Actual source code: pounders.c

      1: #include <../src/tao/leastsquares/impls/pounders/pounders.h>

      3: static PetscErrorCode pounders_h(Tao subtao, Vec v, Mat H, Mat Hpre, void *ctx)
      4: {
      5:   PetscFunctionBegin;
      6:   PetscFunctionReturn(PETSC_SUCCESS);
      7: }

      9: static PetscErrorCode pounders_fg(Tao subtao, Vec x, PetscReal *f, Vec g, void *ctx)
     10: {
     11:   TAO_POUNDERS *mfqP = (TAO_POUNDERS *)ctx;
     12:   PetscReal     d1, d2;

     14:   PetscFunctionBegin;
     15:   /* g = A*x  (add b later)*/
     16:   PetscCall(MatMult(mfqP->subH, x, g));

     18:   /* f = 1/2 * x'*(Ax) + b'*x  */
     19:   PetscCall(VecDot(x, g, &d1));
     20:   PetscCall(VecDot(mfqP->subb, x, &d2));
     21:   *f = 0.5 * d1 + d2;

     23:   /* now  g = g + b */
     24:   PetscCall(VecAXPY(g, 1.0, mfqP->subb));
     25:   PetscFunctionReturn(PETSC_SUCCESS);
     26: }

     28: static PetscErrorCode pounders_feval(Tao tao, Vec x, Vec F, PetscReal *fsum)
     29: {
     30:   TAO_POUNDERS *mfqP = (TAO_POUNDERS *)tao->data;
     31:   PetscInt      i, row, col;
     32:   PetscReal     fr, fc;

     34:   PetscFunctionBegin;
     35:   PetscCall(TaoComputeResidual(tao, x, F));
     36:   if (tao->res_weights_v) {
     37:     PetscCall(VecPointwiseMult(mfqP->workfvec, tao->res_weights_v, F));
     38:     PetscCall(VecDot(mfqP->workfvec, mfqP->workfvec, fsum));
     39:   } else if (tao->res_weights_w) {
     40:     *fsum = 0;
     41:     for (i = 0; i < tao->res_weights_n; i++) {
     42:       row = tao->res_weights_rows[i];
     43:       col = tao->res_weights_cols[i];
     44:       PetscCall(VecGetValues(F, 1, &row, &fr));
     45:       PetscCall(VecGetValues(F, 1, &col, &fc));
     46:       *fsum += tao->res_weights_w[i] * fc * fr;
     47:     }
     48:   } else {
     49:     PetscCall(VecDot(F, F, fsum));
     50:   }
     51:   PetscCall(PetscInfo(tao, "Least-squares residual norm: %20.19e\n", (double)*fsum));
     52:   PetscCheck(!PetscIsInfOrNanReal(*fsum), PETSC_COMM_SELF, PETSC_ERR_USER, "User provided compute function generated Inf or NaN");
     53:   PetscFunctionReturn(PETSC_SUCCESS);
     54: }

     56: static PetscErrorCode gqtwrap(Tao tao, PetscReal *gnorm, PetscReal *qmin)
     57: {
     58: #if defined(PETSC_USE_REAL_SINGLE)
     59:   PetscReal atol = 1.0e-5;
     60: #else
     61:   PetscReal atol = 1.0e-10;
     62: #endif
     63:   PetscInt      info, its;
     64:   TAO_POUNDERS *mfqP = (TAO_POUNDERS *)tao->data;

     66:   PetscFunctionBegin;
     67:   if (!mfqP->usegqt) {
     68:     PetscReal maxval;
     69:     PetscInt  i, j;

     71:     PetscCall(VecSetValues(mfqP->subb, mfqP->n, mfqP->indices, mfqP->Gres, INSERT_VALUES));
     72:     PetscCall(VecAssemblyBegin(mfqP->subb));
     73:     PetscCall(VecAssemblyEnd(mfqP->subb));

     75:     PetscCall(VecSet(mfqP->subx, 0.0));

     77:     PetscCall(VecSet(mfqP->subndel, -1.0));
     78:     PetscCall(VecSet(mfqP->subpdel, +1.0));

     80:     /* Complete the lower triangle of the Hessian matrix */
     81:     for (i = 0; i < mfqP->n; i++) {
     82:       for (j = i + 1; j < mfqP->n; j++) mfqP->Hres[j + mfqP->n * i] = mfqP->Hres[mfqP->n * j + i];
     83:     }
     84:     PetscCall(MatSetValues(mfqP->subH, mfqP->n, mfqP->indices, mfqP->n, mfqP->indices, mfqP->Hres, INSERT_VALUES));
     85:     PetscCall(MatAssemblyBegin(mfqP->subH, MAT_FINAL_ASSEMBLY));
     86:     PetscCall(MatAssemblyEnd(mfqP->subH, MAT_FINAL_ASSEMBLY));

     88:     PetscCall(TaoResetStatistics(mfqP->subtao));
     89:     /* PetscCall(TaoSetTolerances(mfqP->subtao,*gnorm,*gnorm,PETSC_CURRENT)); */
     90:     /* enforce bound constraints -- experimental */
     91:     if (tao->XU && tao->XL) {
     92:       PetscCall(VecCopy(tao->XU, mfqP->subxu));
     93:       PetscCall(VecAXPY(mfqP->subxu, -1.0, tao->solution));
     94:       PetscCall(VecScale(mfqP->subxu, 1.0 / mfqP->delta));
     95:       PetscCall(VecCopy(tao->XL, mfqP->subxl));
     96:       PetscCall(VecAXPY(mfqP->subxl, -1.0, tao->solution));
     97:       PetscCall(VecScale(mfqP->subxl, 1.0 / mfqP->delta));

     99:       PetscCall(VecPointwiseMin(mfqP->subxu, mfqP->subxu, mfqP->subpdel));
    100:       PetscCall(VecPointwiseMax(mfqP->subxl, mfqP->subxl, mfqP->subndel));
    101:     } else {
    102:       PetscCall(VecCopy(mfqP->subpdel, mfqP->subxu));
    103:       PetscCall(VecCopy(mfqP->subndel, mfqP->subxl));
    104:     }
    105:     /* Make sure xu > xl */
    106:     PetscCall(VecCopy(mfqP->subxl, mfqP->subpdel));
    107:     PetscCall(VecAXPY(mfqP->subpdel, -1.0, mfqP->subxu));
    108:     PetscCall(VecMax(mfqP->subpdel, NULL, &maxval));
    109:     PetscCheck(maxval <= 1e-10, PetscObjectComm((PetscObject)tao), PETSC_ERR_ARG_OUTOFRANGE, "upper bound < lower bound in subproblem");
    110:     /* Make sure xu > tao->solution > xl */
    111:     PetscCall(VecCopy(mfqP->subxl, mfqP->subpdel));
    112:     PetscCall(VecAXPY(mfqP->subpdel, -1.0, mfqP->subx));
    113:     PetscCall(VecMax(mfqP->subpdel, NULL, &maxval));
    114:     PetscCheck(maxval <= 1e-10, PetscObjectComm((PetscObject)tao), PETSC_ERR_ARG_OUTOFRANGE, "initial guess < lower bound in subproblem");

    116:     PetscCall(VecCopy(mfqP->subx, mfqP->subpdel));
    117:     PetscCall(VecAXPY(mfqP->subpdel, -1.0, mfqP->subxu));
    118:     PetscCall(VecMax(mfqP->subpdel, NULL, &maxval));
    119:     PetscCheck(maxval <= 1e-10, PetscObjectComm((PetscObject)tao), PETSC_ERR_ARG_OUTOFRANGE, "initial guess > upper bound in subproblem");

    121:     PetscCall(TaoSolve(mfqP->subtao));
    122:     PetscCall(TaoGetSolutionStatus(mfqP->subtao, NULL, qmin, NULL, NULL, NULL, NULL));

    124:     /* test bounds post-solution*/
    125:     PetscCall(VecCopy(mfqP->subxl, mfqP->subpdel));
    126:     PetscCall(VecAXPY(mfqP->subpdel, -1.0, mfqP->subx));
    127:     PetscCall(VecMax(mfqP->subpdel, NULL, &maxval));
    128:     if (maxval > 1e-5) {
    129:       PetscCall(PetscInfo(tao, "subproblem solution < lower bound\n"));
    130:       tao->reason = TAO_DIVERGED_TR_REDUCTION;
    131:     }

    133:     PetscCall(VecCopy(mfqP->subx, mfqP->subpdel));
    134:     PetscCall(VecAXPY(mfqP->subpdel, -1.0, mfqP->subxu));
    135:     PetscCall(VecMax(mfqP->subpdel, NULL, &maxval));
    136:     if (maxval > 1e-5) {
    137:       PetscCall(PetscInfo(tao, "subproblem solution > upper bound\n"));
    138:       tao->reason = TAO_DIVERGED_TR_REDUCTION;
    139:     }
    140:   } else {
    141:     PetscCall(gqt(mfqP->n, mfqP->Hres, mfqP->n, mfqP->Gres, 1.0, mfqP->gqt_rtol, atol, mfqP->gqt_maxits, gnorm, qmin, mfqP->Xsubproblem, &info, &its, mfqP->work, mfqP->work2, mfqP->work3));
    142:   }
    143:   *qmin *= -1;
    144:   PetscFunctionReturn(PETSC_SUCCESS);
    145: }

    147: static PetscErrorCode pounders_update_res(Tao tao)
    148: {
    149:   TAO_POUNDERS *mfqP = (TAO_POUNDERS *)tao->data;
    150:   PetscInt      i, row, col;
    151:   PetscBLASInt  blasn, blasn2, blasm, ione = 1;
    152:   PetscReal     zero = 0.0, one = 1.0, wii, factor;

    154:   PetscFunctionBegin;
    155:   PetscCall(PetscBLASIntCast(mfqP->n, &blasn));
    156:   PetscCall(PetscBLASIntCast(mfqP->m, &blasm));
    157:   PetscCall(PetscBLASIntCast(mfqP->n * mfqP->n, &blasn2));
    158:   for (i = 0; i < mfqP->n; i++) mfqP->Gres[i] = 0;
    159:   for (i = 0; i < mfqP->n * mfqP->n; i++) mfqP->Hres[i] = 0;

    161:   /* Compute Gres= sum_ij[wij * (cjgi + cigj)] */
    162:   if (tao->res_weights_v) {
    163:     /* Vector(diagonal) weights: gres = sum_i(wii*ci*gi) */
    164:     for (i = 0; i < mfqP->m; i++) {
    165:       PetscCall(VecGetValues(tao->res_weights_v, 1, &i, &factor));
    166:       factor = factor * mfqP->C[i];
    167:       PetscCallBLAS("BLASaxpy", BLASaxpy_(&blasn, &factor, &mfqP->Fdiff[blasn * i], &ione, mfqP->Gres, &ione));
    168:     }

    170:     /* compute Hres = sum_ij [wij * (*ci*Hj + cj*Hi + gi gj' + gj gi') ] */
    171:     /* vector(diagonal weights) Hres = sum_i(wii*(ci*Hi + gi * gi')*/
    172:     for (i = 0; i < mfqP->m; i++) {
    173:       PetscCall(VecGetValues(tao->res_weights_v, 1, &i, &wii));
    174:       if (tao->niter > 1) {
    175:         factor = wii * mfqP->C[i];
    176:         /* add wii * ci * Hi */
    177:         PetscCallBLAS("BLASaxpy", BLASaxpy_(&blasn2, &factor, &mfqP->H[i], &blasm, mfqP->Hres, &ione));
    178:       }
    179:       /* add wii * gi * gi' */
    180:       PetscCallBLAS("BLASgemm", BLASgemm_("N", "T", &blasn, &blasn, &ione, &wii, &mfqP->Fdiff[blasn * i], &blasn, &mfqP->Fdiff[blasn * i], &blasn, &one, mfqP->Hres, &blasn));
    181:     }
    182:   } else if (tao->res_weights_w) {
    183:     /* General case: .5 * Gres= sum_ij[wij * (cjgi + cigj)] */
    184:     for (i = 0; i < tao->res_weights_n; i++) {
    185:       row = tao->res_weights_rows[i];
    186:       col = tao->res_weights_cols[i];

    188:       factor = tao->res_weights_w[i] * mfqP->C[col] / 2.0;
    189:       PetscCallBLAS("BLASaxpy", BLASaxpy_(&blasn, &factor, &mfqP->Fdiff[blasn * row], &ione, mfqP->Gres, &ione));
    190:       factor = tao->res_weights_w[i] * mfqP->C[row] / 2.0;
    191:       PetscCallBLAS("BLASaxpy", BLASaxpy_(&blasn, &factor, &mfqP->Fdiff[blasn * col], &ione, mfqP->Gres, &ione));
    192:     }

    194:     /* compute Hres = sum_ij [wij * (*ci*Hj + cj*Hi + gi gj' + gj gi') ] */
    195:     /* .5 * sum_ij [wij * (*ci*Hj + cj*Hi + gi gj' + gj gi') ] */
    196:     for (i = 0; i < tao->res_weights_n; i++) {
    197:       row    = tao->res_weights_rows[i];
    198:       col    = tao->res_weights_cols[i];
    199:       factor = tao->res_weights_w[i] / 2.0;
    200:       /* add wij * gi gj' + wij * gj gi' */
    201:       PetscCallBLAS("BLASgemm", BLASgemm_("N", "T", &blasn, &blasn, &ione, &factor, &mfqP->Fdiff[blasn * row], &blasn, &mfqP->Fdiff[blasn * col], &blasn, &one, mfqP->Hres, &blasn));
    202:       PetscCallBLAS("BLASgemm", BLASgemm_("N", "T", &blasn, &blasn, &ione, &factor, &mfqP->Fdiff[blasn * col], &blasn, &mfqP->Fdiff[blasn * row], &blasn, &one, mfqP->Hres, &blasn));
    203:     }
    204:     if (tao->niter > 1) {
    205:       for (i = 0; i < tao->res_weights_n; i++) {
    206:         row = tao->res_weights_rows[i];
    207:         col = tao->res_weights_cols[i];

    209:         /* add  wij*cj*Hi */
    210:         factor = tao->res_weights_w[i] * mfqP->C[col] / 2.0;
    211:         PetscCallBLAS("BLASaxpy", BLASaxpy_(&blasn2, &factor, &mfqP->H[row], &blasm, mfqP->Hres, &ione));

    213:         /* add wij*ci*Hj */
    214:         factor = tao->res_weights_w[i] * mfqP->C[row] / 2.0;
    215:         PetscCallBLAS("BLASaxpy", BLASaxpy_(&blasn2, &factor, &mfqP->H[col], &blasm, mfqP->Hres, &ione));
    216:       }
    217:     }
    218:   } else {
    219:     /* Default: Gres= sum_i[cigi] = G*c' */
    220:     PetscCall(PetscInfo(tao, "Identity weights\n"));
    221:     PetscCallBLAS("BLASgemv", BLASgemv_("N", &blasn, &blasm, &one, mfqP->Fdiff, &blasn, mfqP->C, &ione, &zero, mfqP->Gres, &ione));

    223:     /* compute Hres = sum_ij [wij * (*ci*Hj + cj*Hi + gi gj' + gj gi') ] */
    224:     /*  Hres = G*G' + 0.5 sum {F(xkin,i)*H(:,:,i)}  */
    225:     PetscCallBLAS("BLASgemm", BLASgemm_("N", "T", &blasn, &blasn, &blasm, &one, mfqP->Fdiff, &blasn, mfqP->Fdiff, &blasn, &zero, mfqP->Hres, &blasn));

    227:     /* sum(F(xkin,i)*H(:,:,i)) */
    228:     if (tao->niter > 1) {
    229:       for (i = 0; i < mfqP->m; i++) {
    230:         factor = mfqP->C[i];
    231:         PetscCallBLAS("BLASaxpy", BLASaxpy_(&blasn2, &factor, &mfqP->H[i], &blasm, mfqP->Hres, &ione));
    232:       }
    233:     }
    234:   }
    235:   PetscFunctionReturn(PETSC_SUCCESS);
    236: }

    238: static PetscErrorCode phi2eval(PetscReal *x, PetscInt n, PetscReal *phi)
    239: {
    240:   /* Phi = .5*[x(1)^2  sqrt(2)*x(1)*x(2) ... sqrt(2)*x(1)*x(n) ... x(2)^2 sqrt(2)*x(2)*x(3) .. x(n)^2] */
    241:   PetscInt  i, j, k;
    242:   PetscReal sqrt2 = PetscSqrtReal(2.0);

    244:   PetscFunctionBegin;
    245:   j = 0;
    246:   for (i = 0; i < n; i++) {
    247:     phi[j] = 0.5 * x[i] * x[i];
    248:     j++;
    249:     for (k = i + 1; k < n; k++) {
    250:       phi[j] = x[i] * x[k] / sqrt2;
    251:       j++;
    252:     }
    253:   }
    254:   PetscFunctionReturn(PETSC_SUCCESS);
    255: }

    257: static PetscErrorCode getquadpounders(TAO_POUNDERS *mfqP)
    258: {
    259:   /* Computes the parameters of the quadratic Q(x) = c + g'*x + 0.5*x*G*x'
    260:    that satisfies the interpolation conditions Q(X[:,j]) = f(j)
    261:    for j=1,...,m and with a Hessian matrix of least Frobenius norm */

    263:   /* NB --we are ignoring c */
    264:   PetscInt     i, j, k, num, np = mfqP->nmodelpoints;
    265:   PetscReal    one = 1.0, zero = 0.0, negone = -1.0;
    266:   PetscBLASInt blasnpmax, blasnplus1, blasnp, blasint, blasint2;
    267:   PetscBLASInt info, ione = 1;
    268:   PetscReal    sqrt2 = PetscSqrtReal(2.0);

    270:   PetscFunctionBegin;
    271:   PetscCall(PetscBLASIntCast(mfqP->npmax, &blasnpmax));
    272:   PetscCall(PetscBLASIntCast(mfqP->n + 1, &blasnplus1));
    273:   PetscCall(PetscBLASIntCast(np, &blasnp));
    274:   PetscCall(PetscBLASIntCast(mfqP->n * (mfqP->n + 1) / 2, &blasint));
    275:   PetscCall(PetscBLASIntCast(np - mfqP->n - 1, &blasint2));
    276:   for (i = 0; i < mfqP->n * mfqP->m; i++) mfqP->Gdel[i] = 0;
    277:   for (i = 0; i < mfqP->n * mfqP->n * mfqP->m; i++) mfqP->Hdel[i] = 0;

    279:   /* factor M */
    280:   PetscCallBLAS("LAPACKgetrf", LAPACKgetrf_(&blasnplus1, &blasnp, mfqP->M, &blasnplus1, mfqP->npmaxiwork, &info));
    281:   PetscCheck(info == 0, PETSC_COMM_SELF, PETSC_ERR_LIB, "LAPACK routine getrf returned with value %" PetscBLASInt_FMT, info);

    283:   if (np == mfqP->n + 1) {
    284:     for (i = 0; i < mfqP->npmax - mfqP->n - 1; i++) mfqP->omega[i] = 0.0;
    285:     for (i = 0; i < mfqP->n * (mfqP->n + 1) / 2; i++) mfqP->beta[i] = 0.0;
    286:   } else {
    287:     /* Let Ltmp = (L'*L) */
    288:     PetscCallBLAS("BLASgemm", BLASgemm_("T", "N", &blasint2, &blasint2, &blasint, &one, &mfqP->L[(mfqP->n + 1) * blasint], &blasint, &mfqP->L[(mfqP->n + 1) * blasint], &blasint, &zero, mfqP->L_tmp, &blasint));

    290:     /* factor Ltmp */
    291:     PetscCallBLAS("LAPACKpotrf", LAPACKpotrf_("L", &blasint2, mfqP->L_tmp, &blasint, &info));
    292:     PetscCheck(info == 0, PETSC_COMM_SELF, PETSC_ERR_LIB, "LAPACK routine potrf returned with value %" PetscBLASInt_FMT, info);
    293:   }

    295:   for (k = 0; k < mfqP->m; k++) {
    296:     if (np != mfqP->n + 1) {
    297:       /* Solve L'*L*Omega = Z' * RESk*/
    298:       PetscCallBLAS("BLASgemv", BLASgemv_("T", &blasnp, &blasint2, &one, mfqP->Z, &blasnpmax, &mfqP->RES[mfqP->npmax * k], &ione, &zero, mfqP->omega, &ione));
    299:       PetscCallBLAS("LAPACKpotrs", LAPACKpotrs_("L", &blasint2, &ione, mfqP->L_tmp, &blasint, mfqP->omega, &blasint2, &info));
    300:       PetscCheck(info == 0, PETSC_COMM_SELF, PETSC_ERR_LIB, "LAPACK routine potrs returned with value %" PetscBLASInt_FMT, info);

    302:       /* Beta = L*Omega */
    303:       PetscCallBLAS("BLASgemv", BLASgemv_("N", &blasint, &blasint2, &one, &mfqP->L[(mfqP->n + 1) * blasint], &blasint, mfqP->omega, &ione, &zero, mfqP->beta, &ione));
    304:     }

    306:     /* solve M'*Alpha = RESk - N'*Beta */
    307:     PetscCallBLAS("BLASgemv", BLASgemv_("T", &blasint, &blasnp, &negone, mfqP->N, &blasint, mfqP->beta, &ione, &one, &mfqP->RES[mfqP->npmax * k], &ione));
    308:     PetscCallBLAS("LAPACKgetrs", LAPACKgetrs_("T", &blasnplus1, &ione, mfqP->M, &blasnplus1, mfqP->npmaxiwork, &mfqP->RES[mfqP->npmax * k], &blasnplus1, &info));
    309:     PetscCheck(info == 0, PETSC_COMM_SELF, PETSC_ERR_LIB, "LAPACK routine getrs returned with value %" PetscBLASInt_FMT, info);

    311:     /* Gdel(:,k) = Alpha(2:n+1) */
    312:     for (i = 0; i < mfqP->n; i++) mfqP->Gdel[i + mfqP->n * k] = mfqP->RES[mfqP->npmax * k + i + 1];

    314:     /* Set Hdels */
    315:     num = 0;
    316:     for (i = 0; i < mfqP->n; i++) {
    317:       /* H[i,i,k] = Beta(num) */
    318:       mfqP->Hdel[(i * mfqP->n + i) * mfqP->m + k] = mfqP->beta[num];
    319:       num++;
    320:       for (j = i + 1; j < mfqP->n; j++) {
    321:         /* H[i,j,k] = H[j,i,k] = Beta(num)/sqrt(2) */
    322:         mfqP->Hdel[(j * mfqP->n + i) * mfqP->m + k] = mfqP->beta[num] / sqrt2;
    323:         mfqP->Hdel[(i * mfqP->n + j) * mfqP->m + k] = mfqP->beta[num] / sqrt2;
    324:         num++;
    325:       }
    326:     }
    327:   }
    328:   PetscFunctionReturn(PETSC_SUCCESS);
    329: }

    331: static PetscErrorCode morepoints(TAO_POUNDERS *mfqP)
    332: {
    333:   /* Assumes mfqP->model_indices[0]  is minimum index
    334:    Finishes adding points to mfqP->model_indices (up to npmax)
    335:    Computes L,Z,M,N
    336:    np is actual number of points in model (should equal npmax?) */
    337:   PetscInt         point, i, j, offset;
    338:   PetscInt         reject;
    339:   PetscBLASInt     blasn, blasnpmax, blasnplus1, info, blasnmax, blasint, blasint2, blasnp, blasmaxmn;
    340:   const PetscReal *x;
    341:   PetscReal        normd;

    343:   PetscFunctionBegin;
    344:   PetscCall(PetscBLASIntCast(mfqP->npmax, &blasnpmax));
    345:   PetscCall(PetscBLASIntCast(mfqP->n, &blasn));
    346:   PetscCall(PetscBLASIntCast(mfqP->nmax, &blasnmax));
    347:   PetscCall(PetscBLASIntCast(mfqP->n + 1, &blasnplus1));
    348:   PetscCall(PetscBLASIntCast(mfqP->n, &blasnp));
    349:   /* Initialize M,N */
    350:   for (i = 0; i < mfqP->n + 1; i++) {
    351:     PetscCall(VecGetArrayRead(mfqP->Xhist[mfqP->model_indices[i]], &x));
    352:     mfqP->M[(mfqP->n + 1) * i] = 1.0;
    353:     for (j = 0; j < mfqP->n; j++) mfqP->M[j + 1 + ((mfqP->n + 1) * i)] = (x[j] - mfqP->xmin[j]) / mfqP->delta;
    354:     PetscCall(VecRestoreArrayRead(mfqP->Xhist[mfqP->model_indices[i]], &x));
    355:     PetscCall(phi2eval(&mfqP->M[1 + ((mfqP->n + 1) * i)], mfqP->n, &mfqP->N[mfqP->n * (mfqP->n + 1) / 2 * i]));
    356:   }

    358:   /* Now we add points until we have npmax starting with the most recent ones */
    359:   point              = mfqP->nHist - 1;
    360:   mfqP->nmodelpoints = mfqP->n + 1;
    361:   while (mfqP->nmodelpoints < mfqP->npmax && point >= 0) {
    362:     /* Reject any points already in the model */
    363:     reject = 0;
    364:     for (j = 0; j < mfqP->n + 1; j++) {
    365:       if (point == mfqP->model_indices[j]) {
    366:         reject = 1;
    367:         break;
    368:       }
    369:     }

    371:     /* Reject if norm(d) >c2 */
    372:     if (!reject) {
    373:       PetscCall(VecCopy(mfqP->Xhist[point], mfqP->workxvec));
    374:       PetscCall(VecAXPY(mfqP->workxvec, -1.0, mfqP->Xhist[mfqP->minindex]));
    375:       PetscCall(VecNorm(mfqP->workxvec, NORM_2, &normd));
    376:       normd /= mfqP->delta;
    377:       if (normd > mfqP->c2) reject = 1;
    378:     }
    379:     if (reject) {
    380:       point--;
    381:       continue;
    382:     }

    384:     PetscCall(VecGetArrayRead(mfqP->Xhist[point], &x));
    385:     mfqP->M[(mfqP->n + 1) * mfqP->nmodelpoints] = 1.0;
    386:     for (j = 0; j < mfqP->n; j++) mfqP->M[j + 1 + ((mfqP->n + 1) * mfqP->nmodelpoints)] = (x[j] - mfqP->xmin[j]) / mfqP->delta;
    387:     PetscCall(VecRestoreArrayRead(mfqP->Xhist[point], &x));
    388:     PetscCall(phi2eval(&mfqP->M[1 + (mfqP->n + 1) * mfqP->nmodelpoints], mfqP->n, &mfqP->N[mfqP->n * (mfqP->n + 1) / 2 * (mfqP->nmodelpoints)]));

    390:     /* Update QR factorization */
    391:     /* Copy M' to Q_tmp */
    392:     for (i = 0; i < mfqP->n + 1; i++) {
    393:       for (j = 0; j < mfqP->npmax; j++) mfqP->Q_tmp[j + mfqP->npmax * i] = mfqP->M[i + (mfqP->n + 1) * j];
    394:     }
    395:     PetscCall(PetscBLASIntCast(mfqP->nmodelpoints + 1, &blasnp));
    396:     /* Q_tmp,R = qr(M') */
    397:     PetscCall(PetscBLASIntCast(PetscMax(mfqP->m, mfqP->n + 1), &blasmaxmn));
    398:     PetscCallBLAS("LAPACKgeqrf", LAPACKgeqrf_(&blasnp, &blasnplus1, mfqP->Q_tmp, &blasnpmax, mfqP->tau_tmp, mfqP->mwork, &blasmaxmn, &info));
    399:     PetscCheck(info == 0, PETSC_COMM_SELF, PETSC_ERR_LIB, "LAPACK routine geqrf returned with value %" PetscBLASInt_FMT, info);

    401:     /* Reject if min(svd(N*Q(:,n+2:np+1)) <= theta2 */
    402:     /* L = N*Qtmp */
    403:     PetscCall(PetscBLASIntCast(mfqP->n * (mfqP->n + 1) / 2, &blasint2));
    404:     /* Copy N to L_tmp */
    405:     for (i = 0; i < mfqP->n * (mfqP->n + 1) / 2 * mfqP->npmax; i++) mfqP->L_tmp[i] = mfqP->N[i];
    406:     /* Copy L_save to L_tmp */

    408:     /* L_tmp = N*Qtmp' */
    409:     PetscCallBLAS("LAPACKormqr", LAPACKormqr_("R", "N", &blasint2, &blasnp, &blasnplus1, mfqP->Q_tmp, &blasnpmax, mfqP->tau_tmp, mfqP->L_tmp, &blasint2, mfqP->npmaxwork, &blasnmax, &info));
    410:     PetscCheck(info == 0, PETSC_COMM_SELF, PETSC_ERR_LIB, "LAPACK routine ormqr returned with value %" PetscBLASInt_FMT, info);

    412:     /* Copy L_tmp to L_save */
    413:     for (i = 0; i < mfqP->npmax * mfqP->n * (mfqP->n + 1) / 2; i++) mfqP->L_save[i] = mfqP->L_tmp[i];

    415:     /* Get svd for L_tmp(:,n+2:np+1) (L_tmp is modified in process) */
    416:     PetscCall(PetscBLASIntCast(mfqP->nmodelpoints - mfqP->n, &blasint));
    417:     PetscCall(PetscFPTrapPush(PETSC_FP_TRAP_OFF));
    418:     PetscCallBLAS("LAPACKgesvd", LAPACKgesvd_("N", "N", &blasint2, &blasint, &mfqP->L_tmp[(mfqP->n + 1) * blasint2], &blasint2, mfqP->beta, mfqP->work, &blasn, mfqP->work, &blasn, mfqP->npmaxwork, &blasnmax, &info));
    419:     PetscCall(PetscFPTrapPop());
    420:     PetscCheck(info == 0, PETSC_COMM_SELF, PETSC_ERR_LIB, "LAPACK routine gesvd returned with value %" PetscBLASInt_FMT, info);

    422:     if (mfqP->beta[PetscMin(blasint, blasint2) - 1] > mfqP->theta2) {
    423:       /* accept point */
    424:       mfqP->model_indices[mfqP->nmodelpoints] = point;
    425:       /* Copy Q_tmp to Q */
    426:       for (i = 0; i < mfqP->npmax * mfqP->npmax; i++) mfqP->Q[i] = mfqP->Q_tmp[i];
    427:       for (i = 0; i < mfqP->npmax; i++) mfqP->tau[i] = mfqP->tau_tmp[i];
    428:       mfqP->nmodelpoints++;
    429:       PetscCall(PetscBLASIntCast(mfqP->nmodelpoints, &blasnp));

    431:       /* Copy L_save to L */
    432:       for (i = 0; i < mfqP->npmax * mfqP->n * (mfqP->n + 1) / 2; i++) mfqP->L[i] = mfqP->L_save[i];
    433:     }
    434:     point--;
    435:   }

    437:   PetscCall(PetscBLASIntCast(mfqP->nmodelpoints, &blasnp));
    438:   /* Copy Q(:,n+2:np) to Z */
    439:   /* First set Q_tmp to I */
    440:   for (i = 0; i < mfqP->npmax * mfqP->npmax; i++) mfqP->Q_tmp[i] = 0.0;
    441:   for (i = 0; i < mfqP->npmax; i++) mfqP->Q_tmp[i + mfqP->npmax * i] = 1.0;

    443:   /* Q_tmp = I * Q */
    444:   PetscCallBLAS("LAPACKormqr", LAPACKormqr_("R", "N", &blasnp, &blasnp, &blasnplus1, mfqP->Q, &blasnpmax, mfqP->tau, mfqP->Q_tmp, &blasnpmax, mfqP->npmaxwork, &blasnmax, &info));
    445:   PetscCheck(info == 0, PETSC_COMM_SELF, PETSC_ERR_LIB, "LAPACK routine ormqr returned with value %" PetscBLASInt_FMT, info);

    447:   /* Copy Q_tmp(:,n+2:np) to Z) */
    448:   offset = mfqP->npmax * (mfqP->n + 1);
    449:   for (i = offset; i < mfqP->npmax * mfqP->npmax; i++) mfqP->Z[i - offset] = mfqP->Q_tmp[i];

    451:   if (mfqP->nmodelpoints == mfqP->n + 1) {
    452:     /* Set L to I_{n+1} */
    453:     for (i = 0; i < mfqP->npmax * mfqP->n * (mfqP->n + 1) / 2; i++) mfqP->L[i] = 0.0;
    454:     for (i = 0; i < mfqP->n; i++) mfqP->L[(mfqP->n * (mfqP->n + 1) / 2) * i + i] = 1.0;
    455:   }
    456:   PetscFunctionReturn(PETSC_SUCCESS);
    457: }

    459: /* Only call from modelimprove, addpoint() needs ->Q_tmp and ->work to be set */
    460: static PetscErrorCode addpoint(Tao tao, TAO_POUNDERS *mfqP, PetscInt index)
    461: {
    462:   PetscFunctionBegin;
    463:   /* Create new vector in history: X[newidx] = X[mfqP->index] + delta*X[index]*/
    464:   PetscCall(VecDuplicate(mfqP->Xhist[0], &mfqP->Xhist[mfqP->nHist]));
    465:   PetscCall(VecSetValues(mfqP->Xhist[mfqP->nHist], mfqP->n, mfqP->indices, &mfqP->Q_tmp[index * mfqP->npmax], INSERT_VALUES));
    466:   PetscCall(VecAssemblyBegin(mfqP->Xhist[mfqP->nHist]));
    467:   PetscCall(VecAssemblyEnd(mfqP->Xhist[mfqP->nHist]));
    468:   PetscCall(VecAYPX(mfqP->Xhist[mfqP->nHist], mfqP->delta, mfqP->Xhist[mfqP->minindex]));

    470:   /* Project into feasible region */
    471:   if (tao->XU && tao->XL) PetscCall(VecMedian(mfqP->Xhist[mfqP->nHist], tao->XL, tao->XU, mfqP->Xhist[mfqP->nHist]));

    473:   /* Compute value of new vector */
    474:   PetscCall(VecDuplicate(mfqP->Fhist[0], &mfqP->Fhist[mfqP->nHist]));
    475:   CHKMEMQ;
    476:   PetscCall(pounders_feval(tao, mfqP->Xhist[mfqP->nHist], mfqP->Fhist[mfqP->nHist], &mfqP->Fres[mfqP->nHist]));

    478:   /* Add new vector to model */
    479:   mfqP->model_indices[mfqP->nmodelpoints] = mfqP->nHist;
    480:   mfqP->nmodelpoints++;
    481:   mfqP->nHist++;
    482:   PetscFunctionReturn(PETSC_SUCCESS);
    483: }

    485: static PetscErrorCode modelimprove(Tao tao, TAO_POUNDERS *mfqP, PetscInt addallpoints)
    486: {
    487:   /* modeld = Q(:,np+1:n)' */
    488:   PetscInt     i, j, minindex = 0;
    489:   PetscReal    dp, half = 0.5, one = 1.0, minvalue = PETSC_INFINITY;
    490:   PetscBLASInt blasn, blasnpmax, blask, info;
    491:   PetscBLASInt blas1 = 1, blasnmax;

    493:   PetscFunctionBegin;
    494:   PetscCall(PetscBLASIntCast(mfqP->n, &blasn));
    495:   PetscCall(PetscBLASIntCast(mfqP->npmax, &blasnpmax));
    496:   PetscCall(PetscBLASIntCast(mfqP->nmodelpoints, &blask));
    497:   PetscCall(PetscBLASIntCast(mfqP->nmax, &blasnmax));

    499:   /* Qtmp = I(n x n) */
    500:   for (i = 0; i < mfqP->n; i++) {
    501:     for (j = 0; j < mfqP->n; j++) mfqP->Q_tmp[i + mfqP->npmax * j] = 0.0;
    502:   }
    503:   for (j = 0; j < mfqP->n; j++) mfqP->Q_tmp[j + mfqP->npmax * j] = 1.0;

    505:   /* Qtmp = Q * I */
    506:   PetscCallBLAS("LAPACKormqr", LAPACKormqr_("R", "N", &blasn, &blasn, &blask, mfqP->Q, &blasnpmax, mfqP->tau, mfqP->Q_tmp, &blasnpmax, mfqP->npmaxwork, &blasnmax, &info));

    508:   for (i = mfqP->nmodelpoints; i < mfqP->n; i++) {
    509:     PetscCallBLAS("BLASdot", dp = BLASdot_(&blasn, &mfqP->Q_tmp[i * mfqP->npmax], &blas1, mfqP->Gres, &blas1));
    510:     if (dp > 0.0) { /* Model says use the other direction! */
    511:       for (j = 0; j < mfqP->n; j++) mfqP->Q_tmp[i * mfqP->npmax + j] *= -1;
    512:     }
    513:     /* mfqP->work[i] = Cres+Modeld(i,:)*(Gres+.5*Hres*Modeld(i,:)') */
    514:     for (j = 0; j < mfqP->n; j++) mfqP->work2[j] = mfqP->Gres[j];
    515:     PetscCallBLAS("BLASgemv", BLASgemv_("N", &blasn, &blasn, &half, mfqP->Hres, &blasn, &mfqP->Q_tmp[i * mfqP->npmax], &blas1, &one, mfqP->work2, &blas1));
    516:     PetscCallBLAS("BLASdot", mfqP->work[i] = BLASdot_(&blasn, &mfqP->Q_tmp[i * mfqP->npmax], &blas1, mfqP->work2, &blas1));
    517:     if (i == mfqP->nmodelpoints || mfqP->work[i] < minvalue) {
    518:       minindex = i;
    519:       minvalue = mfqP->work[i];
    520:     }
    521:     if (addallpoints != 0) PetscCall(addpoint(tao, mfqP, i));
    522:   }
    523:   if (!addallpoints) PetscCall(addpoint(tao, mfqP, minindex));
    524:   PetscFunctionReturn(PETSC_SUCCESS);
    525: }

    527: static PetscErrorCode affpoints(TAO_POUNDERS *mfqP, PetscReal *xmin, PetscReal c)
    528: {
    529:   PetscInt         i, j;
    530:   PetscBLASInt     blasm, blasj, blask, blasn, ione = 1, info;
    531:   PetscBLASInt     blasnpmax, blasmaxmn;
    532:   PetscReal        proj, normd;
    533:   const PetscReal *x;

    535:   PetscFunctionBegin;
    536:   PetscCall(PetscBLASIntCast(mfqP->npmax, &blasnpmax));
    537:   PetscCall(PetscBLASIntCast(mfqP->m, &blasm));
    538:   PetscCall(PetscBLASIntCast(mfqP->n, &blasn));
    539:   for (i = mfqP->nHist - 1; i >= 0; i--) {
    540:     PetscCall(VecGetArrayRead(mfqP->Xhist[i], &x));
    541:     for (j = 0; j < mfqP->n; j++) mfqP->work[j] = (x[j] - xmin[j]) / mfqP->delta;
    542:     PetscCall(VecRestoreArrayRead(mfqP->Xhist[i], &x));
    543:     PetscCallBLAS("BLAScopy", BLAScopy_(&blasn, mfqP->work, &ione, mfqP->work2, &ione));
    544:     PetscCallBLAS("BLASnrm2", normd = BLASnrm2_(&blasn, mfqP->work, &ione));
    545:     if (normd <= c) {
    546:       PetscCall(PetscBLASIntCast(PetscMax(mfqP->n - mfqP->nmodelpoints, 0), &blasj));
    547:       if (!mfqP->q_is_I) {
    548:         /* project D onto null */
    549:         PetscCall(PetscBLASIntCast(mfqP->nmodelpoints, &blask));
    550:         PetscCallBLAS("LAPACKormqr", LAPACKormqr_("R", "N", &ione, &blasn, &blask, mfqP->Q, &blasnpmax, mfqP->tau, mfqP->work2, &ione, mfqP->mwork, &blasm, &info));
    551:         PetscCheck(info >= 0, PETSC_COMM_SELF, PETSC_ERR_LIB, "ormqr returned value %" PetscBLASInt_FMT, info);
    552:       }
    553:       PetscCallBLAS("BLASnrm2", proj = BLASnrm2_(&blasj, &mfqP->work2[mfqP->nmodelpoints], &ione));

    555:       if (proj >= mfqP->theta1) { /* add this index to model */
    556:         mfqP->model_indices[mfqP->nmodelpoints] = i;
    557:         mfqP->nmodelpoints++;
    558:         PetscCallBLAS("BLAScopy", BLAScopy_(&blasn, mfqP->work, &ione, &mfqP->Q_tmp[mfqP->npmax * (mfqP->nmodelpoints - 1)], &ione));
    559:         PetscCall(PetscBLASIntCast(mfqP->npmax * (mfqP->nmodelpoints), &blask));
    560:         PetscCallBLAS("BLAScopy", BLAScopy_(&blask, mfqP->Q_tmp, &ione, mfqP->Q, &ione));
    561:         PetscCall(PetscBLASIntCast(mfqP->nmodelpoints, &blask));
    562:         PetscCall(PetscBLASIntCast(PetscMax(mfqP->m, mfqP->n), &blasmaxmn));
    563:         PetscCallBLAS("LAPACKgeqrf", LAPACKgeqrf_(&blasn, &blask, mfqP->Q, &blasnpmax, mfqP->tau, mfqP->mwork, &blasmaxmn, &info));
    564:         PetscCheck(info >= 0, PETSC_COMM_SELF, PETSC_ERR_LIB, "geqrf returned value %" PetscBLASInt_FMT, info);
    565:         mfqP->q_is_I = 0;
    566:       }
    567:       if (mfqP->nmodelpoints == mfqP->n) break;
    568:     }
    569:   }
    570:   PetscFunctionReturn(PETSC_SUCCESS);
    571: }

    573: static PetscErrorCode TaoSolve_POUNDERS(Tao tao)
    574: {
    575:   TAO_POUNDERS    *mfqP = (TAO_POUNDERS *)tao->data;
    576:   PetscInt         i, ii, j, k, l;
    577:   PetscReal        step = 1.0;
    578:   PetscInt         low, high;
    579:   PetscReal        minnorm;
    580:   PetscReal       *x, *f;
    581:   const PetscReal *xmint, *fmin;
    582:   PetscReal        deltaold;
    583:   PetscReal        gnorm;
    584:   PetscBLASInt     info, ione = 1, iblas;
    585:   PetscBool        valid, same;
    586:   PetscReal        mdec, rho, normxsp;
    587:   PetscReal        one = 1.0, zero = 0.0, ratio;
    588:   PetscBLASInt     blasm, blasn, blasncopy, blasnpmax;
    589:   static PetscBool set = PETSC_FALSE;

    591:   /* n = # of parameters
    592:      m = dimension (components) of function  */
    593:   PetscFunctionBegin;
    594:   PetscCall(PetscCitationsRegister("@article{UNEDF0,\n"
    595:                                    "title = {Nuclear energy density optimization},\n"
    596:                                    "author = {Kortelainen, M.  and Lesinski, T.  and Mor\'e, J.  and Nazarewicz, W.\n"
    597:                                    "          and Sarich, J.  and Schunck, N.  and Stoitsov, M. V. and Wild, S. },\n"
    598:                                    "journal = {Phys. Rev. C},\n"
    599:                                    "volume = {82},\n"
    600:                                    "number = {2},\n"
    601:                                    "pages = {024313},\n"
    602:                                    "numpages = {18},\n"
    603:                                    "year = {2010},\n"
    604:                                    "month = {Aug},\n"
    605:                                    "doi = {10.1103/PhysRevC.82.024313}\n}\n",
    606:                                    &set));
    607:   tao->niter = 0;
    608:   if (tao->XL && tao->XU) {
    609:     /* Check x0 <= XU */
    610:     PetscReal val;

    612:     PetscCall(VecCopy(tao->solution, mfqP->Xhist[0]));
    613:     PetscCall(VecAXPY(mfqP->Xhist[0], -1.0, tao->XU));
    614:     PetscCall(VecMax(mfqP->Xhist[0], NULL, &val));
    615:     PetscCheck(val <= 1e-10, PetscObjectComm((PetscObject)tao), PETSC_ERR_ARG_OUTOFRANGE, "X0 > upper bound");

    617:     /* Check x0 >= xl */
    618:     PetscCall(VecCopy(tao->XL, mfqP->Xhist[0]));
    619:     PetscCall(VecAXPY(mfqP->Xhist[0], -1.0, tao->solution));
    620:     PetscCall(VecMax(mfqP->Xhist[0], NULL, &val));
    621:     PetscCheck(val <= 1e-10, PetscObjectComm((PetscObject)tao), PETSC_ERR_ARG_OUTOFRANGE, "X0 < lower bound");

    623:     /* Check x0 + delta < XU  -- should be able to get around this eventually */

    625:     PetscCall(VecSet(mfqP->Xhist[0], mfqP->delta));
    626:     PetscCall(VecAXPY(mfqP->Xhist[0], 1.0, tao->solution));
    627:     PetscCall(VecAXPY(mfqP->Xhist[0], -1.0, tao->XU));
    628:     PetscCall(VecMax(mfqP->Xhist[0], NULL, &val));
    629:     PetscCheck(val <= 1e-10, PetscObjectComm((PetscObject)tao), PETSC_ERR_ARG_OUTOFRANGE, "X0 + delta > upper bound");
    630:   }

    632:   PetscCall(PetscBLASIntCast(mfqP->m, &blasm));
    633:   PetscCall(PetscBLASIntCast(mfqP->n, &blasn));
    634:   PetscCall(PetscBLASIntCast(mfqP->npmax, &blasnpmax));
    635:   for (i = 0; i < mfqP->n * mfqP->n * mfqP->m; ++i) mfqP->H[i] = 0;

    637:   PetscCall(VecCopy(tao->solution, mfqP->Xhist[0]));

    639:   /* This provides enough information to approximate the gradient of the objective */
    640:   /* using a forward difference scheme. */

    642:   PetscCall(PetscInfo(tao, "Initialize simplex; delta = %10.9e\n", (double)mfqP->delta));
    643:   PetscCall(pounders_feval(tao, mfqP->Xhist[0], mfqP->Fhist[0], &mfqP->Fres[0]));
    644:   mfqP->minindex = 0;
    645:   minnorm        = mfqP->Fres[0];

    647:   PetscCall(VecGetOwnershipRange(mfqP->Xhist[0], &low, &high));
    648:   for (i = 1; i < mfqP->n + 1; ++i) {
    649:     PetscCall(VecCopy(mfqP->Xhist[0], mfqP->Xhist[i]));

    651:     if (i - 1 >= low && i - 1 < high) {
    652:       PetscCall(VecGetArray(mfqP->Xhist[i], &x));
    653:       x[i - 1 - low] += mfqP->delta;
    654:       PetscCall(VecRestoreArray(mfqP->Xhist[i], &x));
    655:     }
    656:     CHKMEMQ;
    657:     PetscCall(pounders_feval(tao, mfqP->Xhist[i], mfqP->Fhist[i], &mfqP->Fres[i]));
    658:     if (mfqP->Fres[i] < minnorm) {
    659:       mfqP->minindex = i;
    660:       minnorm        = mfqP->Fres[i];
    661:     }
    662:   }
    663:   PetscCall(VecCopy(mfqP->Xhist[mfqP->minindex], tao->solution));
    664:   PetscCall(VecCopy(mfqP->Fhist[mfqP->minindex], tao->ls_res));
    665:   PetscCall(PetscInfo(tao, "Finalize simplex; minnorm = %10.9e\n", (double)minnorm));

    667:   /* Gather mpi vecs to one big local vec */

    669:   /* Begin serial code */

    671:   /* Disp[i] = Xi-xmin, i=1,..,mfqP->minindex-1,mfqP->minindex+1,..,n */
    672:   /* Fdiff[i] = (Fi-Fmin)', i=1,..,mfqP->minindex-1,mfqP->minindex+1,..,n */
    673:   /* (Column oriented for blas calls) */
    674:   ii = 0;

    676:   PetscCall(PetscInfo(tao, "Build matrix: %" PetscInt_FMT "\n", (PetscInt)mfqP->size));
    677:   if (1 == mfqP->size) {
    678:     PetscCall(VecGetArrayRead(mfqP->Xhist[mfqP->minindex], &xmint));
    679:     for (i = 0; i < mfqP->n; i++) mfqP->xmin[i] = xmint[i];
    680:     PetscCall(VecRestoreArrayRead(mfqP->Xhist[mfqP->minindex], &xmint));
    681:     PetscCall(VecGetArrayRead(mfqP->Fhist[mfqP->minindex], &fmin));
    682:     for (i = 0; i < mfqP->n + 1; i++) {
    683:       if (i == mfqP->minindex) continue;

    685:       PetscCall(VecGetArray(mfqP->Xhist[i], &x));
    686:       for (j = 0; j < mfqP->n; j++) mfqP->Disp[ii + mfqP->npmax * j] = (x[j] - mfqP->xmin[j]) / mfqP->delta;
    687:       PetscCall(VecRestoreArray(mfqP->Xhist[i], &x));

    689:       PetscCall(VecGetArray(mfqP->Fhist[i], &f));
    690:       for (j = 0; j < mfqP->m; j++) mfqP->Fdiff[ii + mfqP->n * j] = f[j] - fmin[j];
    691:       PetscCall(VecRestoreArray(mfqP->Fhist[i], &f));

    693:       mfqP->model_indices[ii++] = i;
    694:     }
    695:     for (j = 0; j < mfqP->m; j++) mfqP->C[j] = fmin[j];
    696:     PetscCall(VecRestoreArrayRead(mfqP->Fhist[mfqP->minindex], &fmin));
    697:   } else {
    698:     PetscCall(VecSet(mfqP->localxmin, 0));
    699:     PetscCall(VecScatterBegin(mfqP->scatterx, mfqP->Xhist[mfqP->minindex], mfqP->localxmin, INSERT_VALUES, SCATTER_FORWARD));
    700:     PetscCall(VecScatterEnd(mfqP->scatterx, mfqP->Xhist[mfqP->minindex], mfqP->localxmin, INSERT_VALUES, SCATTER_FORWARD));

    702:     PetscCall(VecGetArrayRead(mfqP->localxmin, &xmint));
    703:     for (i = 0; i < mfqP->n; i++) mfqP->xmin[i] = xmint[i];
    704:     PetscCall(VecRestoreArrayRead(mfqP->localxmin, &xmint));

    706:     PetscCall(VecScatterBegin(mfqP->scatterf, mfqP->Fhist[mfqP->minindex], mfqP->localfmin, INSERT_VALUES, SCATTER_FORWARD));
    707:     PetscCall(VecScatterEnd(mfqP->scatterf, mfqP->Fhist[mfqP->minindex], mfqP->localfmin, INSERT_VALUES, SCATTER_FORWARD));
    708:     PetscCall(VecGetArrayRead(mfqP->localfmin, &fmin));
    709:     for (i = 0; i < mfqP->n + 1; i++) {
    710:       if (i == mfqP->minindex) continue;

    712:       PetscCall(VecScatterBegin(mfqP->scatterx, mfqP->Xhist[ii], mfqP->localx, INSERT_VALUES, SCATTER_FORWARD));
    713:       PetscCall(VecScatterEnd(mfqP->scatterx, mfqP->Xhist[ii], mfqP->localx, INSERT_VALUES, SCATTER_FORWARD));
    714:       PetscCall(VecGetArray(mfqP->localx, &x));
    715:       for (j = 0; j < mfqP->n; j++) mfqP->Disp[ii + mfqP->npmax * j] = (x[j] - mfqP->xmin[j]) / mfqP->delta;
    716:       PetscCall(VecRestoreArray(mfqP->localx, &x));

    718:       PetscCall(VecScatterBegin(mfqP->scatterf, mfqP->Fhist[ii], mfqP->localf, INSERT_VALUES, SCATTER_FORWARD));
    719:       PetscCall(VecScatterEnd(mfqP->scatterf, mfqP->Fhist[ii], mfqP->localf, INSERT_VALUES, SCATTER_FORWARD));
    720:       PetscCall(VecGetArray(mfqP->localf, &f));
    721:       for (j = 0; j < mfqP->m; j++) mfqP->Fdiff[ii + mfqP->n * j] = f[j] - fmin[j];
    722:       PetscCall(VecRestoreArray(mfqP->localf, &f));

    724:       mfqP->model_indices[ii++] = i;
    725:     }
    726:     for (j = 0; j < mfqP->m; j++) mfqP->C[j] = fmin[j];
    727:     PetscCall(VecRestoreArrayRead(mfqP->localfmin, &fmin));
    728:   }

    730:   /* Determine the initial quadratic models */
    731:   /* G = D(ModelIn,:) \ (F(ModelIn,1:m)-repmat(F(xkin,1:m),n,1)); */
    732:   /* D (nxn) Fdiff (nxm)  => G (nxm) */
    733:   blasncopy = blasn;
    734:   PetscCallBLAS("LAPACKgesv", LAPACKgesv_(&blasn, &blasm, mfqP->Disp, &blasnpmax, mfqP->iwork, mfqP->Fdiff, &blasncopy, &info));
    735:   PetscCall(PetscInfo(tao, "Linear solve return: %" PetscInt_FMT "\n", (PetscInt)info));

    737:   PetscCall(pounders_update_res(tao));

    739:   valid = PETSC_TRUE;

    741:   PetscCall(VecSetValues(tao->gradient, mfqP->n, mfqP->indices, mfqP->Gres, INSERT_VALUES));
    742:   PetscCall(VecAssemblyBegin(tao->gradient));
    743:   PetscCall(VecAssemblyEnd(tao->gradient));
    744:   PetscCall(VecNorm(tao->gradient, NORM_2, &gnorm));
    745:   gnorm *= mfqP->delta;
    746:   PetscCall(VecCopy(mfqP->Xhist[mfqP->minindex], tao->solution));

    748:   tao->reason = TAO_CONTINUE_ITERATING;
    749:   PetscCall(TaoLogConvergenceHistory(tao, minnorm, gnorm, 0.0, tao->ksp_its));
    750:   PetscCall(TaoMonitor(tao, tao->niter, minnorm, gnorm, 0.0, step));
    751:   PetscUseTypeMethod(tao, convergencetest, tao->cnvP);

    753:   mfqP->nHist        = mfqP->n + 1;
    754:   mfqP->nmodelpoints = mfqP->n + 1;
    755:   PetscCall(PetscInfo(tao, "Initial gradient: %20.19e\n", (double)gnorm));

    757:   while (tao->reason == TAO_CONTINUE_ITERATING) {
    758:     PetscReal gnm = 1e-4;
    759:     /* Call general purpose update function */
    760:     PetscTryTypeMethod(tao, update, tao->niter, tao->user_update);
    761:     tao->niter++;
    762:     /* Solve the subproblem min{Q(s): ||s|| <= 1.0} */
    763:     PetscCall(gqtwrap(tao, &gnm, &mdec));
    764:     /* Evaluate the function at the new point */

    766:     for (i = 0; i < mfqP->n; i++) mfqP->work[i] = mfqP->Xsubproblem[i] * mfqP->delta + mfqP->xmin[i];
    767:     PetscCall(VecDuplicate(tao->solution, &mfqP->Xhist[mfqP->nHist]));
    768:     PetscCall(VecDuplicate(tao->ls_res, &mfqP->Fhist[mfqP->nHist]));
    769:     PetscCall(VecSetValues(mfqP->Xhist[mfqP->nHist], mfqP->n, mfqP->indices, mfqP->work, INSERT_VALUES));
    770:     PetscCall(VecAssemblyBegin(mfqP->Xhist[mfqP->nHist]));
    771:     PetscCall(VecAssemblyEnd(mfqP->Xhist[mfqP->nHist]));

    773:     PetscCall(pounders_feval(tao, mfqP->Xhist[mfqP->nHist], mfqP->Fhist[mfqP->nHist], &mfqP->Fres[mfqP->nHist]));

    775:     rho = (mfqP->Fres[mfqP->minindex] - mfqP->Fres[mfqP->nHist]) / mdec;
    776:     mfqP->nHist++;

    778:     /* Update the center */
    779:     if ((rho >= mfqP->eta1) || (rho > mfqP->eta0 && valid == PETSC_TRUE)) {
    780:       /* Update model to reflect new base point */
    781:       for (i = 0; i < mfqP->n; i++) mfqP->work[i] = (mfqP->work[i] - mfqP->xmin[i]) / mfqP->delta;
    782:       for (j = 0; j < mfqP->m; j++) {
    783:         /* C(j) = C(j) + work*G(:,j) + .5*work*H(:,:,j)*work';
    784:          G(:,j) = G(:,j) + H(:,:,j)*work' */
    785:         for (k = 0; k < mfqP->n; k++) {
    786:           mfqP->work2[k] = 0.0;
    787:           for (l = 0; l < mfqP->n; l++) mfqP->work2[k] += mfqP->H[j + mfqP->m * (k + l * mfqP->n)] * mfqP->work[l];
    788:         }
    789:         for (i = 0; i < mfqP->n; i++) {
    790:           mfqP->C[j] += mfqP->work[i] * (mfqP->Fdiff[i + mfqP->n * j] + 0.5 * mfqP->work2[i]);
    791:           mfqP->Fdiff[i + mfqP->n * j] += mfqP->work2[i];
    792:         }
    793:       }
    794:       /* Cres += work*Gres + .5*work*Hres*work';
    795:        Gres += Hres*work'; */

    797:       PetscCallBLAS("BLASgemv", BLASgemv_("N", &blasn, &blasn, &one, mfqP->Hres, &blasn, mfqP->work, &ione, &zero, mfqP->work2, &ione));
    798:       for (i = 0; i < mfqP->n; i++) mfqP->Gres[i] += mfqP->work2[i];
    799:       mfqP->minindex = mfqP->nHist - 1;
    800:       minnorm        = mfqP->Fres[mfqP->minindex];
    801:       PetscCall(VecCopy(mfqP->Fhist[mfqP->minindex], tao->ls_res));
    802:       /* Change current center */
    803:       PetscCall(VecGetArrayRead(mfqP->Xhist[mfqP->minindex], &xmint));
    804:       for (i = 0; i < mfqP->n; i++) mfqP->xmin[i] = xmint[i];
    805:       PetscCall(VecRestoreArrayRead(mfqP->Xhist[mfqP->minindex], &xmint));
    806:     }

    808:     /* Evaluate at a model-improving point if necessary */
    809:     if (valid == PETSC_FALSE) {
    810:       mfqP->q_is_I       = 1;
    811:       mfqP->nmodelpoints = 0;
    812:       PetscCall(affpoints(mfqP, mfqP->xmin, mfqP->c1));
    813:       if (mfqP->nmodelpoints < mfqP->n) {
    814:         PetscCall(PetscInfo(tao, "Model not valid -- model-improving\n"));
    815:         PetscCall(modelimprove(tao, mfqP, 1));
    816:       }
    817:     }

    819:     /* Update the trust region radius */
    820:     deltaold = mfqP->delta;
    821:     normxsp  = 0;
    822:     for (i = 0; i < mfqP->n; i++) normxsp += mfqP->Xsubproblem[i] * mfqP->Xsubproblem[i];
    823:     normxsp = PetscSqrtReal(normxsp);
    824:     if (rho >= mfqP->eta1 && normxsp > 0.5 * mfqP->delta) {
    825:       mfqP->delta = PetscMin(mfqP->delta * mfqP->gamma1, mfqP->deltamax);
    826:     } else if (valid == PETSC_TRUE) {
    827:       mfqP->delta = PetscMax(mfqP->delta * mfqP->gamma0, mfqP->deltamin);
    828:     }

    830:     /* Compute the next interpolation set */
    831:     mfqP->q_is_I       = 1;
    832:     mfqP->nmodelpoints = 0;
    833:     PetscCall(PetscInfo(tao, "Affine Points: xmin = %20.19e, c1 = %20.19e\n", (double)*mfqP->xmin, (double)mfqP->c1));
    834:     PetscCall(affpoints(mfqP, mfqP->xmin, mfqP->c1));
    835:     if (mfqP->nmodelpoints == mfqP->n) {
    836:       valid = PETSC_TRUE;
    837:     } else {
    838:       valid = PETSC_FALSE;
    839:       PetscCall(PetscInfo(tao, "Affine Points: xmin = %20.19e, c2 = %20.19e\n", (double)*mfqP->xmin, (double)mfqP->c2));
    840:       PetscCall(affpoints(mfqP, mfqP->xmin, mfqP->c2));
    841:       if (mfqP->n > mfqP->nmodelpoints) {
    842:         PetscCall(PetscInfo(tao, "Model not valid -- adding geometry points\n"));
    843:         PetscCall(modelimprove(tao, mfqP, mfqP->n - mfqP->nmodelpoints));
    844:       }
    845:     }
    846:     for (i = mfqP->nmodelpoints; i > 0; i--) mfqP->model_indices[i] = mfqP->model_indices[i - 1];
    847:     mfqP->nmodelpoints++;
    848:     mfqP->model_indices[0] = mfqP->minindex;
    849:     PetscCall(morepoints(mfqP));
    850:     for (i = 0; i < mfqP->nmodelpoints; i++) {
    851:       PetscCall(VecGetArray(mfqP->Xhist[mfqP->model_indices[i]], &x));
    852:       for (j = 0; j < mfqP->n; j++) mfqP->Disp[i + mfqP->npmax * j] = (x[j] - mfqP->xmin[j]) / deltaold;
    853:       PetscCall(VecRestoreArray(mfqP->Xhist[mfqP->model_indices[i]], &x));
    854:       PetscCall(VecGetArray(mfqP->Fhist[mfqP->model_indices[i]], &f));
    855:       for (j = 0; j < mfqP->m; j++) {
    856:         for (k = 0; k < mfqP->n; k++) {
    857:           mfqP->work[k] = 0.0;
    858:           for (l = 0; l < mfqP->n; l++) mfqP->work[k] += mfqP->H[j + mfqP->m * (k + mfqP->n * l)] * mfqP->Disp[i + mfqP->npmax * l];
    859:         }
    860:         PetscCallBLAS("BLASdot", mfqP->RES[j * mfqP->npmax + i] = -mfqP->C[j] - BLASdot_(&blasn, &mfqP->Fdiff[j * mfqP->n], &ione, &mfqP->Disp[i], &blasnpmax) - 0.5 * BLASdot_(&blasn, mfqP->work, &ione, &mfqP->Disp[i], &blasnpmax) + f[j]);
    861:       }
    862:       PetscCall(VecRestoreArray(mfqP->Fhist[mfqP->model_indices[i]], &f));
    863:     }

    865:     /* Update the quadratic model */
    866:     PetscCall(PetscInfo(tao, "Get Quad, size: %" PetscInt_FMT ", points: %" PetscInt_FMT "\n", mfqP->n, mfqP->nmodelpoints));
    867:     PetscCall(getquadpounders(mfqP));
    868:     PetscCall(VecGetArrayRead(mfqP->Fhist[mfqP->minindex], &fmin));
    869:     PetscCallBLAS("BLAScopy", BLAScopy_(&blasm, fmin, &ione, mfqP->C, &ione));
    870:     /* G = G*(delta/deltaold) + Gdel */
    871:     ratio = mfqP->delta / deltaold;
    872:     iblas = blasm * blasn;
    873:     PetscCallBLAS("BLASscal", BLASscal_(&iblas, &ratio, mfqP->Fdiff, &ione));
    874:     PetscCallBLAS("BLASaxpy", BLASaxpy_(&iblas, &one, mfqP->Gdel, &ione, mfqP->Fdiff, &ione));
    875:     /* H = H*(delta/deltaold)^2 + Hdel */
    876:     iblas = blasm * blasn * blasn;
    877:     ratio *= ratio;
    878:     PetscCallBLAS("BLASscal", BLASscal_(&iblas, &ratio, mfqP->H, &ione));
    879:     PetscCallBLAS("BLASaxpy", BLASaxpy_(&iblas, &one, mfqP->Hdel, &ione, mfqP->H, &ione));

    881:     /* Get residuals */
    882:     PetscCall(pounders_update_res(tao));

    884:     /* Export solution and gradient residual to TAO */
    885:     PetscCall(VecCopy(mfqP->Xhist[mfqP->minindex], tao->solution));
    886:     PetscCall(VecSetValues(tao->gradient, mfqP->n, mfqP->indices, mfqP->Gres, INSERT_VALUES));
    887:     PetscCall(VecAssemblyBegin(tao->gradient));
    888:     PetscCall(VecAssemblyEnd(tao->gradient));
    889:     PetscCall(VecNorm(tao->gradient, NORM_2, &gnorm));
    890:     gnorm *= mfqP->delta;
    891:     /*  final criticality test */
    892:     PetscCall(TaoLogConvergenceHistory(tao, minnorm, gnorm, 0.0, tao->ksp_its));
    893:     PetscCall(TaoMonitor(tao, tao->niter, minnorm, gnorm, 0.0, step));
    894:     PetscUseTypeMethod(tao, convergencetest, tao->cnvP);
    895:     /* test for repeated model */
    896:     if (mfqP->nmodelpoints == mfqP->last_nmodelpoints) {
    897:       same = PETSC_TRUE;
    898:     } else {
    899:       same = PETSC_FALSE;
    900:     }
    901:     for (i = 0; i < mfqP->nmodelpoints; i++) {
    902:       if (same) {
    903:         if (mfqP->model_indices[i] == mfqP->last_model_indices[i]) {
    904:           same = PETSC_TRUE;
    905:         } else {
    906:           same = PETSC_FALSE;
    907:         }
    908:       }
    909:       mfqP->last_model_indices[i] = mfqP->model_indices[i];
    910:     }
    911:     mfqP->last_nmodelpoints = mfqP->nmodelpoints;
    912:     if (same && mfqP->delta == deltaold) {
    913:       PetscCall(PetscInfo(tao, "Identical model used in successive iterations\n"));
    914:       tao->reason = TAO_CONVERGED_STEPTOL;
    915:     }
    916:   }
    917:   PetscFunctionReturn(PETSC_SUCCESS);
    918: }

    920: static PetscErrorCode TaoSetUp_POUNDERS(Tao tao)
    921: {
    922:   TAO_POUNDERS *mfqP = (TAO_POUNDERS *)tao->data;
    923:   PetscInt      i, j;
    924:   IS            isfloc, isfglob, isxloc, isxglob;

    926:   PetscFunctionBegin;
    927:   if (!tao->gradient) PetscCall(VecDuplicate(tao->solution, &tao->gradient));
    928:   if (!tao->stepdirection) PetscCall(VecDuplicate(tao->solution, &tao->stepdirection));
    929:   PetscCall(VecGetSize(tao->solution, &mfqP->n));
    930:   PetscCall(VecGetSize(tao->ls_res, &mfqP->m));
    931:   mfqP->c1 = PetscSqrtReal((PetscReal)mfqP->n);
    932:   if (mfqP->npmax == PETSC_CURRENT) mfqP->npmax = 2 * mfqP->n + 1;
    933:   mfqP->npmax = PetscMin((mfqP->n + 1) * (mfqP->n + 2) / 2, mfqP->npmax);
    934:   mfqP->npmax = PetscMax(mfqP->npmax, mfqP->n + 2);

    936:   PetscCall(PetscMalloc1(tao->max_funcs + 100, &mfqP->Xhist));
    937:   PetscCall(PetscMalloc1(tao->max_funcs + 100, &mfqP->Fhist));
    938:   for (i = 0; i < mfqP->n + 1; i++) {
    939:     PetscCall(VecDuplicate(tao->solution, &mfqP->Xhist[i]));
    940:     PetscCall(VecDuplicate(tao->ls_res, &mfqP->Fhist[i]));
    941:   }
    942:   PetscCall(VecDuplicate(tao->solution, &mfqP->workxvec));
    943:   PetscCall(VecDuplicate(tao->ls_res, &mfqP->workfvec));
    944:   mfqP->nHist = 0;

    946:   PetscCall(PetscMalloc1(tao->max_funcs + 100, &mfqP->Fres));
    947:   PetscCall(PetscMalloc1(mfqP->npmax * mfqP->m, &mfqP->RES));
    948:   PetscCall(PetscMalloc1(mfqP->n, &mfqP->work));
    949:   PetscCall(PetscMalloc1(mfqP->n, &mfqP->work2));
    950:   PetscCall(PetscMalloc1(mfqP->n, &mfqP->work3));
    951:   PetscCall(PetscMalloc1(PetscMax(mfqP->m, mfqP->n + 1), &mfqP->mwork));
    952:   PetscCall(PetscMalloc1(mfqP->npmax - mfqP->n - 1, &mfqP->omega));
    953:   PetscCall(PetscMalloc1(mfqP->n * (mfqP->n + 1) / 2, &mfqP->beta));
    954:   PetscCall(PetscMalloc1(mfqP->n + 1, &mfqP->alpha));

    956:   PetscCall(PetscMalloc1(mfqP->n * mfqP->n * mfqP->m, &mfqP->H));
    957:   PetscCall(PetscMalloc1(mfqP->npmax * mfqP->npmax, &mfqP->Q));
    958:   PetscCall(PetscMalloc1(mfqP->npmax * mfqP->npmax, &mfqP->Q_tmp));
    959:   PetscCall(PetscMalloc1(mfqP->n * (mfqP->n + 1) / 2 * (mfqP->npmax), &mfqP->L));
    960:   PetscCall(PetscMalloc1(mfqP->n * (mfqP->n + 1) / 2 * (mfqP->npmax), &mfqP->L_tmp));
    961:   PetscCall(PetscMalloc1(mfqP->n * (mfqP->n + 1) / 2 * (mfqP->npmax), &mfqP->L_save));
    962:   PetscCall(PetscMalloc1(mfqP->n * (mfqP->n + 1) / 2 * (mfqP->npmax), &mfqP->N));
    963:   PetscCall(PetscMalloc1(mfqP->npmax * (mfqP->n + 1), &mfqP->M));
    964:   PetscCall(PetscMalloc1(mfqP->npmax * (mfqP->npmax - mfqP->n - 1), &mfqP->Z));
    965:   PetscCall(PetscMalloc1(mfqP->npmax, &mfqP->tau));
    966:   PetscCall(PetscMalloc1(mfqP->npmax, &mfqP->tau_tmp));
    967:   mfqP->nmax = PetscMax(5 * mfqP->npmax, mfqP->n * (mfqP->n + 1) / 2);
    968:   PetscCall(PetscMalloc1(mfqP->nmax, &mfqP->npmaxwork));
    969:   PetscCall(PetscMalloc1(mfqP->nmax, &mfqP->npmaxiwork));
    970:   PetscCall(PetscMalloc1(mfqP->n, &mfqP->xmin));
    971:   PetscCall(PetscMalloc1(mfqP->m, &mfqP->C));
    972:   PetscCall(PetscMalloc1(mfqP->m * mfqP->n, &mfqP->Fdiff));
    973:   PetscCall(PetscMalloc1(mfqP->npmax * mfqP->n, &mfqP->Disp));
    974:   PetscCall(PetscMalloc1(mfqP->n, &mfqP->Gres));
    975:   PetscCall(PetscMalloc1(mfqP->n * mfqP->n, &mfqP->Hres));
    976:   PetscCall(PetscMalloc1(mfqP->n * mfqP->n, &mfqP->Gpoints));
    977:   PetscCall(PetscMalloc1(mfqP->npmax, &mfqP->model_indices));
    978:   PetscCall(PetscMalloc1(mfqP->npmax, &mfqP->last_model_indices));
    979:   PetscCall(PetscMalloc1(mfqP->n, &mfqP->Xsubproblem));
    980:   PetscCall(PetscMalloc1(mfqP->m * mfqP->n, &mfqP->Gdel));
    981:   PetscCall(PetscMalloc1(mfqP->n * mfqP->n * mfqP->m, &mfqP->Hdel));
    982:   PetscCall(PetscMalloc1(PetscMax(mfqP->m, mfqP->n), &mfqP->indices));
    983:   PetscCall(PetscMalloc1(mfqP->n, &mfqP->iwork));
    984:   PetscCall(PetscMalloc1(mfqP->m * mfqP->m, &mfqP->w));
    985:   for (i = 0; i < mfqP->m; i++) {
    986:     for (j = 0; j < mfqP->m; j++) {
    987:       if (i == j) {
    988:         mfqP->w[i + mfqP->m * j] = 1.0;
    989:       } else {
    990:         mfqP->w[i + mfqP->m * j] = 0.0;
    991:       }
    992:     }
    993:   }
    994:   for (i = 0; i < PetscMax(mfqP->m, mfqP->n); i++) mfqP->indices[i] = i;
    995:   PetscCallMPI(MPI_Comm_size(((PetscObject)tao)->comm, &mfqP->size));
    996:   if (mfqP->size > 1) {
    997:     PetscCall(VecCreateSeq(PETSC_COMM_SELF, mfqP->n, &mfqP->localx));
    998:     PetscCall(VecCreateSeq(PETSC_COMM_SELF, mfqP->n, &mfqP->localxmin));
    999:     PetscCall(VecCreateSeq(PETSC_COMM_SELF, mfqP->m, &mfqP->localf));
    1000:     PetscCall(VecCreateSeq(PETSC_COMM_SELF, mfqP->m, &mfqP->localfmin));
    1001:     PetscCall(ISCreateStride(MPI_COMM_SELF, mfqP->n, 0, 1, &isxloc));
    1002:     PetscCall(ISCreateStride(MPI_COMM_SELF, mfqP->n, 0, 1, &isxglob));
    1003:     PetscCall(ISCreateStride(MPI_COMM_SELF, mfqP->m, 0, 1, &isfloc));
    1004:     PetscCall(ISCreateStride(MPI_COMM_SELF, mfqP->m, 0, 1, &isfglob));

    1006:     PetscCall(VecScatterCreate(tao->solution, isxglob, mfqP->localx, isxloc, &mfqP->scatterx));
    1007:     PetscCall(VecScatterCreate(tao->ls_res, isfglob, mfqP->localf, isfloc, &mfqP->scatterf));

    1009:     PetscCall(ISDestroy(&isxloc));
    1010:     PetscCall(ISDestroy(&isxglob));
    1011:     PetscCall(ISDestroy(&isfloc));
    1012:     PetscCall(ISDestroy(&isfglob));
    1013:   }

    1015:   if (!mfqP->usegqt) {
    1016:     KSP ksp;
    1017:     PC  pc;
    1018:     PetscCall(VecCreateSeqWithArray(PETSC_COMM_SELF, mfqP->n, mfqP->n, mfqP->Xsubproblem, &mfqP->subx));
    1019:     PetscCall(VecCreateSeq(PETSC_COMM_SELF, mfqP->n, &mfqP->subxl));
    1020:     PetscCall(VecDuplicate(mfqP->subxl, &mfqP->subb));
    1021:     PetscCall(VecDuplicate(mfqP->subxl, &mfqP->subxu));
    1022:     PetscCall(VecDuplicate(mfqP->subxl, &mfqP->subpdel));
    1023:     PetscCall(VecDuplicate(mfqP->subxl, &mfqP->subndel));
    1024:     PetscCall(TaoCreate(PETSC_COMM_SELF, &mfqP->subtao));
    1025:     PetscCall(PetscObjectIncrementTabLevel((PetscObject)mfqP->subtao, (PetscObject)tao, 1));
    1026:     PetscCall(TaoSetType(mfqP->subtao, TAOBNTR));
    1027:     PetscCall(TaoSetOptionsPrefix(mfqP->subtao, "pounders_subsolver_"));
    1028:     PetscCall(TaoSetSolution(mfqP->subtao, mfqP->subx));
    1029:     PetscCall(TaoSetObjectiveAndGradient(mfqP->subtao, NULL, pounders_fg, (void *)mfqP));
    1030:     PetscCall(TaoSetMaximumIterations(mfqP->subtao, mfqP->gqt_maxits));
    1031:     PetscCall(TaoSetFromOptions(mfqP->subtao));
    1032:     PetscCall(TaoGetKSP(mfqP->subtao, &ksp));
    1033:     if (ksp) {
    1034:       PetscCall(KSPGetPC(ksp, &pc));
    1035:       PetscCall(PCSetType(pc, PCNONE));
    1036:     }
    1037:     PetscCall(TaoSetVariableBounds(mfqP->subtao, mfqP->subxl, mfqP->subxu));
    1038:     PetscCall(MatCreateSeqDense(PETSC_COMM_SELF, mfqP->n, mfqP->n, mfqP->Hres, &mfqP->subH));
    1039:     PetscCall(TaoSetHessian(mfqP->subtao, mfqP->subH, mfqP->subH, pounders_h, (void *)mfqP));
    1040:   }
    1041:   PetscFunctionReturn(PETSC_SUCCESS);
    1042: }

    1044: static PetscErrorCode TaoDestroy_POUNDERS(Tao tao)
    1045: {
    1046:   TAO_POUNDERS *mfqP = (TAO_POUNDERS *)tao->data;
    1047:   PetscInt      i;

    1049:   PetscFunctionBegin;
    1050:   if (!mfqP->usegqt) {
    1051:     PetscCall(TaoDestroy(&mfqP->subtao));
    1052:     PetscCall(VecDestroy(&mfqP->subx));
    1053:     PetscCall(VecDestroy(&mfqP->subxl));
    1054:     PetscCall(VecDestroy(&mfqP->subxu));
    1055:     PetscCall(VecDestroy(&mfqP->subb));
    1056:     PetscCall(VecDestroy(&mfqP->subpdel));
    1057:     PetscCall(VecDestroy(&mfqP->subndel));
    1058:     PetscCall(MatDestroy(&mfqP->subH));
    1059:   }
    1060:   PetscCall(PetscFree(mfqP->Fres));
    1061:   PetscCall(PetscFree(mfqP->RES));
    1062:   PetscCall(PetscFree(mfqP->work));
    1063:   PetscCall(PetscFree(mfqP->work2));
    1064:   PetscCall(PetscFree(mfqP->work3));
    1065:   PetscCall(PetscFree(mfqP->mwork));
    1066:   PetscCall(PetscFree(mfqP->omega));
    1067:   PetscCall(PetscFree(mfqP->beta));
    1068:   PetscCall(PetscFree(mfqP->alpha));
    1069:   PetscCall(PetscFree(mfqP->H));
    1070:   PetscCall(PetscFree(mfqP->Q));
    1071:   PetscCall(PetscFree(mfqP->Q_tmp));
    1072:   PetscCall(PetscFree(mfqP->L));
    1073:   PetscCall(PetscFree(mfqP->L_tmp));
    1074:   PetscCall(PetscFree(mfqP->L_save));
    1075:   PetscCall(PetscFree(mfqP->N));
    1076:   PetscCall(PetscFree(mfqP->M));
    1077:   PetscCall(PetscFree(mfqP->Z));
    1078:   PetscCall(PetscFree(mfqP->tau));
    1079:   PetscCall(PetscFree(mfqP->tau_tmp));
    1080:   PetscCall(PetscFree(mfqP->npmaxwork));
    1081:   PetscCall(PetscFree(mfqP->npmaxiwork));
    1082:   PetscCall(PetscFree(mfqP->xmin));
    1083:   PetscCall(PetscFree(mfqP->C));
    1084:   PetscCall(PetscFree(mfqP->Fdiff));
    1085:   PetscCall(PetscFree(mfqP->Disp));
    1086:   PetscCall(PetscFree(mfqP->Gres));
    1087:   PetscCall(PetscFree(mfqP->Hres));
    1088:   PetscCall(PetscFree(mfqP->Gpoints));
    1089:   PetscCall(PetscFree(mfqP->model_indices));
    1090:   PetscCall(PetscFree(mfqP->last_model_indices));
    1091:   PetscCall(PetscFree(mfqP->Xsubproblem));
    1092:   PetscCall(PetscFree(mfqP->Gdel));
    1093:   PetscCall(PetscFree(mfqP->Hdel));
    1094:   PetscCall(PetscFree(mfqP->indices));
    1095:   PetscCall(PetscFree(mfqP->iwork));
    1096:   PetscCall(PetscFree(mfqP->w));
    1097:   for (i = 0; i < mfqP->nHist; i++) {
    1098:     PetscCall(VecDestroy(&mfqP->Xhist[i]));
    1099:     PetscCall(VecDestroy(&mfqP->Fhist[i]));
    1100:   }
    1101:   PetscCall(VecDestroy(&mfqP->workxvec));
    1102:   PetscCall(VecDestroy(&mfqP->workfvec));
    1103:   PetscCall(PetscFree(mfqP->Xhist));
    1104:   PetscCall(PetscFree(mfqP->Fhist));

    1106:   if (mfqP->size > 1) {
    1107:     PetscCall(VecDestroy(&mfqP->localx));
    1108:     PetscCall(VecDestroy(&mfqP->localxmin));
    1109:     PetscCall(VecDestroy(&mfqP->localf));
    1110:     PetscCall(VecDestroy(&mfqP->localfmin));
    1111:   }
    1112:   PetscCall(PetscFree(tao->data));
    1113:   PetscFunctionReturn(PETSC_SUCCESS);
    1114: }

    1116: static PetscErrorCode TaoSetFromOptions_POUNDERS(Tao tao, PetscOptionItems *PetscOptionsObject)
    1117: {
    1118:   TAO_POUNDERS *mfqP = (TAO_POUNDERS *)tao->data;

    1120:   PetscFunctionBegin;
    1121:   PetscOptionsHeadBegin(PetscOptionsObject, "POUNDERS method for least-squares optimization");
    1122:   PetscCall(PetscOptionsReal("-tao_pounders_delta", "initial delta", "", mfqP->delta, &mfqP->delta0, NULL));
    1123:   mfqP->delta = mfqP->delta0;
    1124:   PetscCall(PetscOptionsInt("-tao_pounders_npmax", "max number of points in model", "", mfqP->npmax, &mfqP->npmax, NULL));
    1125:   PetscCall(PetscOptionsBool("-tao_pounders_gqt", "use gqt algorithm for subproblem", "", mfqP->usegqt, &mfqP->usegqt, NULL));
    1126:   PetscOptionsHeadEnd();
    1127:   PetscFunctionReturn(PETSC_SUCCESS);
    1128: }

    1130: static PetscErrorCode TaoView_POUNDERS(Tao tao, PetscViewer viewer)
    1131: {
    1132:   TAO_POUNDERS *mfqP = (TAO_POUNDERS *)tao->data;
    1133:   PetscBool     isascii;

    1135:   PetscFunctionBegin;
    1136:   PetscCall(PetscObjectTypeCompare((PetscObject)viewer, PETSCVIEWERASCII, &isascii));
    1137:   if (isascii) {
    1138:     PetscCall(PetscViewerASCIIPrintf(viewer, "initial delta: %g\n", (double)mfqP->delta0));
    1139:     PetscCall(PetscViewerASCIIPrintf(viewer, "final delta: %g\n", (double)mfqP->delta));
    1140:     PetscCall(PetscViewerASCIIPrintf(viewer, "model points: %" PetscInt_FMT "\n", mfqP->nmodelpoints));
    1141:     if (mfqP->usegqt) {
    1142:       PetscCall(PetscViewerASCIIPrintf(viewer, "subproblem solver: gqt\n"));
    1143:     } else {
    1144:       PetscCall(TaoView(mfqP->subtao, viewer));
    1145:     }
    1146:   }
    1147:   PetscFunctionReturn(PETSC_SUCCESS);
    1148: }
    1149: /*MC
    1150:   TAOPOUNDERS - POUNDERS derivate-free model-based algorithm for nonlinear least squares

    1152:   Options Database Keys:
    1153: + -tao_pounders_delta - initial step length
    1154: . -tao_pounders_npmax - maximum number of points in model
    1155: - -tao_pounders_gqt - use gqt algorithm for subproblem instead of TRON

    1157:   Level: beginner

    1159: M*/

    1161: PETSC_EXTERN PetscErrorCode TaoCreate_POUNDERS(Tao tao)
    1162: {
    1163:   TAO_POUNDERS *mfqP = (TAO_POUNDERS *)tao->data;

    1165:   PetscFunctionBegin;
    1166:   tao->ops->setup          = TaoSetUp_POUNDERS;
    1167:   tao->ops->solve          = TaoSolve_POUNDERS;
    1168:   tao->ops->view           = TaoView_POUNDERS;
    1169:   tao->ops->setfromoptions = TaoSetFromOptions_POUNDERS;
    1170:   tao->ops->destroy        = TaoDestroy_POUNDERS;

    1172:   PetscCall(PetscNew(&mfqP));
    1173:   tao->data = (void *)mfqP;

    1175:   /* Override default settings (unless already changed) */
    1176:   PetscCall(TaoParametersInitialize(tao));
    1177:   PetscObjectParameterSetDefault(tao, max_it, 2000);
    1178:   PetscObjectParameterSetDefault(tao, max_funcs, 4000);

    1180:   mfqP->npmax      = PETSC_CURRENT;
    1181:   mfqP->delta0     = 0.1;
    1182:   mfqP->delta      = 0.1;
    1183:   mfqP->deltamax   = 1e3;
    1184:   mfqP->deltamin   = 1e-6;
    1185:   mfqP->c2         = 10.0;
    1186:   mfqP->theta1     = 1e-5;
    1187:   mfqP->theta2     = 1e-4;
    1188:   mfqP->gamma0     = 0.5;
    1189:   mfqP->gamma1     = 2.0;
    1190:   mfqP->eta0       = 0.0;
    1191:   mfqP->eta1       = 0.1;
    1192:   mfqP->usegqt     = PETSC_FALSE;
    1193:   mfqP->gqt_rtol   = 0.001;
    1194:   mfqP->gqt_maxits = 50;
    1195:   mfqP->workxvec   = NULL;
    1196:   PetscFunctionReturn(PETSC_SUCCESS);
    1197: }