetFromOptions(tao->ksp);
    133:     break;

    135:   default:
    136:     /* Use the method set by the ksp_type */
    137:     break;
    138:   }

    140:   /* Initialize trust-region radius when using nash, stcg, or gltr
    141:    Will be reset during the first iteration */
    142:   if (NLS_KSP_NASH == nlsP->ksp_type) {
    143:     KSPNASHSetRadius(tao->ksp,nlsP->max_radius);
    144:   } else if (NLS_KSP_STCG == nlsP->ksp_type) {
    145:     KSPSTCGSetRadius(tao->ksp,nlsP->max_radius);
    146:   } else if (NLS_KSP_GLTR == nlsP->ksp_type) {
    147:     KSPGLTRSetRadius(tao->ksp,nlsP->max_radius);
    148:   }

    150:   if (NLS_KSP_NASH == nlsP->ksp_type || NLS_KSP_STCG == nlsP->ksp_type || NLS_KSP_GLTR == nlsP->ksp_type) {
    151:     tao->trust = tao->trust0;

    153:     if (tao->trust < 0.0) SETERRQ(PETSC_COMM_SELF,1, "Initial radius negative");

    155:     /* Modify the radius if it is too large or small */
    156:     tao->trust = PetscMax(tao->trust, nlsP->min_radius);
    157:     tao->trust = PetscMin(tao->trust, nlsP->max_radius);
    158:   }

    160:   /* Get vectors we will need */
    161:   if (NLS_PC_BFGS == nlsP->pc_type && !nlsP->M) {
    162:     VecGetLocalSize(tao->solution,&n);
    163:     VecGetSize(tao->solution,&N);
    164:     MatCreateLMVM(((PetscObject)tao)->comm,n,N,&nlsP->M);
    165:     MatLMVMAllocateVectors(nlsP->M,tao->solution);
    166:   }

    168:   /* Check convergence criteria */
    169:   TaoComputeObjectiveAndGradient(tao, tao->solution, &f, tao->gradient);
    170:   TaoGradientNorm(tao, tao->gradient,NORM_2,&gnorm);
    171:   if (PetscIsInfOrNanReal(f) || PetscIsInfOrNanReal(gnorm)) SETERRQ(PETSC_COMM_SELF,1, "User provided compute function generated Inf or NaN");

    173:   TaoMonitor(tao, tao->niter, f, gnorm, 0.0, 1.0, &reason);
    174:   if (reason != TAO_CONTINUE_ITERATING) return(0);

    176:   /* create vectors for the limited memory preconditioner */
    177:   if ((NLS_PC_BFGS == nlsP->pc_type) && (BFGS_SCALE_BFGS != nlsP->bfgs_scale_type)) {
    178:     if (!nlsP->Diag) {
    179:       VecDuplicate(tao->solution,&nlsP->Diag);
    180:     }
    181:   }

    183:   /* Modify the preconditioner to use the bfgs approximation */
    184:   KSPGetPC(tao->ksp, &pc);
    185:   switch(nlsP->pc_type) {
    186:   case NLS_PC_NONE:
    187:     PCSetType(pc, PCNONE);
    188:     PCSetFromOptions(pc);
    189:     break;

    191:   case NLS_PC_AHESS:
    192:     PCSetType(pc, PCJACOBI);
    193:     PCSetFromOptions(pc);
    194:     PCJacobiSetUseAbs(pc,PETSC_TRUE);
    195:     break;

    197:   case NLS_PC_BFGS:
    198:     PCSetType(pc, PCSHELL);
    199:     PCSetFromOptions(pc);
    200:     PCShellSetName(pc, "bfgs");
    201:     PCShellSetContext(pc, nlsP->M);
    202:     PCShellSetApply(pc, MatLMVMSolveShell);
    203:     break;

    205:   default:
    206:     /* Use the pc method set by pc_type */
    207:     break;
    208:   }

    210:   /* Initialize trust-region radius.  The initialization is only performed
    211:      when we are using Nash, Steihaug-Toint or the Generalized Lanczos method. */
    212:   if (NLS_KSP_NASH == nlsP->ksp_type || NLS_KSP_STCG == nlsP->ksp_type || NLS_KSP_GLTR == nlsP->ksp_type) {
    213:     switch(nlsP->init_type) {
    214:     case NLS_INIT_CONSTANT:
    215:       /* Use the initial radius specified */
    216:       break;

    218:     case NLS_INIT_INTERPOLATION:
    219:       /* Use the initial radius specified */
    220:       max_radius = 0.0;

    222:       for (j = 0; j < j_max; ++j) {
    223:         fmin = f;
    224:         sigma = 0.0;

    226:         if (needH) {
    227:           TaoComputeHessian(tao, tao->solution,tao->hessian,tao->hessian_pre);
    228:           needH = 0;
    229:         }

    231:         for (i = 0; i < i_max; ++i) {
    232:           VecCopy(tao->solution,nlsP->W);
    233:           VecAXPY(nlsP->W,-tao->trust/gnorm,tao->gradient);
    234:           TaoComputeObjective(tao, nlsP->W, &ftrial);
    235:           if (PetscIsInfOrNanReal(ftrial)) {
    236:             tau = nlsP->gamma1_i;
    237:           } else {
    238:             if (ftrial < fmin) {
    239:               fmin = ftrial;
    240:               sigma = -tao->trust / gnorm;
    241:             }

    243:             MatMult(tao->hessian, tao->gradient, nlsP->D);
    244:             VecDot(tao->gradient, nlsP->D, &prered);

    246:             prered = tao->trust * (gnorm - 0.5 * tao->trust * prered / (gnorm * gnorm));
    247:             actred = f - ftrial;
    248:             if ((PetscAbsScalar(actred) <= nlsP->epsilon) && (PetscAbsScalar(prered) <= nlsP->epsilon)) {
    249:               kappa = 1.0;
    250:             } else {
    251:               kappa = actred / prered;
    252:             }

    254:             tau_1 = nlsP->theta_i * gnorm * tao->trust / (nlsP->theta_i * gnorm * tao->trust + (1.0 - nlsP->theta_i) * prered - actred);
    255:             tau_2 = nlsP->theta_i * gnorm * tao->trust / (nlsP->theta_i * gnorm * tao->trust - (1.0 + nlsP->theta_i) * prered + actred);
    256:             tau_min = PetscMin(tau_1, tau_2);
    257:             tau_max = PetscMax(tau_1, tau_2);

    259:             if (PetscAbsScalar(kappa - 1.0) <= nlsP->mu1_i) {
    260:               /* Great agreement */
    261:               max_radius = PetscMax(max_radius, tao->trust);

    263:               if (tau_max < 1.0) {
    264:                 tau = nlsP->gamma3_i;
    265:               } else if (tau_max > nlsP->gamma4_i) {
    266:                 tau = nlsP->gamma4_i;
    267:               } else if (tau_1 >= 1.0 && tau_1 <= nlsP->gamma4_i && tau_2 < 1.0) {
    268:                 tau = tau_1;
    269:               } else if (tau_2 >= 1.0 && tau_2 <= nlsP->gamma4_i && tau_1 < 1.0) {
    270:                 tau = tau_2;
    271:               } else {
    272:                 tau = tau_max;
    273:               }
    274:             } else if (PetscAbsScalar(kappa - 1.0) <= nlsP->mu2_i) {
    275:               /* Good agreement */
    276:               max_radius = PetscMax(max_radius, tao->trust);

    278:               if (tau_max < nlsP->gamma2_i) {
    279:                 tau = nlsP->gamma2_i;
    280:               } else if (tau_max > nlsP->gamma3_i) {
    281:                 tau = nlsP->gamma3_i;
    282:               } else {
    283:                 tau = tau_max;
    284:               }
    285:             } else {
    286:               /* Not good agreement */
    287:               if (tau_min > 1.0) {
    288:                 tau = nlsP->gamma2_i;
    289:               } else if (tau_max < nlsP->gamma1_i) {
    290:                 tau = nlsP->gamma1_i;
    291:               } else if ((tau_min < nlsP->gamma1_i) && (tau_max >= 1.0)) {
    292:                 tau = nlsP->gamma1_i;
    293:               } else if ((tau_1 >= nlsP->gamma1_i) && (tau_1 < 1.0) && ((tau_2 < nlsP->gamma1_i) || (tau_2 >= 1.0))) {
    294:                 tau = tau_1;
    295:               } else if ((tau_2 >= nlsP->gamma1_i) && (tau_2 < 1.0) && ((tau_1 < nlsP->gamma1_i) || (tau_2 >= 1.0))) {
    296:                 tau = tau_2;
    297:               } else {
    298:                 tau = tau_max;
    299:               }
    300:             }
    301:           }
    302:           tao->trust = tau * tao->trust;
    303:         }

    305:         if (fmin < f) {
    306:           f = fmin;
    307:           VecAXPY(tao->solution,sigma,tao->gradient);
    308:           TaoComputeGradient(tao,tao->solution,tao->gradient);

    310:           TaoGradientNorm(tao, tao->gradient,NORM_2,&gnorm);
    311:           if (PetscIsInfOrNanReal(gnorm)) SETERRQ(PETSC_COMM_SELF,1, "User provided compute gradient generated Inf or NaN");
    312:           needH = 1;

    314:           TaoMonitor(tao, tao->niter, f, gnorm, 0.0, 1.0, &reason);
    315:           if (reason != TAO_CONTINUE_ITERATING) return(0);
    316:         }
    317:       }
    318:       tao->trust = PetscMax(tao->trust, max_radius);

    320:       /* Modify the radius if it is too large or small */
    321:       tao->trust = PetscMax(tao->trust, nlsP->min_radius);
    322:       tao->trust = PetscMin(tao->trust, nlsP->max_radius);
    323:       break;

    325:     default:
    326:       /* Norm of the first direction will initialize radius */
    327:       tao->trust = 0.0;
    328:       break;
    329:     }
    330:   }

    332:   /* Set initial scaling for the BFGS preconditioner
    333:      This step is done after computing the initial trust-region radius
    334:      since the function value may have decreased */
    335:   if (NLS_PC_BFGS == nlsP->pc_type) {
    336:     if (f != 0.0) {
    337:       delta = 2.0 * PetscAbsScalar(f) / (gnorm*gnorm);
    338:     } else {
    339:       delta = 2.0 / (gnorm*gnorm);
    340:     }
    341:     MatLMVMSetDelta(nlsP->M,delta);
    342:   }

    344:   /* Set counter for gradient/reset steps*/
    345:   nlsP->newt = 0;
    346:   nlsP->bfgs = 0;
    347:   nlsP->sgrad = 0;
    348:   nlsP->grad = 0;

    350:   /* Have not converged; continue with Newton method */
    351:   while (reason == TAO_CONTINUE_ITERATING) {
    352:     ++tao->niter;
    353:     tao->ksp_its=0;

    355:     /* Compute the Hessian */
    356:     if (needH) {
    357:       TaoComputeHessian(tao,tao->solution,tao->hessian,tao->hessian_pre);
    358:     }

    360:     if ((NLS_PC_BFGS == nlsP->pc_type) && (BFGS_SCALE_AHESS == nlsP->bfgs_scale_type)) {
    361:       /* Obtain diagonal for the bfgs preconditioner  */
    362:       MatGetDiagonal(tao->hessian, nlsP->Diag);
    363:       VecAbs(nlsP->Diag);
    364:       VecReciprocal(nlsP->Diag);
    365:       MatLMVMSetScale(nlsP->M,nlsP->Diag);
    366:     }

    368:     /* Shift the Hessian matrix */
    369:     if (pert > 0) {
    370:       MatShift(tao->hessian, pert);
    371:       if (tao->hessian != tao->hessian_pre) {
    372:         MatShift(tao->hessian_pre, pert);
    373:       }
    374:     }

    376:     if (NLS_PC_BFGS == nlsP->pc_type) {
    377:       if (BFGS_SCALE_PHESS == nlsP->bfgs_scale_type) {
    378:         /* Obtain diagonal for the bfgs preconditioner  */
    379:         MatGetDiagonal(tao->hessian, nlsP->Diag);
    380:         VecAbs(nlsP->Diag);
    381:         VecReciprocal(nlsP->Diag);
    382:         MatLMVMSetScale(nlsP->M,nlsP->Diag);
    383:       }
    384:       /* Update the limited memory preconditioner */
    385:       MatLMVMUpdate(nlsP->M, tao->solution, tao->gradient);
    386:       ++bfgsUpdates;
    387:     }

    389:     /* Solve the Newton system of equations */
    390:     KSPSetOperators(tao->ksp,tao->hessian,tao->hessian_pre);
    391:     if (NLS_KSP_NASH == nlsP->ksp_type || NLS_KSP_STCG == nlsP->ksp_type ||  NLS_KSP_GLTR == nlsP->ksp_type) {

    393:       if (NLS_KSP_NASH == nlsP->ksp_type) {
    394:         KSPNASHSetRadius(tao->ksp,nlsP->max_radius);
    395:       } else if (NLS_KSP_STCG == nlsP->ksp_type) {
    396:          KSPSTCGSetRadius(tao->ksp,nlsP->max_radius);
    397:       } else if (NLS_KSP_GLTR == nlsP->ksp_type) {
    398:         KSPGLTRSetRadius(tao->ksp,nlsP->max_radius);
    399:       }

    401:       KSPSolve(tao->ksp, tao->gradient, nlsP->D);
    402:       KSPGetIterationNumber(tao->ksp,&kspits);
    403:       tao->ksp_its+=kspits;
    404:       tao->ksp_tot_its+=kspits;

    406:       if (NLS_KSP_NASH == nlsP->ksp_type) {
    407:         KSPNASHGetNormD(tao->ksp,&norm_d);
    408:       } else if (NLS_KSP_STCG == nlsP->ksp_type) {
    409:          KSPSTCGGetNormD(tao->ksp,&norm_d);
    410:       } else if (NLS_KSP_GLTR == nlsP->ksp_type) {
    411:         KSPGLTRGetNormD(tao->ksp,&norm_d);
    412:       }

    414:       if (0.0 == tao->trust) {
    415:         /* Radius was uninitialized; use the norm of the direction */
    416:         if (norm_d > 0.0) {
    417:           tao->trust = norm_d;

    419:           /* Modify the radius if it is too large or small */
    420:           tao->trust = PetscMax(tao->trust, nlsP->min_radius);
    421:           tao->trust = PetscMin(tao->trust, nlsP->max_radius);
    422:         } else {
    423:           /* The direction was bad; set radius to default value and re-solve
    424:              the trust-region subproblem to get a direction */
    425:           tao->trust = tao->trust0;

    427:           /* Modify the radius if it is too large or small */
    428:           tao->trust = PetscMax(tao->trust, nlsP->min_radius);
    429:           tao->trust = PetscMin(tao->trust, nlsP->max_radius);

    431:           if (NLS_KSP_NASH == nlsP->ksp_type) {
    432:             KSPNASHSetRadius(tao->ksp,nlsP->max_radius);
    433:           } else if (NLS_KSP_STCG == nlsP->ksp_type) {
    434:             KSPSTCGSetRadius(tao->ksp,nlsP->max_radius);
    435:           } else if (NLS_KSP_GLTR == nlsP->ksp_type) {
    436:             KSPGLTRSetRadius(tao->ksp,nlsP->max_radius);
    437:           }

    439:           KSPSolve(tao->ksp, tao->gradient, nlsP->D);
    440:           KSPGetIterationNumber(tao->ksp,&kspits);
    441:           tao->ksp_its+=kspits;
    442:           tao->ksp_tot_its+=kspits;
    443:           if (NLS_KSP_NASH == nlsP->ksp_type) {
    444:             KSPNASHGetNormD(tao->ksp,&norm_d);
    445:           } else if (NLS_KSP_STCG == nlsP->ksp_type) {
    446:             KSPSTCGGetNormD(tao->ksp,&norm_d);
    447:           } else if (NLS_KSP_GLTR == nlsP->ksp_type) {
    448:             KSPGLTRGetNormD(tao->ksp,&norm_d);
    449:           }

    451:           if (norm_d == 0.0) SETERRQ(PETSC_COMM_SELF,1, "Initial direction zero");
    452:         }
    453:       }
    454:     } else {
    455:       KSPSolve(tao->ksp, tao->gradient, nlsP->D);
    456:       KSPGetIterationNumber(tao->ksp, &kspits);
    457:       tao->ksp_its += kspits;
    458:       tao->ksp_tot_its+=kspits;
    459:     }
    460:     VecScale(nlsP->D, -1.0);
    461:     KSPGetConvergedReason(tao->ksp, &ksp_reason);
    462:     if ((KSP_DIVERGED_INDEFINITE_PC == ksp_reason) &&  (NLS_PC_BFGS == nlsP->pc_type) && (bfgsUpdates > 1)) {
    463:       /* Preconditioner is numerically indefinite; reset the
    464:          approximate if using BFGS preconditioning. */

    466:       if (f != 0.0) {
    467:         delta = 2.0 * PetscAbsScalar(f) / (gnorm*gnorm);
    468:       } else {
    469:         delta = 2.0 / (gnorm*gnorm);
    470:       }
    471:       MatLMVMSetDelta(nlsP->M,delta);
    472:       MatLMVMReset(nlsP->M);
    473:       MatLMVMUpdate(nlsP->M, tao->solution, tao->gradient);
    474:       bfgsUpdates = 1;
    475:     }

    477:     if (KSP_CONVERGED_ATOL == ksp_reason) {
    478:       ++nlsP->ksp_atol;
    479:     } else if (KSP_CONVERGED_RTOL == ksp_reason) {
    480:       ++nlsP->ksp_rtol;
    481:     } else if (KSP_CONVERGED_CG_CONSTRAINED == ksp_reason) {
    482:       ++nlsP->ksp_ctol;
    483:     } else if (KSP_CONVERGED_CG_NEG_CURVE == ksp_reason) {
    484:       ++nlsP->ksp_negc;
    485:     } else if (KSP_DIVERGED_DTOL == ksp_reason) {
    486:       ++nlsP->ksp_dtol;
    487:     } else if (KSP_DIVERGED_ITS == ksp_reason) {
    488:       ++nlsP->ksp_iter;
    489:     } else {
    490:       ++nlsP->ksp_othr;
    491:     }

    493:     /* Check for success (descent direction) */
    494:     VecDot(nlsP->D, tao->gradient, &gdx);
    495:     if ((gdx >= 0.0) || PetscIsInfOrNanReal(gdx)) {
    496:       /* Newton step is not descent or direction produced Inf or NaN
    497:          Update the perturbation for next time */
    498:       if (pert <= 0.0) {
    499:         /* Initialize the perturbation */
    500:         pert = PetscMin(nlsP->imax, PetscMax(nlsP->imin, nlsP->imfac * gnorm));
    501:         if (NLS_KSP_GLTR == nlsP->ksp_type) {
    502:           KSPGLTRGetMinEig(tao->ksp,&e_min);
    503:           pert = PetscMax(pert, -e_min);
    504:         }
    505:       } else {
    506:         /* Increase the perturbation */
    507:         pert = PetscMin(nlsP->pmax, PetscMax(nlsP->pgfac * pert, nlsP->pmgfac * gnorm));
    508:       }

    510:       if (NLS_PC_BFGS != nlsP->pc_type) {
    511:         /* We don't have the bfgs matrix around and updated
    512:            Must use gradient direction in this case */
    513:         VecCopy(tao->gradient, nlsP->D);
    514:         VecScale(nlsP->D, -1.0);
    515:         ++nlsP->grad;
    516:         stepType = NLS_GRADIENT;
    517:       } else {
    518:         /* Attempt to use the BFGS direction */
    519:         MatLMVMSolve(nlsP->M, tao->gradient, nlsP->D);
    520:         VecScale(nlsP->D, -1.0);

    522:         /* Check for success (descent direction) */
    523:         VecDot(tao->gradient, nlsP->D, &gdx);
    524:         if ((gdx >= 0) || PetscIsInfOrNanReal(gdx)) {
    525:           /* BFGS direction is not descent or direction produced not a number
    526:              We can assert bfgsUpdates > 1 in this case because
    527:              the first solve produces the scaled gradient direction,
    528:              which is guaranteed to be descent */

    530:           /* Use steepest descent direction (scaled) */

    532:           if (f != 0.0) {
    533:             delta = 2.0 * PetscAbsScalar(f) / (gnorm*gnorm);
    534:           } else {
    535:             delta = 2.0 / (gnorm*gnorm);
    536:           }
    537:           MatLMVMSetDelta(nlsP->M, delta);
    538:           MatLMVMReset(nlsP->M);
    539:           MatLMVMUpdate(nlsP->M, tao->solution, tao->gradient);
    540:           MatLMVMSolve(nlsP->M, tao->gradient, nlsP->D);
    541:           VecScale(nlsP->D, -1.0);

    543:           bfgsUpdates = 1;
    544:           ++nlsP->sgrad;
    545:           stepType = NLS_SCALED_GRADIENT;
    546:         } else {
    547:           if (1 == bfgsUpdates) {
    548:             /* The first BFGS direction is always the scaled gradient */
    549:             ++nlsP->sgrad;
    550:             stepType = NLS_SCALED_GRADIENT;
    551:           } else {
    552:             ++nlsP->bfgs;
    553:             stepType = NLS_BFGS;
    554:           }
    555:         }
    556:       }
    557:     } else {
    558:       /* Computed Newton step is descent */
    559:       switch (ksp_reason) {
    560:       case KSP_DIVERGED_NANORINF:
    561:       case KSP_DIVERGED_BREAKDOWN:
    562:       case KSP_DIVERGED_INDEFINITE_MAT:
    563:       case KSP_DIVERGED_INDEFINITE_PC:
    564:       case KSP_CONVERGED_CG_NEG_CURVE:
    565:         /* Matrix or preconditioner is indefinite; increase perturbation */
    566:         if (pert <= 0.0) {
    567:           /* Initialize the perturbation */
    568:           pert = PetscMin(nlsP->imax, PetscMax(nlsP->imin, nlsP->imfac * gnorm));
    569:           if (NLS_KSP_GLTR == nlsP->ksp_type) {
    570:             KSPGLTRGetMinEig(tao->ksp, &e_min);
    571:             pert = PetscMax(pert, -e_min);
    572:           }
    573:         } else {
    574:           /* Increase the perturbation */
    575:           pert = PetscMin(nlsP->pmax, PetscMax(nlsP->pgfac * pert, nlsP->pmgfac * gnorm));
    576:         }
    577:         break;

    579:       default:
    580:         /* Newton step computation is good; decrease perturbation */
    581:         pert = PetscMin(nlsP->psfac * pert, nlsP->pmsfac * gnorm);
    582:         if (pert < nlsP->pmin) {
    583:           pert = 0.0;
    584:         }
    585:         break;
    586:       }

    588:       ++nlsP->newt;
    589:       stepType = NLS_NEWTON;
    590:     }

    592:     /* Perform the linesearch */
    593:     fold = f;
    594:     VecCopy(tao->solution, nlsP->Xold);
    595:     VecCopy(tao->gradient, nlsP->Gold);

    597:     TaoLineSearchApply(tao->linesearch, tao->solution, &f, tao->gradient, nlsP->D, &step, &ls_reason);
    598:     TaoAddLineSearchCounts(tao);

    600:     while (ls_reason != TAOLINESEARCH_SUCCESS && ls_reason != TAOLINESEARCH_SUCCESS_USER && stepType != NLS_GRADIENT) {      /* Linesearch failed */
    601:       f = fold;
    602:       VecCopy(nlsP->Xold, tao->solution);
    603:       VecCopy(nlsP->Gold, tao->gradient);

    605:       switch(stepType) {
    606:       case NLS_NEWTON:
    607:         /* Failed to obtain acceptable iterate with Newton 1step
    608:            Update the perturbation for next time */
    609:         if (pert <= 0.0) {
    610:           /* Initialize the perturbation */
    611:           pert = PetscMin(nlsP->imax, PetscMax(nlsP->imin, nlsP->imfac * gnorm));
    612:           if (NLS_KSP_GLTR == nlsP->ksp_type) {
    613:             KSPGLTRGetMinEig(tao->ksp,&e_min);
    614:             pert = PetscMax(pert, -e_min);
    615:           }
    616:         } else {
    617:           /* Increase the perturbation */
    618:           pert = PetscMin(nlsP->pmax, PetscMax(nlsP->pgfac * pert, nlsP->pmgfac * gnorm));
    619:         }

    621:         if (NLS_PC_BFGS != nlsP->pc_type) {
    622:           /* We don't have the bfgs matrix around and being updated
    623:              Must use gradient direction in this case */
    624:           VecCopy(tao->gradient, nlsP->D);
    625:           ++nlsP->grad;
    626:           stepType = NLS_GRADIENT;
    627:         } else {
    628:           /* Attempt to use the BFGS direction */
    629:           MatLMVMSolve(nlsP->M, tao->gradient, nlsP->D);
    630:           /* Check for success (descent direction) */
    631:           VecDot(tao->solution, nlsP->D, &gdx);
    632:           if ((gdx <= 0) || PetscIsInfOrNanReal(gdx)) {
    633:             /* BFGS direction is not descent or direction produced not a number
    634:                We can assert bfgsUpdates > 1 in this case
    635:                Use steepest descent direction (scaled) */

    637:             if (f != 0.0) {
    638:               delta = 2.0 * PetscAbsScalar(f) / (gnorm*gnorm);
    639:             } else {
    640:               delta = 2.0 / (gnorm*gnorm);
    641:             }
    642:             MatLMVMSetDelta(nlsP->M, delta);
    643:             MatLMVMReset(nlsP->M);
    644:             MatLMVMUpdate(nlsP->M, tao->solution, tao->gradient);
    645:             MatLMVMSolve(nlsP->M, tao->gradient, nlsP->D);

    647:             bfgsUpdates = 1;
    648:             ++nlsP->sgrad;
    649:             stepType = NLS_SCALED_GRADIENT;
    650:           } else {
    651:             if (1 == bfgsUpdates) {
    652:               /* The first BFGS direction is always the scaled gradient */
    653:               ++nlsP->sgrad;
    654:               stepType = NLS_SCALED_GRADIENT;
    655:             } else {
    656:               ++nlsP->bfgs;
    657:               stepType = NLS_BFGS;
    658:             }
    659:           }
    660:         }
    661:         break;

    663:       case NLS_BFGS:
    664:         /* Can only enter if pc_type == NLS_PC_BFGS
    665:            Failed to obtain acceptable iterate with BFGS step
    666:            Attempt to use the scaled gradient direction */

    668:         if (f != 0.0) {
    669:           delta = 2.0 * PetscAbsScalar(f) / (gnorm*gnorm);
    670:         } else {
    671:           delta = 2.0 / (gnorm*gnorm);
    672:         }
    673:         MatLMVMSetDelta(nlsP->M, delta);
    674:         MatLMVMReset(nlsP->M);
    675:         MatLMVMUpdate(nlsP->M, tao->solution, tao->gradient);
    676:         MatLMVMSolve(nlsP->M, tao->gradient, nlsP->D);

    678:         bfgsUpdates = 1;
    679:         ++nlsP->sgrad;
    680:         stepType = NLS_SCALED_GRADIENT;
    681:         break;

    683:       case NLS_SCALED_GRADIENT:
    684:         /* Can only enter if pc_type == NLS_PC_BFGS
    685:            The scaled gradient step did not produce a new iterate;
    686:            attemp to use the gradient direction.
    687:            Need to make sure we are not using a different diagonal scaling */

    689:         MatLMVMSetScale(nlsP->M,0);
    690:         MatLMVMSetDelta(nlsP->M,1.0);
    691:         MatLMVMReset(nlsP->M);
    692:         MatLMVMUpdate(nlsP->M, tao->solution, tao->gradient);
    693:         MatLMVMSolve(nlsP->M, tao->gradient, nlsP->D);

    695:         bfgsUpdates = 1;
    696:         ++nlsP->grad;
    697:         stepType = NLS_GRADIENT;
    698:         break;
    699:       }
    700:       VecScale(nlsP->D, -1.0);

    702:       TaoLineSearchApply(tao->linesearch, tao->solution, &f, tao->gradient, nlsP->D, &step, &ls_reason);
    703:       TaoLineSearchGetFullStepObjective(tao->linesearch, &f_full);
    704:       TaoAddLineSearchCounts(tao);
    705:     }

    707:     if (ls_reason != TAOLINESEARCH_SUCCESS && ls_reason != TAOLINESEARCH_SUCCESS_USER) {
    708:       /* Failed to find an improving point */
    709:       f = fold;
    710:       VecCopy(nlsP->Xold, tao->solution);
    711:       VecCopy(nlsP->Gold, tao->gradient);
    712:       step = 0.0;
    713:       reason = TAO_DIVERGED_LS_FAILURE;
    714:       tao->reason = TAO_DIVERGED_LS_FAILURE;
    715:       break;
    716:     }

    718:     /* Update trust region radius */
    719:     if (NLS_KSP_NASH == nlsP->ksp_type || NLS_KSP_STCG == nlsP->ksp_type || NLS_KSP_GLTR == nlsP->ksp_type) {
    720:       switch(nlsP->update_type) {
    721:       case NLS_UPDATE_STEP:
    722:         if (stepType == NLS_NEWTON) {
    723:           if (step < nlsP->nu1) {
    724:             /* Very bad step taken; reduce radius */
    725:             tao->trust = nlsP->omega1 * PetscMin(norm_d, tao->trust);
    726:           } else if (step < nlsP->nu2) {
    727:             /* Reasonably bad step taken; reduce radius */
    728:             tao->trust = nlsP->omega2 * PetscMin(norm_d, tao->trust);
    729:           } else if (step < nlsP->nu3) {
    730:             /*  Reasonable step was taken; leave radius alone */
    731:             if (nlsP->omega3 < 1.0) {
    732:               tao->trust = nlsP->omega3 * PetscMin(norm_d, tao->trust);
    733:             } else if (nlsP->omega3 > 1.0) {
    734:               tao->trust = PetscMax(nlsP->omega3 * norm_d, tao->trust);
    735:             }
    736:           } else if (step < nlsP->nu4) {
    737:             /*  Full step taken; increase the radius */
    738:             tao->trust = PetscMax(nlsP->omega4 * norm_d, tao->trust);
    739:           } else {
    740:             /*  More than full step taken; increase the radius */
    741:             tao->trust = PetscMax(nlsP->omega5 * norm_d, tao->trust);
    742:           }
    743:         } else {
    744:           /*  Newton step was not good; reduce the radius */
    745:           tao->trust = nlsP->omega1 * PetscMin(norm_d, tao->trust);
    746:         }
    747:         break;

    749:       case NLS_UPDATE_REDUCTION:
    750:         if (stepType == NLS_NEWTON) {
    751:           /*  Get predicted reduction */

    753:           if (NLS_KSP_STCG == nlsP->ksp_type) {
    754:             KSPSTCGGetObjFcn(tao->ksp,&prered);
    755:           } else if (NLS_KSP_NASH == nlsP->ksp_type)  {
    756:             KSPNASHGetObjFcn(tao->ksp,&prered);
    757:           } else {
    758:             KSPGLTRGetObjFcn(tao->ksp,&prered);
    759:           }

    761:           if (prered >= 0.0) {
    762:             /*  The predicted reduction has the wrong sign.  This cannot */
    763:             /*  happen in infinite precision arithmetic.  Step should */
    764:             /*  be rejected! */
    765:             tao->trust = nlsP->alpha1 * PetscMin(tao->trust, norm_d);
    766:           } else {
    767:             if (PetscIsInfOrNanReal(f_full)) {
    768:               tao->trust = nlsP->alpha1 * PetscMin(tao->trust, norm_d);
    769:             } else {
    770:               /*  Compute and actual reduction */
    771:               actred = fold - f_full;
    772:               prered = -prered;
    773:               if ((PetscAbsScalar(actred) <= nlsP->epsilon) &&
    774:                   (PetscAbsScalar(prered) <= nlsP->epsilon)) {
    775:                 kappa = 1.0;
    776:               } else {
    777:                 kappa = actred / prered;
    778:               }

    780:               /*  Accept of reject the step and update radius */
    781:               if (kappa < nlsP->eta1) {
    782:                 /*  Very bad step */
    783:                 tao->trust = nlsP->alpha1 * PetscMin(tao->trust, norm_d);
    784:               } else if (kappa < nlsP->eta2) {
    785:                 /*  Marginal bad step */
    786:                 tao->trust = nlsP->alpha2 * PetscMin(tao->trust, norm_d);
    787:               } else if (kappa < nlsP->eta3) {
    788:                 /*  Reasonable step */
    789:                 if (nlsP->alpha3 < 1.0) {
    790:                   tao->trust = nlsP->alpha3 * PetscMin(norm_d, tao->trust);
    791:                 } else if (nlsP->alpha3 > 1.0) {
    792:                   tao->trust = PetscMax(nlsP->alpha3 * norm_d, tao->trust);
    793:                 }
    794:               } else if (kappa < nlsP->eta4) {
    795:                 /*  Good step */
    796:                 tao->trust = PetscMax(nlsP->alpha4 * norm_d, tao->trust);
    797:               } else {
    798:                 /*  Very good step */
    799:                 tao->trust = PetscMax(nlsP->alpha5 * norm_d, tao->trust);
    800:               }
    801:             }
    802:           }
    803:         } else {
    804:           /*  Newton step was not good; reduce the radius */
    805:           tao->trust = nlsP->alpha1 * PetscMin(norm_d, tao->trust);
    806:         }
    807:         break;

    809:       default:
    810:         if (stepType == NLS_NEWTON) {

    812:           if (NLS_KSP_STCG == nlsP->ksp_type) {
    813:               KSPSTCGGetObjFcn(tao->ksp,&prered);
    814:           } else if (NLS_KSP_NASH == nlsP->ksp_type)  {
    815:               KSPNASHGetObjFcn(tao->ksp,&prered);
    816:           } else {
    817:               KSPGLTRGetObjFcn(tao->ksp,&prered);
    818:           }
    819:           if (prered >= 0.0) {
    820:             /*  The predicted reduction has the wrong sign.  This cannot */
    821:             /*  happen in infinite precision arithmetic.  Step should */
    822:             /*  be rejected! */
    823:             tao->trust = nlsP->gamma1 * PetscMin(tao->trust, norm_d);
    824:           } else {
    825:             if (PetscIsInfOrNanReal(f_full)) {
    826:               tao->trust = nlsP->gamma1 * PetscMin(tao->trust, norm_d);
    827:             } else {
    828:               actred = fold - f_full;
    829:               prered = -prered;
    830:               if ((PetscAbsScalar(actred) <= nlsP->epsilon) && (PetscAbsScalar(prered) <= nlsP->epsilon)) {
    831:                 kappa = 1.0;
    832:               } else {
    833:                 kappa = actred / prered;
    834:               }

    836:               tau_1 = nlsP->theta * gdx / (nlsP->theta * gdx - (1.0 - nlsP->theta) * prered + actred);
    837:               tau_2 = nlsP->theta * gdx / (nlsP->theta * gdx + (1.0 + nlsP->theta) * prered - actred);
    838:               tau_min = PetscMin(tau_1, tau_2);
    839:               tau_max = PetscMax(tau_1, tau_2);

    841:               if (kappa >= 1.0 - nlsP->mu1) {
    842:                 /*  Great agreement */
    843:                 if (tau_max < 1.0) {
    844:                   tao->trust = PetscMax(tao->trust, nlsP->gamma3 * norm_d);
    845:                 } else if (tau_max > nlsP->gamma4) {
    846:                   tao->trust = PetscMax(tao->trust, nlsP->gamma4 * norm_d);
    847:                 } else {
    848:                   tao->trust = PetscMax(tao->trust, tau_max * norm_d);
    849:                 }
    850:               } else if (kappa >= 1.0 - nlsP->mu2) {
    851:                 /*  Good agreement */

    853:                 if (tau_max < nlsP->gamma2) {
    854:                   tao->trust = nlsP->gamma2 * PetscMin(tao->trust, norm_d);
    855:                 } else if (tau_max > nlsP->gamma3) {
    856:                   tao->trust = PetscMax(tao->trust, nlsP->gamma3 * norm_d);
    857:                 } else if (tau_max < 1.0) {
    858:                   tao->trust = tau_max * PetscMin(tao->trust, norm_d);
    859:                 } else {
    860:                   tao->trust = PetscMax(tao->trust, tau_max * norm_d);
    861:                 }
    862:               } else {
    863:                 /*  Not good agreement */
    864:                 if (tau_min > 1.0) {
    865:                   tao->trust = nlsP->gamma2 * PetscMin(tao->trust, norm_d);
    866:                 } else if (tau_max < nlsP->gamma1) {
    867:                   tao->trust = nlsP->gamma1 * PetscMin(tao->trust, norm_d);
    868:                 } else if ((tau_min < nlsP->gamma1) && (tau_max >= 1.0)) {
    869:                   tao->trust = nlsP->gamma1 * PetscMin(tao->trust, norm_d);
    870:                 } else if ((tau_1 >= nlsP->gamma1) && (tau_1 < 1.0) && ((tau_2 < nlsP->gamma1) || (tau_2 >= 1.0))) {
    871:                   tao->trust = tau_1 * PetscMin(tao->trust, norm_d);
    872:                 } else if ((tau_2 >= nlsP->gamma1) && (tau_2 < 1.0) && ((tau_1 < nlsP->gamma1) || (tau_2 >= 1.0))) {
    873:                   tao->trust = tau_2 * PetscMin(tao->trust, norm_d);
    874:                 } else {
    875:                   tao->trust = tau_max * PetscMin(tao->trust, norm_d);
    876:                 }
    877:               }
    878:             }
    879:           }
    880:         } else {
    881:           /*  Newton step was not good; reduce the radius */
    882:           tao->trust = nlsP->gamma1 * PetscMin(norm_d, tao->trust);
    883:         }
    884:         break;
    885:       }

    887:       /*  The radius may have been increased; modify if it is too large */
    888:       tao->trust = PetscMin(tao->trust, nlsP->max_radius);
    889:     }

    891:     /*  Check for termination */
    892:     TaoGradientNorm(tao, tao->gradient,NORM_2,&gnorm);
    893:     if (PetscIsInfOrNanReal(f) || PetscIsInfOrNanReal(gnorm)) SETERRQ(PETSC_COMM_SELF,1,"User provided compute function generated Not-a-Number");
    894:     needH = 1;
    895:     TaoMonitor(tao, tao->niter, f, gnorm, 0.0, step, &reason);
    896:   }
    897:   return(0);
    898: }

    900: /* ---------------------------------------------------------- */
    903: static PetscErrorCode TaoSetUp_NLS(Tao tao)
    904: {
    905:   TAO_NLS        *nlsP = (TAO_NLS *)tao->data;

    909:   if (!tao->gradient) {VecDuplicate(tao->solution,&tao->gradient);}
    910:   if (!tao->stepdirection) {VecDuplicate(tao->solution,&tao->stepdirection);}
    911:   if (!nlsP->W) {VecDuplicate(tao->solution,&nlsP->W);}
    912:   if (!nlsP->D) {VecDuplicate(tao->solution,&nlsP->D);}
    913:   if (!nlsP->Xold) {VecDuplicate(tao->solution,&nlsP->Xold);}
    914:   if (!nlsP->Gold) {VecDuplicate(tao->solution,&nlsP->Gold);}
    915:   nlsP->Diag = 0;
    916:   nlsP->M = 0;
    917:   return(0);
    918: }

    920: /*------------------------------------------------------------*/
    923: static PetscErrorCode TaoDestroy_NLS(Tao tao)
    924: {
    925:   TAO_NLS        *nlsP = (TAO_NLS *)tao->data;

    929:   if (tao->setupcalled) {
    930:     VecDestroy(&nlsP->D);
    931:     VecDestroy(&nlsP->W);
    932:     VecDestroy(&nlsP->Xold);
    933:     VecDestroy(&nlsP->Gold);
    934:   }
    935:   VecDestroy(&nlsP->Diag);
    936:   MatDestroy(&nlsP->M);
    937:   PetscFree(tao->data);
    938:   return(0);
    939: }

    941: /*------------------------------------------------------------*/
    944: static PetscErrorCode TaoSetFromOptions_NLS(PetscOptionItems *PetscOptionsObject,Tao tao)
    945: {
    946:   TAO_NLS        *nlsP = (TAO_NLS *)tao->data;

    950:   PetscOptionsHead(PetscOptionsObject,"Newton line search method for unconstrained optimization");
    951:   PetscOptionsEList("-tao_nls_ksp_type", "ksp type", "", NLS_KSP, NLS_KSP_TYPES, NLS_KSP[nlsP->ksp_type], &nlsP->ksp_type, 0);
    952:   PetscOptionsEList("-tao_nls_pc_type", "pc type", "", NLS_PC, NLS_PC_TYPES, NLS_PC[nlsP->pc_type], &nlsP->pc_type, 0);
    953:   PetscOptionsEList("-tao_nls_bfgs_scale_type", "bfgs scale type", "", BFGS_SCALE, BFGS_SCALE_TYPES, BFGS_SCALE[nlsP->bfgs_scale_type], &nlsP->bfgs_scale_type, 0);
    954:   PetscOptionsEList("-tao_nls_init_type", "radius initialization type", "", NLS_INIT, NLS_INIT_TYPES, NLS_INIT[nlsP->init_type], &nlsP->init_type, 0);
    955:   PetscOptionsEList("-tao_nls_update_type", "radius update type", "", NLS_UPDATE, NLS_UPDATE_TYPES, NLS_UPDATE[nlsP->update_type], &nlsP->update_type, 0);
    956:   PetscOptionsReal("-tao_nls_sval", "perturbation starting value", "", nlsP->sval, &nlsP->sval,NULL);
    957:   PetscOptionsReal("-tao_nls_imin", "minimum initial perturbation", "", nlsP->imin, &nlsP->imin,NULL);
    958:   PetscOptionsReal("-tao_nls_imax", "maximum initial perturbation", "", nlsP->imax, &nlsP->imax,NULL);
    959:   PetscOptionsReal("-tao_nls_imfac", "initial merit factor", "", nlsP->imfac, &nlsP->imfac,NULL);
    960:   PetscOptionsReal("-tao_nls_pmin", "minimum perturbation", "", nlsP->pmin, &nlsP->pmin,NULL);
    961:   PetscOptionsReal("-tao_nls_pmax", "maximum perturbation", "", nlsP->pmax, &nlsP->pmax,NULL);
    962:   PetscOptionsReal("-tao_nls_pgfac", "growth factor", "", nlsP->pgfac, &nlsP->pgfac,NULL);
    963:   PetscOptionsReal("-tao_nls_psfac", "shrink factor", "", nlsP->psfac, &nlsP->psfac,NULL);
    964:   PetscOptionsReal("-tao_nls_pmgfac", "merit growth factor", "", nlsP->pmgfac, &nlsP->pmgfac,NULL);
    965:   PetscOptionsReal("-tao_nls_pmsfac", "merit shrink factor", "", nlsP->pmsfac, &nlsP->pmsfac,NULL);
    966:   PetscOptionsReal("-tao_nls_eta1", "poor steplength; reduce radius", "", nlsP->eta1, &nlsP->eta1,NULL);
    967:   PetscOptionsReal("-tao_nls_eta2", "reasonable steplength; leave radius alone", "", nlsP->eta2, &nlsP->eta2,NULL);
    968:   PetscOptionsReal("-tao_nls_eta3", "good steplength; increase radius", "", nlsP->eta3, &nlsP->eta3,NULL);
    969:   PetscOptionsReal("-tao_nls_eta4", "excellent steplength; greatly increase radius", "", nlsP->eta4, &nlsP->eta4,NULL);
    970:   PetscOptionsReal("-tao_nls_alpha1", "", "", nlsP->alpha1, &nlsP->alpha1,NULL);
    971:   PetscOptionsReal("-tao_nls_alpha2", "", "", nlsP->alpha2, &nlsP->alpha2,NULL);
    972:   PetscOptionsReal("-tao_nls_alpha3", "", "", nlsP->alpha3, &nlsP->alpha3,NULL);
    973:   PetscOptionsReal("-tao_nls_alpha4", "", "", nlsP->alpha4, &nlsP->alpha4,NULL);
    974:   PetscOptionsReal("-tao_nls_alpha5", "", "", nlsP->alpha5, &nlsP->alpha5,NULL);
    975:   PetscOptionsReal("-tao_nls_nu1", "poor steplength; reduce radius", "", nlsP->nu1, &nlsP->nu1,NULL);
    976:   PetscOptionsReal("-tao_nls_nu2", "reasonable steplength; leave radius alone", "", nlsP->nu2, &nlsP->nu2,NULL);
    977:   PetscOptionsReal("-tao_nls_nu3", "good steplength; increase radius", "", nlsP->nu3, &nlsP->nu3,NULL);
    978:   PetscOptionsReal("-tao_nls_nu4", "excellent steplength; greatly increase radius", "", nlsP->nu4, &nlsP->nu4,NULL);
    979:   PetscOptionsReal("-tao_nls_omega1", "", "", nlsP->omega1, &nlsP->omega1,NULL);
    980:   PetscOptionsReal("-tao_nls_omega2", "", "", nlsP->omega2, &nlsP->omega2,NULL);
    981:   PetscOptionsReal("-tao_nls_omega3", "", "", nlsP->omega3, &nlsP->omega3,NULL);
    982:   PetscOptionsReal("-tao_nls_omega4", "", "", nlsP->omega4, &nlsP->omega4,NULL);
    983:   PetscOptionsReal("-tao_nls_omega5", "", "", nlsP->omega5, &nlsP->omega5,NULL);
    984:   PetscOptionsReal("-tao_nls_mu1_i", "", "", nlsP->mu1_i, &nlsP->mu1_i,NULL);
    985:   PetscOptionsReal("-tao_nls_mu2_i", "", "", nlsP->mu2_i, &nlsP->mu2_i,NULL);
    986:   PetscOptionsReal("-tao_nls_gamma1_i", "", "", nlsP->gamma1_i, &nlsP->gamma1_i,NULL);
    987:   PetscOptionsReal("-tao_nls_gamma2_i", "", "", nlsP->gamma2_i, &nlsP->gamma2_i,NULL);
    988:   PetscOptionsReal("-tao_nls_gamma3_i", "", "", nlsP->gamma3_i, &nlsP->gamma3_i,NULL);
    989:   PetscOptionsReal("-tao_nls_gamma4_i", "", "", nlsP->gamma4_i, &nlsP->gamma4_i,NULL);
    990:   PetscOptionsReal("-tao_nls_theta_i", "", "", nlsP->theta_i, &nlsP->theta_i,NULL);
    991:   PetscOptionsReal("-tao_nls_mu1", "", "", nlsP->mu1, &nlsP->mu1,NULL);
    992:   PetscOptionsReal("-tao_nls_mu2", "", "", nlsP->mu2, &nlsP->mu2,NULL);
    993:   PetscOptionsReal("-tao_nls_gamma1", "", "", nlsP->gamma1, &nlsP->gamma1,NULL);
    994:   PetscOptionsReal("-tao_nls_gamma2", "", "", nlsP->gamma2, &nlsP->gamma2,NULL);
    995:   PetscOptionsReal("-tao_nls_gamma3", "", "", nlsP->gamma3, &nlsP->gamma3,NULL);
    996:   PetscOptionsReal("-tao_nls_gamma4", "", "", nlsP->gamma4, &nlsP->gamma4,NULL);
    997:   PetscOptionsReal("-tao_nls_theta", "", "", nlsP->theta, &nlsP->theta,NULL);
    998:   PetscOptionsReal("-tao_nls_min_radius", "lower bound on initial radius", "", nlsP->min_radius, &nlsP->min_radius,NULL);
    999:   PetscOptionsReal("-tao_nls_max_radius", "upper bound on radius", "", nlsP->max_radius, &nlsP->max_radius,NULL);
    1000:   PetscOptionsReal("-tao_nls_epsilon", "tolerance used when computing actual and predicted reduction", "", nlsP->epsilon, &nlsP->epsilon,NULL);
    1001:   PetscOptionsTail();
    1002:   TaoLineSearchSetFromOptions(tao->linesearch);
    1003:   KSPSetFromOptions(tao->ksp);
    1004:   return(0);
    1005: }

    1008: /*------------------------------------------------------------*/
    1011: static PetscErrorCode TaoView_NLS(Tao tao, PetscViewer viewer)
    1012: {
    1013:   TAO_NLS        *nlsP = (TAO_NLS *)tao->data;
    1014:   PetscInt       nrejects;
    1015:   PetscBool      isascii;

    1019:   PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&isascii);
    1020:   if (isascii) {
    1021:     PetscViewerASCIIPushTab(viewer);
    1022:     if (NLS_PC_BFGS == nlsP->pc_type && nlsP->M) {
    1023:       MatLMVMGetRejects(nlsP->M,&nrejects);
    1024:       PetscViewerASCIIPrintf(viewer, "Rejected matrix updates: %D\n",nrejects);
    1025:     }
    1026:     PetscViewerASCIIPrintf(viewer, "Newton steps: %D\n", nlsP->newt);
    1027:     PetscViewerASCIIPrintf(viewer, "BFGS steps: %D\n", nlsP->bfgs);
    1028:     PetscViewerASCIIPrintf(viewer, "Scaled gradient steps: %D\n", nlsP->sgrad);
    1029:     PetscViewerASCIIPrintf(viewer, "Gradient steps: %D\n", nlsP->grad);

    1031:     PetscViewerASCIIPrintf(viewer, "nls ksp atol: %D\n", nlsP->ksp_atol);
    1032:     PetscViewerASCIIPrintf(viewer, "nls ksp rtol: %D\n", nlsP->ksp_rtol);
    1033:     PetscViewerASCIIPrintf(viewer, "nls ksp ctol: %D\n", nlsP->ksp_ctol);
    1034:     PetscViewerASCIIPrintf(viewer, "nls ksp negc: %D\n", nlsP->ksp_negc);
    1035:     PetscViewerASCIIPrintf(viewer, "nls ksp dtol: %D\n", nlsP->ksp_dtol);
    1036:     PetscViewerASCIIPrintf(viewer, "nls ksp iter: %D\n", nlsP->ksp_iter);
    1037:     PetscViewerASCIIPrintf(viewer, "nls ksp othr: %D\n", nlsP->ksp_othr);
    1038:     PetscViewerASCIIPopTab(viewer);
    1039:   }
    1040:   return(0);
    1041: }

    1043: /* ---------------------------------------------------------- */
    1044: /*MC
    1045:   TAONLS - Newton's method with linesearch for unconstrained minimization.
    1046:   At each iteration, the Newton line search method solves the symmetric
    1047:   system of equations to obtain the step diretion dk:
    1048:               Hk dk = -gk
    1049:   a More-Thuente line search is applied on the direction dk to approximately
    1050:   solve
    1051:               min_t f(xk + t d_k)

    1053:     Options Database Keys:
    1054: + -tao_nls_ksp_type - "cg","nash","stcg","gltr","petsc"
    1055: . -tao_nls_pc_type - "none","ahess","bfgs","petsc"
    1056: . -tao_nls_bfgs_scale_type - "ahess","phess","bfgs"
    1057: . -tao_nls_init_type - "constant","direction","interpolation"
    1058: . -tao_nls_update_type - "step","direction","interpolation"
    1059: . -tao_nls_sval - perturbation starting value
    1060: . -tao_nls_imin - minimum initial perturbation
    1061: . -tao_nls_imax - maximum initial perturbation
    1062: . -tao_nls_pmin - minimum perturbation
    1063: . -tao_nls_pmax - maximum perturbation
    1064: . -tao_nls_pgfac - growth factor
    1065: . -tao_nls_psfac - shrink factor
    1066: . -tao_nls_imfac - initial merit factor
    1067: . -tao_nls_pmgfac - merit growth factor
    1068: . -tao_nls_pmsfac - merit shrink factor
    1069: . -tao_nls_eta1 - poor steplength; reduce radius
    1070: . -tao_nls_eta2 - reasonable steplength; leave radius
    1071: . -tao_nls_eta3 - good steplength; increase readius
    1072: . -tao_nls_eta4 - excellent steplength; greatly increase radius
    1073: . -tao_nls_alpha1 - alpha1 reduction
    1074: . -tao_nls_alpha2 - alpha2 reduction
    1075: . -tao_nls_alpha3 - alpha3 reduction
    1076: . -tao_nls_alpha4 - alpha4 reduction
    1077: . -tao_nls_alpha - alpha5 reduction
    1078: . -tao_nls_mu1 - mu1 interpolation update
    1079: . -tao_nls_mu2 - mu2 interpolation update
    1080: . -tao_nls_gamma1 - gamma1 interpolation update
    1081: . -tao_nls_gamma2 - gamma2 interpolation update
    1082: . -tao_nls_gamma3 - gamma3 interpolation update
    1083: . -tao_nls_gamma4 - gamma4 interpolation update
    1084: . -tao_nls_theta - theta interpolation update
    1085: . -tao_nls_omega1 - omega1 step update
    1086: . -tao_nls_omega2 - omega2 step update
    1087: . -tao_nls_omega3 - omega3 step update
    1088: . -tao_nls_omega4 - omega4 step update
    1089: . -tao_nls_omega5 - omega5 step update
    1090: . -tao_nls_mu1_i -  mu1 interpolation init factor
    1091: . -tao_nls_mu2_i -  mu2 interpolation init factor
    1092: . -tao_nls_gamma1_i -  gamma1 interpolation init factor
    1093: . -tao_nls_gamma2_i -  gamma2 interpolation init factor
    1094: . -tao_nls_gamma3_i -  gamma3 interpolation init factor
    1095: . -tao_nls_gamma4_i -  gamma4 interpolation init factor
    1096: - -tao_nls_theta_i -  theta interpolation init factor

    1098:   Level: beginner
    1099: M*/

    1103: PETSC_EXTERN PetscErrorCode TaoCreate_NLS(Tao tao)
    1104: {
    1105:   TAO_NLS        *nlsP;
    1106:   const char     *morethuente_type = TAOLINESEARCHMT;

    1110:   PetscNewLog(tao,&nlsP);

    1112:   tao->ops->setup = TaoSetUp_NLS;
    1113:   tao->ops->solve = TaoSolve_NLS;
    1114:   tao->ops->view = TaoView_NLS;
    1115:   tao->ops->setfromoptions = TaoSetFromOptions_NLS;
    1116:   tao->ops->destroy = TaoDestroy_NLS;

    1118:   /* Override default settings (unless already changed) */
    1119:   if (!tao->max_it_changed) tao->max_it = 50;
    1120:   if (!tao->trust0_changed) tao->trust0 = 100.0;

    1122:   tao->data = (void*)nlsP;

    1124:   nlsP->sval   = 0.0;
    1125:   nlsP->imin   = 1.0e-4;
    1126:   nlsP->imax   = 1.0e+2;
    1127:   nlsP->imfac  = 1.0e-1;

    1129:   nlsP->pmin   = 1.0e-12;
    1130:   nlsP->pmax   = 1.0e+2;
    1131:   nlsP->pgfac  = 1.0e+1;
    1132:   nlsP->psfac  = 4.0e-1;
    1133:   nlsP->pmgfac = 1.0e-1;
    1134:   nlsP->pmsfac = 1.0e-1;

    1136:   /*  Default values for trust-region radius update based on steplength */
    1137:   nlsP->nu1 = 0.25;
    1138:   nlsP->nu2 = 0.50;
    1139:   nlsP->nu3 = 1.00;
    1140:   nlsP->nu4 = 1.25;

    1142:   nlsP->omega1 = 0.25;
    1143:   nlsP->omega2 = 0.50;
    1144:   nlsP->omega3 = 1.00;
    1145:   nlsP->omega4 = 2.00;
    1146:   nlsP->omega5 = 4.00;

    1148:   /*  Default values for trust-region radius update based on reduction */
    1149:   nlsP->eta1 = 1.0e-4;
    1150:   nlsP->eta2 = 0.25;
    1151:   nlsP->eta3 = 0.50;
    1152:   nlsP->eta4 = 0.90;

    1154:   nlsP->alpha1 = 0.25;
    1155:   nlsP->alpha2 = 0.50;
    1156:   nlsP->alpha3 = 1.00;
    1157:   nlsP->alpha4 = 2.00;
    1158:   nlsP->alpha5 = 4.00;

    1160:   /*  Default values for trust-region radius update based on interpolation */
    1161:   nlsP->mu1 = 0.10;
    1162:   nlsP->mu2 = 0.50;

    1164:   nlsP->gamma1 = 0.25;
    1165:   nlsP->gamma2 = 0.50;
    1166:   nlsP->gamma3 = 2.00;
    1167:   nlsP->gamma4 = 4.00;

    1169:   nlsP->theta = 0.05;

    1171:   /*  Default values for trust region initialization based on interpolation */
    1172:   nlsP->mu1_i = 0.35;
    1173:   nlsP->mu2_i = 0.50;

    1175:   nlsP->gamma1_i = 0.0625;
    1176:   nlsP->gamma2_i = 0.5;
    1177:   nlsP->gamma3_i = 2.0;
    1178:   nlsP->gamma4_i = 5.0;

    1180:   nlsP->theta_i = 0.25;

    1182:   /*  Remaining parameters */
    1183:   nlsP->min_radius = 1.0e-10;
    1184:   nlsP->max_radius = 1.0e10;
    1185:   nlsP->epsilon = 1.0e-6;

    1187:   nlsP->ksp_type        = NLS_KSP_STCG;
    1188:   nlsP->pc_type         = NLS_PC_BFGS;
    1189:   nlsP->bfgs_scale_type = BFGS_SCALE_PHESS;
    1190:   nlsP->init_type       = NLS_INIT_INTERPOLATION;
    1191:   nlsP->update_type     = NLS_UPDATE_STEP;

    1193:   TaoLineSearchCreate(((PetscObject)tao)->comm,&tao->linesearch);
    1194:   TaoLineSearchSetType(tao->linesearch,morethuente_type);
    1195:   TaoLineSearchUseTaoRoutines(tao->linesearch,tao);
    1196:   TaoLineSearchSetOptionsPrefix(tao->linesearch,tao->hdr.prefix);

    1198:   /*  Set linear solver to default for symmetric matrices */
    1199:   KSPCreate(((PetscObject)tao)->comm,&tao->ksp);
    1200:   KSPSetOptionsPrefix(tao->ksp, tao->hdr.prefix);
    1201:   return(0);
    1202: }

    1206: static PetscErrorCode MatLMVMSolveShell(PC pc, Vec b, Vec x)
    1207: {
    1209:   Mat            M;

    1215:   PCShellGetContext(pc,(void**)&M);
    1216:   MatLMVMSolve(M, b, x);
    1217:   return(0);
    1218: }