Actual source code: pounders.c

petsc-3.8.4 2018-03-24
Report Typos and Errors
  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: {
  6:   return(0);
  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;

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

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

 24:   /* now  g = g + b */
 25:   VecAXPY(g, 1.0, mfqP->subb);
 26:   return(0);
 27: }

 29: static PetscErrorCode pounders_feval(Tao tao, Vec x, Vec F, PetscReal *fsum)
 30: {
 32:   TAO_POUNDERS   *mfqP = (TAO_POUNDERS*)tao->data;
 33:   PetscInt i,row,col;
 34:   PetscReal fr,fc;
 36:   TaoComputeSeparableObjective(tao,x,F);
 37:   if (tao->sep_weights_v) {
 38:     VecPointwiseMult(mfqP->workfvec,tao->sep_weights_v,F);
 39:     VecNorm(mfqP->workfvec,NORM_2,fsum);
 40:     *fsum*=(*fsum);
 41:   } else if (tao->sep_weights_w) {
 42:     *fsum=0;
 43:     for (i=0;i<tao->sep_weights_n;i++) {
 44:       row=tao->sep_weights_rows[i];
 45:       col=tao->sep_weights_cols[i];
 46:       VecGetValues(F,1,&row,&fr);
 47:       VecGetValues(F,1,&col,&fc);
 48:       *fsum += tao->sep_weights_w[i]*fc*fr;
 49:     }
 50:   } else {
 51:     VecNorm(F,NORM_2,fsum);
 52:     *fsum*=(*fsum);
 53:   }
 54:   if (PetscIsInfOrNanReal(*fsum)) SETERRQ(PETSC_COMM_SELF,1, "User provided compute function generated Inf or NaN");
 55:   return(0);
 56: }

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

 70:   if (!mfqP->usegqt) {
 71:     PetscReal maxval;
 72:     PetscInt  i,j;

 74:     VecSetValues(mfqP->subb,mfqP->n,mfqP->indices,mfqP->Gres,INSERT_VALUES);
 75:     VecAssemblyBegin(mfqP->subb);
 76:     VecAssemblyEnd(mfqP->subb);

 78:     VecSet(mfqP->subx,0.0);

 80:     VecSet(mfqP->subndel,-mfqP->delta);
 81:     VecSet(mfqP->subpdel,mfqP->delta);

 83:     for (i=0;i<mfqP->n;i++) {
 84:       for (j=i;j<mfqP->n;j++) {
 85:         mfqP->Hres[j+mfqP->n*i] = mfqP->Hres[mfqP->n*j+i];
 86:       }
 87:     }
 88:     MatSetValues(mfqP->subH,mfqP->n,mfqP->indices,mfqP->n,mfqP->indices,mfqP->Hres,INSERT_VALUES);
 89:     MatAssemblyBegin(mfqP->subH,MAT_FINAL_ASSEMBLY);
 90:     MatAssemblyEnd(mfqP->subH,MAT_FINAL_ASSEMBLY);

 92:     TaoResetStatistics(mfqP->subtao);
 93:     /* TaoSetTolerances(mfqP->subtao,*gnorm,*gnorm,PETSC_DEFAULT); */
 94:     /* enforce bound constraints -- experimental */
 95:     if (tao->XU && tao->XL) {
 96:       VecCopy(tao->XU,mfqP->subxu);
 97:       VecAXPY(mfqP->subxu,-1.0,tao->solution);
 98:       VecScale(mfqP->subxu,1.0/mfqP->delta);
 99:       VecCopy(tao->XL,mfqP->subxl);
100:       VecAXPY(mfqP->subxl,-1.0,tao->solution);
101:       VecScale(mfqP->subxl,1.0/mfqP->delta);

103:       VecPointwiseMin(mfqP->subxu,mfqP->subxu,mfqP->subpdel);
104:       VecPointwiseMax(mfqP->subxl,mfqP->subxl,mfqP->subndel);
105:     } else {
106:       VecCopy(mfqP->subpdel,mfqP->subxu);
107:       VecCopy(mfqP->subndel,mfqP->subxl);
108:     }
109:     /* Make sure xu > xl */
110:     VecCopy(mfqP->subxl,mfqP->subpdel);
111:     VecAXPY(mfqP->subpdel,-1.0,mfqP->subxu);
112:     VecMax(mfqP->subpdel,NULL,&maxval);
113:     if (maxval > 1e-10) SETERRQ(PETSC_COMM_WORLD,1,"upper bound < lower bound in subproblem");
114:     /* Make sure xu > tao->solution > xl */
115:     VecCopy(mfqP->subxl,mfqP->subpdel);
116:     VecAXPY(mfqP->subpdel,-1.0,mfqP->subx);
117:     VecMax(mfqP->subpdel,NULL,&maxval);
118:     if (maxval > 1e-10) SETERRQ(PETSC_COMM_WORLD,1,"initial guess < lower bound in subproblem");

120:     VecCopy(mfqP->subx,mfqP->subpdel);
121:     VecAXPY(mfqP->subpdel,-1.0,mfqP->subxu);
122:     VecMax(mfqP->subpdel,NULL,&maxval);
123:     if (maxval > 1e-10) SETERRQ(PETSC_COMM_WORLD,1,"initial guess > upper bound in subproblem");

125:     TaoSolve(mfqP->subtao);
126:     TaoGetSolutionStatus(mfqP->subtao,NULL,qmin,NULL,NULL,NULL,NULL);

128:     /* test bounds post-solution*/
129:     VecCopy(mfqP->subxl,mfqP->subpdel);
130:     VecAXPY(mfqP->subpdel,-1.0,mfqP->subx);
131:     VecMax(mfqP->subpdel,NULL,&maxval);
132:     if (maxval > 1e-5) {
133:       PetscInfo(tao,"subproblem solution < lower bound\n");
134:       tao->reason = TAO_DIVERGED_TR_REDUCTION;
135:     }

137:     VecCopy(mfqP->subx,mfqP->subpdel);
138:     VecAXPY(mfqP->subpdel,-1.0,mfqP->subxu);
139:     VecMax(mfqP->subpdel,NULL,&maxval);
140:     if (maxval > 1e-5) {
141:       PetscInfo(tao,"subproblem solution > upper bound\n");
142:       tao->reason = TAO_DIVERGED_TR_REDUCTION;
143:     }
144:   } else {
145:     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);
146:   }
147:   *qmin *= -1;
148:   return(0);
149: }

151: static PetscErrorCode pounders_update_res(Tao tao)
152: {
153:   TAO_POUNDERS   *mfqP = (TAO_POUNDERS*)tao->data;
154:   PetscInt i,row,col;
155:   PetscBLASInt blasn=mfqP->n,blasn2=blasn*blasn,blasm=mfqP->m,ione=1;
156:   PetscReal zero=0.0,one=1.0,wii,factor;

160:   for (i=0;i<mfqP->n;i++) {
161:     mfqP->Gres[i]=0;
162:   }
163:   for (i=0;i<mfqP->n*mfqP->n;i++) {
164:     mfqP->Hres[i]=0;
165:   }

167:   /* Compute Gres= sum_ij[wij * (cjgi + cigj)] */
168:   if (tao->sep_weights_v) {
169:     /* Vector(diagonal) weights: gres = sum_i(wii*ci*gi) */
170:     for (i=0;i<mfqP->m;i++) {
171:       VecGetValues(tao->sep_weights_v,1,&i,&factor);
172:       factor=factor*mfqP->C[i];
173:       PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&blasn,&factor,&mfqP->Fdiff[blasn*i],&ione,mfqP->Gres,&ione));
174:     }
175:   } else if (tao->sep_weights_n) {
176:     /* general case: .5 * Gres= sum_ij[wij * (cjgi + cigj)] */
177:     for (i=0;i<tao->sep_weights_n;i++) {
178:       row=tao->sep_weights_rows[i];
179:       col=tao->sep_weights_cols[i];

181:       factor = tao->sep_weights_w[i]*mfqP->C[col]/2.0;
182:       PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&blasn,&factor,&mfqP->Fdiff[blasn*row],&ione,mfqP->Gres,&ione));
183:       factor = tao->sep_weights_w[i]*mfqP->C[row]/2.0;
184:       PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&blasn,&factor,&mfqP->Fdiff[blasn*col],&ione,mfqP->Gres,&ione));
185:     }
186:   } else {
187:     /* default: Gres= sum_i[cigi] = G*c' */
188:     PetscStackCallBLAS("BLASgemv",BLASgemv_("N",&blasn,&blasm,&one,mfqP->Fdiff,&blasn,mfqP->C,&ione,&zero,mfqP->Gres,&ione));
189:   }

191:   /* compute Hres = sum_ij [wij * (*ci*Hj + cj*Hi + gi gj' + gj gi') ] */
192:   if (tao->sep_weights_v) {
193:     /* vector(diagonal weights) Hres = sum_i(wii*(ci*Hi + gi * gi' )*/
194:     for (i=0;i<mfqP->m;i++) {
195:       VecGetValues(tao->sep_weights_v,1,&i,&wii);
196:       if (tao->niter>1) {
197:         factor=wii*mfqP->C[i];
198:         /* add wii * ci * Hi */
199:         PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&blasn2,&factor,&mfqP->H[i],&blasn2,mfqP->Hres,&ione));
200:       }
201:       /* add wii * gi * gi' */
202:       PetscStackCallBLAS("BLASgemm_",BLASgemm_("N","T",&blasn,&blasn,&ione,&wii,&mfqP->Fdiff[blasn*i],&blasn,&mfqP->Fdiff[blasn*i],&blasn,&one,mfqP->Hres,&blasn));
203:     }
204:   } else if (tao->sep_weights_w) {
205:     /* .5 * sum_ij [wij * (*ci*Hj + cj*Hi + gi gj' + gj gi') ] */
206:     for (i=0;i<tao->sep_weights_n;i++) {
207:       row=tao->sep_weights_rows[i];
208:       col=tao->sep_weights_cols[i];
209:       factor=tao->sep_weights_w[i]/2.0;
210:       /* add wij * gi gj' + wij * gj gi' */
211:       PetscStackCallBLAS("BLASgemm_",BLASgemm_("N","T",&blasn,&blasn,&ione,&factor,&mfqP->Fdiff[blasn*row],&blasn,&mfqP->Fdiff[blasn*col],&blasn,&one,mfqP->Hres,&blasn));
212:       PetscStackCallBLAS("BLASgemm_",BLASgemm_("N","T",&blasn,&blasn,&ione,&factor,&mfqP->Fdiff[blasn*col],&blasn,&mfqP->Fdiff[blasn*row],&blasn,&one,mfqP->Hres,&blasn));
213:     }
214:     if (tao->niter > 1) {
215:       for (i=0;i<tao->sep_weights_n;i++) {
216:         row=tao->sep_weights_rows[i];
217:         col=tao->sep_weights_cols[i];

219:         /* add  wij*cj*Hi */
220:         factor = tao->sep_weights_w[i]*mfqP->C[col]/2.0;
221:         PetscStackCallBLAS("BLASaxpy_",BLASaxpy_(&blasn2,&factor,&mfqP->H[row],&blasn2,mfqP->Hres,&ione));

223:         /* add wij*ci*Hj */
224:         factor = tao->sep_weights_w[i]*mfqP->C[row]/2.0;
225:         PetscStackCallBLAS("BLASaxpy_",BLASaxpy_(&blasn2,&factor,&mfqP->H[col],&blasn2,mfqP->Hres,&ione));
226:       }
227:     }
228:   } else {
229:     /*  Hres = G*G' + 0.5 sum {F(xkin,i)*H(:,:,i)}  */
230:     PetscStackCallBLAS("BLASgemm",BLASgemm_("N","T",&blasn,&blasn,&blasm,&one,mfqP->Fdiff, &blasn,mfqP->Fdiff, &blasn,&zero,mfqP->Hres,&blasn));

232:     /* sum(F(xkin,i)*H(:,:,i)) */
233:     if (tao->niter>1) {
234:       for (i=0;i<mfqP->m;i++) {
235:         factor = mfqP->C[i];
236:         PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&blasn2,&factor,&mfqP->H[i],&blasn2,mfqP->Hres,&ione));
237:       }
238:     }
239:   }
240:   return(0);
241: }

243: PetscErrorCode phi2eval(PetscReal *x, PetscInt n, PetscReal *phi)
244: {
245: /* 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] */
246:   PetscInt  i,j,k;
247:   PetscReal sqrt2 = PetscSqrtReal(2.0);

250:   j=0;
251:   for (i=0;i<n;i++) {
252:     phi[j] = 0.5 * x[i]*x[i];
253:     j++;
254:     for (k=i+1;k<n;k++) {
255:       phi[j]  = x[i]*x[k]/sqrt2;
256:       j++;
257:     }
258:   }
259:   return(0);
260: }

262: PetscErrorCode getquadpounders(TAO_POUNDERS *mfqP)
263: {
264: /* Computes the parameters of the quadratic Q(x) = c + g'*x + 0.5*x*G*x'
265:    that satisfies the interpolation conditions Q(X[:,j]) = f(j)
266:    for j=1,...,m and with a Hessian matrix of least Frobenius norm */

268:     /* NB --we are ignoring c */
269:   PetscInt     i,j,k,num,np = mfqP->nmodelpoints;
270:   PetscReal    one = 1.0,zero=0.0,negone=-1.0;
271:   PetscBLASInt blasnpmax = mfqP->npmax;
272:   PetscBLASInt blasnplus1 = mfqP->n+1;
273:   PetscBLASInt blasnp = np;
274:   PetscBLASInt blasint = mfqP->n*(mfqP->n+1) / 2;
275:   PetscBLASInt blasint2 = np - mfqP->n-1;
276:   PetscBLASInt info,ione=1;
277:   PetscReal    sqrt2 = PetscSqrtReal(2.0);

280:   for (i=0;i<mfqP->n*mfqP->m;i++) {
281:     mfqP->Gdel[i] = 0;
282:   }
283:   for (i=0;i<mfqP->n*mfqP->n*mfqP->m;i++) {
284:     mfqP->Hdel[i] = 0;
285:   }

287:     /* factor M */
288:   PetscStackCallBLAS("LAPACKgetrf",LAPACKgetrf_(&blasnplus1,&blasnp,mfqP->M,&blasnplus1,mfqP->npmaxiwork,&info));
289:   if (info != 0) SETERRQ1(PETSC_COMM_SELF,1,"LAPACK routine getrf returned with value %d\n",info);

291:   if (np == mfqP->n+1) {
292:     for (i=0;i<mfqP->npmax-mfqP->n-1;i++) {
293:       mfqP->omega[i]=0.0;
294:     }
295:     for (i=0;i<mfqP->n*(mfqP->n+1)/2;i++) {
296:       mfqP->beta[i]=0.0;
297:     }
298:   } else {
299:     /* Let Ltmp = (L'*L) */
300:     PetscStackCallBLAS("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));

302:     /* factor Ltmp */
303:     PetscStackCallBLAS("LAPACKpotrf",LAPACKpotrf_("L",&blasint2,mfqP->L_tmp,&blasint,&info));
304:     if (info != 0) SETERRQ1(PETSC_COMM_SELF,1,"LAPACK routine potrf returned with value %d\n",info);
305:   }

307:   for (k=0;k<mfqP->m;k++) {
308:     if (np != mfqP->n+1) {
309:       /* Solve L'*L*Omega = Z' * RESk*/
310:       PetscStackCallBLAS("BLASgemv",BLASgemv_("T",&blasnp,&blasint2,&one,mfqP->Z,&blasnpmax,&mfqP->RES[mfqP->npmax*k],&ione,&zero,mfqP->omega,&ione));
311:       PetscStackCallBLAS("LAPACKpotrs",LAPACKpotrs_("L",&blasint2,&ione,mfqP->L_tmp,&blasint,mfqP->omega,&blasint2,&info));
312:       if (info != 0) SETERRQ1(PETSC_COMM_SELF,1,"LAPACK routine potrs returned with value %d\n",info);

314:       /* Beta = L*Omega */
315:       PetscStackCallBLAS("BLASgemv",BLASgemv_("N",&blasint,&blasint2,&one,&mfqP->L[(mfqP->n+1)*blasint],&blasint,mfqP->omega,&ione,&zero,mfqP->beta,&ione));
316:     }

318:     /* solve M'*Alpha = RESk - N'*Beta */
319:     PetscStackCallBLAS("BLASgemv",BLASgemv_("T",&blasint,&blasnp,&negone,mfqP->N,&blasint,mfqP->beta,&ione,&one,&mfqP->RES[mfqP->npmax*k],&ione));
320:     PetscStackCallBLAS("LAPACKgetrs",LAPACKgetrs_("T",&blasnplus1,&ione,mfqP->M,&blasnplus1,mfqP->npmaxiwork,&mfqP->RES[mfqP->npmax*k],&blasnplus1,&info));
321:     if (info != 0) SETERRQ1(PETSC_COMM_SELF,1,"LAPACK routine getrs returned with value %d\n",info);

323:     /* Gdel(:,k) = Alpha(2:n+1) */
324:     for (i=0;i<mfqP->n;i++) {
325:       mfqP->Gdel[i + mfqP->n*k] = mfqP->RES[mfqP->npmax*k + i+1];
326:     }

328:     /* Set Hdels */
329:     num=0;
330:     for (i=0;i<mfqP->n;i++) {
331:       /* H[i,i,k] = Beta(num) */
332:       mfqP->Hdel[(i*mfqP->n + i)*mfqP->m + k] = mfqP->beta[num];
333:       num++;
334:       for (j=i+1;j<mfqP->n;j++) {
335:         /* H[i,j,k] = H[j,i,k] = Beta(num)/sqrt(2) */
336:         mfqP->Hdel[(j*mfqP->n + i)*mfqP->m + k] = mfqP->beta[num]/sqrt2;
337:         mfqP->Hdel[(i*mfqP->n + j)*mfqP->m + k] = mfqP->beta[num]/sqrt2;
338:         num++;
339:       }
340:     }
341:   }
342:   return(0);
343: }

345: PetscErrorCode morepoints(TAO_POUNDERS *mfqP)
346: {
347:   /* Assumes mfqP->model_indices[0]  is minimum index
348:    Finishes adding points to mfqP->model_indices (up to npmax)
349:    Computes L,Z,M,N
350:    np is actual number of points in model (should equal npmax?) */
351:   PetscInt        point,i,j,offset;
352:   PetscInt        reject;
353:   PetscBLASInt    blasn=mfqP->n,blasnpmax=mfqP->npmax,blasnplus1=mfqP->n+1,info,blasnmax=mfqP->nmax,blasint,blasint2,blasnp,blasmaxmn;
354:   const PetscReal *x;
355:   PetscReal       normd;
356:   PetscErrorCode  ierr;

359:   /* Initialize M,N */
360:   for (i=0;i<mfqP->n+1;i++) {
361:     VecGetArrayRead(mfqP->Xhist[mfqP->model_indices[i]],&x);
362:     mfqP->M[(mfqP->n+1)*i] = 1.0;
363:     for (j=0;j<mfqP->n;j++) {
364:       mfqP->M[j+1+((mfqP->n+1)*i)] = (x[j]  - mfqP->xmin[j]) / mfqP->delta;
365:     }
366:     VecRestoreArrayRead(mfqP->Xhist[mfqP->model_indices[i]],&x);
367:     phi2eval(&mfqP->M[1+((mfqP->n+1)*i)],mfqP->n,&mfqP->N[mfqP->n*(mfqP->n+1)/2 * i]);
368:   }

370:   /* Now we add points until we have npmax starting with the most recent ones */
371:   point = mfqP->nHist-1;
372:   mfqP->nmodelpoints = mfqP->n+1;
373:   while (mfqP->nmodelpoints < mfqP->npmax && point>=0) {
374:     /* Reject any points already in the model */
375:     reject = 0;
376:     for (j=0;j<mfqP->n+1;j++) {
377:       if (point == mfqP->model_indices[j]) {
378:         reject = 1;
379:         break;
380:       }
381:     }

383:     /* Reject if norm(d) >c2 */
384:     if (!reject) {
385:       VecCopy(mfqP->Xhist[point],mfqP->workxvec);
386:       VecAXPY(mfqP->workxvec,-1.0,mfqP->Xhist[mfqP->minindex]);
387:       VecNorm(mfqP->workxvec,NORM_2,&normd);
388:       normd /= mfqP->delta;
389:       if (normd > mfqP->c2) {
390:         reject =1;
391:       }
392:     }
393:     if (reject){
394:       point--;
395:       continue;
396:     }

398:     VecGetArrayRead(mfqP->Xhist[point],&x);
399:     mfqP->M[(mfqP->n+1)*mfqP->nmodelpoints] = 1.0;
400:     for (j=0;j<mfqP->n;j++) {
401:       mfqP->M[j+1+((mfqP->n+1)*mfqP->nmodelpoints)] = (x[j]  - mfqP->xmin[j]) / mfqP->delta;
402:     }
403:     VecRestoreArrayRead(mfqP->Xhist[point],&x);
404:     phi2eval(&mfqP->M[1+(mfqP->n+1)*mfqP->nmodelpoints],mfqP->n,&mfqP->N[mfqP->n*(mfqP->n+1)/2 * (mfqP->nmodelpoints)]);

406:     /* Update QR factorization */
407:     /* Copy M' to Q_tmp */
408:     for (i=0;i<mfqP->n+1;i++) {
409:       for (j=0;j<mfqP->npmax;j++) {
410:         mfqP->Q_tmp[j+mfqP->npmax*i] = mfqP->M[i+(mfqP->n+1)*j];
411:       }
412:     }
413:     blasnp = mfqP->nmodelpoints+1;
414:     /* Q_tmp,R = qr(M') */
415:     blasmaxmn=PetscMax(mfqP->m,mfqP->n+1);
416:     PetscStackCallBLAS("LAPACKgeqrf",LAPACKgeqrf_(&blasnp,&blasnplus1,mfqP->Q_tmp,&blasnpmax,mfqP->tau_tmp,mfqP->mwork,&blasmaxmn,&info));
417:     if (info != 0) SETERRQ1(PETSC_COMM_SELF,1,"LAPACK routine geqrf returned with value %d\n",info);

419:     /* Reject if min(svd(N*Q(:,n+2:np+1)) <= theta2 */
420:     /* L = N*Qtmp */
421:     blasint2 = mfqP->n * (mfqP->n+1) / 2;
422:     /* Copy N to L_tmp */
423:     for (i=0;i<mfqP->n*(mfqP->n+1)/2 * mfqP->npmax;i++) {
424:       mfqP->L_tmp[i]= mfqP->N[i];
425:     }
426:     /* Copy L_save to L_tmp */

428:     /* L_tmp = N*Qtmp' */
429:     PetscStackCallBLAS("LAPACKormqr",LAPACKormqr_("R","N",&blasint2,&blasnp,&blasnplus1,mfqP->Q_tmp,&blasnpmax,mfqP->tau_tmp,mfqP->L_tmp,&blasint2,mfqP->npmaxwork,&blasnmax,&info));
430:     if (info != 0) SETERRQ1(PETSC_COMM_SELF,1,"LAPACK routine ormqr returned with value %d\n",info);

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

437:     /* Get svd for L_tmp(:,n+2:np+1) (L_tmp is modified in process) */
438:     blasint = mfqP->nmodelpoints - mfqP->n;
439:     PetscFPTrapPush(PETSC_FP_TRAP_OFF);
440:     PetscStackCallBLAS("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));
441:     PetscFPTrapPop();
442:     if (info != 0) SETERRQ1(PETSC_COMM_SELF,1,"LAPACK routine gesvd returned with value %d\n",info);

444:     if (mfqP->beta[PetscMin(blasint,blasint2)-1] > mfqP->theta2) {
445:       /* accept point */
446:       mfqP->model_indices[mfqP->nmodelpoints] = point;
447:       /* Copy Q_tmp to Q */
448:       for (i=0;i<mfqP->npmax* mfqP->npmax;i++) {
449:         mfqP->Q[i] = mfqP->Q_tmp[i];
450:       }
451:       for (i=0;i<mfqP->npmax;i++){
452:         mfqP->tau[i] = mfqP->tau_tmp[i];
453:       }
454:       mfqP->nmodelpoints++;
455:       blasnp = mfqP->nmodelpoints;

457:       /* Copy L_save to L */
458:       for (i=0;i<mfqP->npmax * mfqP->n*(mfqP->n+1)/2;i++) {
459:         mfqP->L[i] = mfqP->L_save[i];
460:       }
461:     }
462:     point--;
463:   }

465:   blasnp = mfqP->nmodelpoints;
466:   /* Copy Q(:,n+2:np) to Z */
467:   /* First set Q_tmp to I */
468:   for (i=0;i<mfqP->npmax*mfqP->npmax;i++) {
469:     mfqP->Q_tmp[i] = 0.0;
470:   }
471:   for (i=0;i<mfqP->npmax;i++) {
472:     mfqP->Q_tmp[i + mfqP->npmax*i] = 1.0;
473:   }

475:   /* Q_tmp = I * Q */
476:   PetscStackCallBLAS("LAPACKormqr",LAPACKormqr_("R","N",&blasnp,&blasnp,&blasnplus1,mfqP->Q,&blasnpmax,mfqP->tau,mfqP->Q_tmp,&blasnpmax,mfqP->npmaxwork,&blasnmax,&info));
477:   if (info != 0) SETERRQ1(PETSC_COMM_SELF,1,"LAPACK routine ormqr returned with value %d\n",info);

479:   /* Copy Q_tmp(:,n+2:np) to Z) */
480:   offset = mfqP->npmax * (mfqP->n+1);
481:   for (i=offset;i<mfqP->npmax*mfqP->npmax;i++) {
482:     mfqP->Z[i-offset] = mfqP->Q_tmp[i];
483:   }

485:   if (mfqP->nmodelpoints == mfqP->n + 1) {
486:     /* Set L to I_{n+1} */
487:     for (i=0;i<mfqP->npmax * mfqP->n*(mfqP->n+1)/2;i++) {
488:       mfqP->L[i] = 0.0;
489:     }
490:     for (i=0;i<mfqP->n;i++) {
491:       mfqP->L[(mfqP->n*(mfqP->n+1)/2)*i + i] = 1.0;
492:     }
493:   }
494:   return(0);
495: }

497: /* Only call from modelimprove, addpoint() needs ->Q_tmp and ->work to be set */
498: PetscErrorCode addpoint(Tao tao, TAO_POUNDERS *mfqP, PetscInt index)
499: {

503:   /* Create new vector in history: X[newidx] = X[mfqP->index] + delta*X[index]*/
504:   VecDuplicate(mfqP->Xhist[0],&mfqP->Xhist[mfqP->nHist]);
505:   VecSetValues(mfqP->Xhist[mfqP->nHist],mfqP->n,mfqP->indices,&mfqP->Q_tmp[index*mfqP->npmax],INSERT_VALUES);
506:   VecAssemblyBegin(mfqP->Xhist[mfqP->nHist]);
507:   VecAssemblyEnd(mfqP->Xhist[mfqP->nHist]);
508:   VecAYPX(mfqP->Xhist[mfqP->nHist],mfqP->delta,mfqP->Xhist[mfqP->minindex]);

510:   /* Project into feasible region */
511:   if (tao->XU && tao->XL) {
512:     VecMedian(mfqP->Xhist[mfqP->nHist], tao->XL, tao->XU, mfqP->Xhist[mfqP->nHist]);
513:   }

515:   /* Compute value of new vector */
516:   VecDuplicate(mfqP->Fhist[0],&mfqP->Fhist[mfqP->nHist]);
517:   CHKMEMQ;
518:   pounders_feval(tao,mfqP->Xhist[mfqP->nHist],mfqP->Fhist[mfqP->nHist],&mfqP->Fres[mfqP->nHist]);

520:   /* Add new vector to model */
521:   mfqP->model_indices[mfqP->nmodelpoints] = mfqP->nHist;
522:   mfqP->nmodelpoints++;
523:   mfqP->nHist++;
524:   return(0);
525: }

527: PetscErrorCode modelimprove(Tao tao, TAO_POUNDERS *mfqP, PetscInt addallpoints)
528: {
529:   /* modeld = Q(:,np+1:n)' */
531:   PetscInt       i,j,minindex=0;
532:   PetscReal      dp,half=0.5,one=1.0,minvalue=PETSC_INFINITY;
533:   PetscBLASInt   blasn=mfqP->n,  blasnpmax = mfqP->npmax, blask,info;
534:   PetscBLASInt   blas1=1,blasnmax = mfqP->nmax;

536:   blask = mfqP->nmodelpoints;
537:   /* Qtmp = I(n x n) */
538:   for (i=0;i<mfqP->n;i++) {
539:     for (j=0;j<mfqP->n;j++) {
540:       mfqP->Q_tmp[i + mfqP->npmax*j] = 0.0;
541:     }
542:   }
543:   for (j=0;j<mfqP->n;j++) {
544:     mfqP->Q_tmp[j + mfqP->npmax*j] = 1.0;
545:   }

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

550:   for (i=mfqP->nmodelpoints;i<mfqP->n;i++) {
551:     dp = BLASdot_(&blasn,&mfqP->Q_tmp[i*mfqP->npmax],&blas1,mfqP->Gres,&blas1);
552:     if (dp>0.0) { /* Model says use the other direction! */
553:       for (j=0;j<mfqP->n;j++) {
554:         mfqP->Q_tmp[i*mfqP->npmax+j] *= -1;
555:       }
556:     }
557:     /* mfqP->work[i] = Cres+Modeld(i,:)*(Gres+.5*Hres*Modeld(i,:)') */
558:     for (j=0;j<mfqP->n;j++) {
559:       mfqP->work2[j] = mfqP->Gres[j];
560:     }
561:     PetscStackCallBLAS("BLASgemv",BLASgemv_("N",&blasn,&blasn,&half,mfqP->Hres,&blasn,&mfqP->Q_tmp[i*mfqP->npmax], &blas1, &one, mfqP->work2,&blas1));
562:     mfqP->work[i] = BLASdot_(&blasn,&mfqP->Q_tmp[i*mfqP->npmax],&blas1,mfqP->work2,&blas1);
563:     if (i==mfqP->nmodelpoints || mfqP->work[i] < minvalue) {
564:       minindex=i;
565:       minvalue = mfqP->work[i];
566:     }
567:     if (addallpoints != 0) {
568:       addpoint(tao,mfqP,i);
569:     }
570:   }
571:   if (!addallpoints) {
572:     addpoint(tao,mfqP,minindex);
573:   }
574:   return(0);
575: }


578: PetscErrorCode affpoints(TAO_POUNDERS *mfqP, PetscReal *xmin,PetscReal c)
579: {
580:   PetscInt        i,j;
581:   PetscBLASInt    blasm=mfqP->m,blasj,blask,blasn=mfqP->n,ione=1,info;
582:   PetscBLASInt    blasnpmax = mfqP->npmax,blasmaxmn;
583:   PetscReal       proj,normd;
584:   const PetscReal *x;
585:   PetscErrorCode  ierr;

588:   for (i=mfqP->nHist-1;i>=0;i--) {
589:     VecGetArrayRead(mfqP->Xhist[i],&x);
590:     for (j=0;j<mfqP->n;j++) {
591:       mfqP->work[j] = (x[j] - xmin[j])/mfqP->delta;
592:     }
593:     VecRestoreArrayRead(mfqP->Xhist[i],&x);
594:     PetscStackCallBLAS("BLAScopy",BLAScopy_(&blasn,mfqP->work,&ione,mfqP->work2,&ione));
595:     normd = BLASnrm2_(&blasn,mfqP->work,&ione);
596:     if (normd <= c*c) {
597:       blasj=PetscMax((mfqP->n - mfqP->nmodelpoints),0);
598:       if (!mfqP->q_is_I) {
599:         /* project D onto null */
600:         blask=(mfqP->nmodelpoints);
601:         PetscStackCallBLAS("LAPACKormqr",LAPACKormqr_("R","N",&ione,&blasn,&blask,mfqP->Q,&blasnpmax,mfqP->tau,mfqP->work2,&ione,mfqP->mwork,&blasm,&info));
602:         if (info < 0) SETERRQ1(PETSC_COMM_SELF,1,"ormqr returned value %d\n",info);
603:       }
604:       proj = BLASnrm2_(&blasj,&mfqP->work2[mfqP->nmodelpoints],&ione);

606:       if (proj >= mfqP->theta1) { /* add this index to model */
607:         mfqP->model_indices[mfqP->nmodelpoints]=i;
608:         mfqP->nmodelpoints++;
609:         PetscStackCallBLAS("BLAScopy",BLAScopy_(&blasn,mfqP->work,&ione,&mfqP->Q_tmp[mfqP->npmax*(mfqP->nmodelpoints-1)],&ione));
610:         blask=mfqP->npmax*(mfqP->nmodelpoints);
611:         PetscStackCallBLAS("BLAScopy",BLAScopy_(&blask,mfqP->Q_tmp,&ione,mfqP->Q,&ione));
612:         blask = mfqP->nmodelpoints;
613:         blasmaxmn = PetscMax(mfqP->m,mfqP->n);
614:         PetscStackCallBLAS("LAPACKgeqrf",LAPACKgeqrf_(&blasn,&blask,mfqP->Q,&blasnpmax,mfqP->tau,mfqP->mwork,&blasmaxmn,&info));
615:         if (info < 0) SETERRQ1(PETSC_COMM_SELF,1,"geqrf returned value %d\n",info);
616:         mfqP->q_is_I = 0;
617:       }
618:       if (mfqP->nmodelpoints == mfqP->n)  {
619:         break;
620:       }
621:     }
622:   }

624:   return(0);
625: }

627: static PetscErrorCode TaoSolve_POUNDERS(Tao tao)
628: {
629:   TAO_POUNDERS       *mfqP = (TAO_POUNDERS *)tao->data;
630:   PetscInt           i,ii,j,k,l;
631:   PetscReal          step=1.0;
632:   TaoConvergedReason reason = TAO_CONTINUE_ITERATING;
633:   PetscInt           low,high;
634:   PetscReal          minnorm;
635:   PetscReal          *x,*f;
636:   const PetscReal    *xmint,*fmin;
637:   PetscReal          cres,deltaold;
638:   PetscReal          gnorm;
639:   PetscBLASInt       info,ione=1,iblas;
640:   PetscBool          valid,same;
641:   PetscReal          mdec, rho, normxsp;
642:   PetscReal          one=1.0,zero=0.0,ratio;
643:   PetscBLASInt       blasm,blasn,blasncopy,blasnpmax;
644:   PetscErrorCode     ierr;
645:   static PetscBool   set = PETSC_FALSE;

647:   /* n = # of parameters
648:      m = dimension (components) of function  */
650:   PetscCitationsRegister("@article{UNEDF0,\n"
651:                                 "title = {Nuclear energy density optimization},\n"
652:                                 "author = {Kortelainen, M.  and Lesinski, T.  and Mor\'e, J.  and Nazarewicz, W.\n"
653:                                 "          and Sarich, J.  and Schunck, N.  and Stoitsov, M. V. and Wild, S. },\n"
654:                                 "journal = {Phys. Rev. C},\n"
655:                                 "volume = {82},\n"
656:                                 "number = {2},\n"
657:                                 "pages = {024313},\n"
658:                                 "numpages = {18},\n"
659:                                 "year = {2010},\n"
660:                                 "month = {Aug},\n"
661:                                 "doi = {10.1103/PhysRevC.82.024313}\n}\n",&set);
662:   tao->niter=0;
663:   if (tao->XL && tao->XU) {
664:     /* Check x0 <= XU */
665:     PetscReal val;
666:     VecCopy(tao->solution,mfqP->Xhist[0]);
667:     VecAXPY(mfqP->Xhist[0],-1.0,tao->XU);
668:     VecMax(mfqP->Xhist[0],NULL,&val);
669:     if (val > 1e-10) SETERRQ(PETSC_COMM_WORLD,1,"X0 > upper bound");

671:     /* Check x0 >= xl */
672:     VecCopy(tao->XL,mfqP->Xhist[0]);
673:     VecAXPY(mfqP->Xhist[0],-1.0,tao->solution);
674:     VecMax(mfqP->Xhist[0],NULL,&val);
675:     if (val > 1e-10) SETERRQ(PETSC_COMM_WORLD,1,"X0 < lower bound");

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

679:     VecSet(mfqP->Xhist[0],mfqP->delta);
680:     VecAXPY(mfqP->Xhist[0],1.0,tao->solution);
681:     VecAXPY(mfqP->Xhist[0],-1.0,tao->XU);
682:     VecMax(mfqP->Xhist[0],NULL,&val);
683:     if (val > 1e-10) SETERRQ(PETSC_COMM_WORLD,1,"X0 + delta > upper bound");
684:   }

686:   CHKMEMQ;
687:   blasm = mfqP->m; blasn=mfqP->n; blasnpmax = mfqP->npmax;
688:   for (i=0;i<mfqP->n*mfqP->n*mfqP->m;i++) mfqP->H[i]=0;

690:   VecCopy(tao->solution,mfqP->Xhist[0]);
691:   CHKMEMQ;
692:   pounders_feval(tao,tao->solution,mfqP->Fhist[0],&mfqP->Fres[0]);
693:   mfqP->minindex = 0;
694:   minnorm = mfqP->Fres[0];
695:   TaoMonitor(tao, tao->niter, minnorm, PETSC_INFINITY, 0.0, step, &reason);
696:   tao->niter++;

698:   VecGetOwnershipRange(mfqP->Xhist[0],&low,&high);
699:   for (i=1;i<mfqP->n+1;i++) {
700:     VecCopy(tao->solution,mfqP->Xhist[i]);
701:     if (i-1 >= low && i-1 < high) {
702:       VecGetArray(mfqP->Xhist[i],&x);
703:       x[i-1-low] += mfqP->delta;
704:       VecRestoreArray(mfqP->Xhist[i],&x);
705:     }
706:     CHKMEMQ;
707:     pounders_feval(tao,mfqP->Xhist[i],mfqP->Fhist[i],&mfqP->Fres[i]);
708:     if (mfqP->Fres[i] < minnorm) {
709:       mfqP->minindex = i;
710:       minnorm = mfqP->Fres[i];
711:     }
712:   }
713:   VecCopy(mfqP->Xhist[mfqP->minindex],tao->solution);
714:   VecCopy(mfqP->Fhist[mfqP->minindex],tao->sep_objective);
715:   /* Gather mpi vecs to one big local vec */

717:   /* Begin serial code */

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

724:   if (mfqP->size == 1) {
725:     VecGetArrayRead(mfqP->Xhist[mfqP->minindex],&xmint);
726:     for (i=0;i<mfqP->n;i++) mfqP->xmin[i] = xmint[i];
727:     VecRestoreArrayRead(mfqP->Xhist[mfqP->minindex],&xmint);
728:     VecGetArrayRead(mfqP->Fhist[mfqP->minindex],&fmin);
729:     for (i=0;i<mfqP->n+1;i++) {
730:       if (i == mfqP->minindex) continue;

732:       VecGetArray(mfqP->Xhist[i],&x);
733:       for (j=0;j<mfqP->n;j++) {
734:         mfqP->Disp[ii+mfqP->npmax*j] = (x[j] - mfqP->xmin[j])/mfqP->delta;
735:       }
736:       VecRestoreArray(mfqP->Xhist[i],&x);

738:       VecGetArray(mfqP->Fhist[i],&f);
739:       for (j=0;j<mfqP->m;j++) {
740:         mfqP->Fdiff[ii+mfqP->n*j] = f[j] - fmin[j];
741:       }
742:       VecRestoreArray(mfqP->Fhist[i],&f);
743:       mfqP->model_indices[ii++] = i;

745:     }
746:     for (j=0;j<mfqP->m;j++) {
747:       mfqP->C[j] = fmin[j];
748:     }
749:     VecRestoreArrayRead(mfqP->Fhist[mfqP->minindex],&fmin);
750:   } else {
751:     VecScatterBegin(mfqP->scatterx,mfqP->Xhist[mfqP->minindex],mfqP->localxmin,INSERT_VALUES,SCATTER_FORWARD);
752:     VecScatterEnd(mfqP->scatterx,mfqP->Xhist[mfqP->minindex],mfqP->localxmin,INSERT_VALUES,SCATTER_FORWARD);
753:     VecGetArrayRead(mfqP->localxmin,&xmint);
754:     for (i=0;i<mfqP->n;i++) mfqP->xmin[i] = xmint[i];
755:     VecRestoreArrayRead(mfqP->localxmin,&xmint);

757:     VecScatterBegin(mfqP->scatterf,mfqP->Fhist[mfqP->minindex],mfqP->localfmin,INSERT_VALUES,SCATTER_FORWARD);
758:     VecScatterEnd(mfqP->scatterf,mfqP->Fhist[mfqP->minindex],mfqP->localfmin,INSERT_VALUES,SCATTER_FORWARD);
759:     VecGetArrayRead(mfqP->localfmin,&fmin);
760:     for (i=0;i<mfqP->n+1;i++) {
761:       if (i == mfqP->minindex) continue;

763:       mfqP->model_indices[ii++] = i;
764:       VecScatterBegin(mfqP->scatterx,mfqP->Xhist[ii],mfqP->localx,INSERT_VALUES, SCATTER_FORWARD);
765:       VecScatterEnd(mfqP->scatterx,mfqP->Xhist[ii],mfqP->localx,INSERT_VALUES, SCATTER_FORWARD);
766:       VecGetArray(mfqP->localx,&x);
767:       for (j=0;j<mfqP->n;j++) {
768:         mfqP->Disp[i+mfqP->npmax*j] = (x[j] - mfqP->xmin[j])/mfqP->delta;
769:       }
770:       VecRestoreArray(mfqP->localx,&x);

772:       VecScatterBegin(mfqP->scatterf,mfqP->Fhist[ii],mfqP->localf,INSERT_VALUES, SCATTER_FORWARD);
773:       VecScatterEnd(mfqP->scatterf,mfqP->Fhist[ii],mfqP->localf,INSERT_VALUES, SCATTER_FORWARD);
774:       VecGetArray(mfqP->localf,&f);
775:       for (j=0;j<mfqP->m;j++) {
776:         mfqP->Fdiff[i*mfqP->n+j] = f[j] - fmin[j];
777:       }
778:       VecRestoreArray(mfqP->localf,&f);
779:     }
780:     for (j=0;j<mfqP->m;j++) {
781:       mfqP->C[j] = fmin[j];
782:     }
783:     VecRestoreArrayRead(mfqP->localfmin,&fmin);
784:   }

786:   /* Determine the initial quadratic models */
787:   /* G = D(ModelIn,:) \ (F(ModelIn,1:m)-repmat(F(xkin,1:m),n,1)); */
788:   /* D (nxn) Fdiff (nxm)  => G (nxm) */
789:   blasncopy = blasn;
790:   PetscStackCallBLAS("LAPACKgesv",LAPACKgesv_(&blasn,&blasm,mfqP->Disp,&blasnpmax,mfqP->iwork,mfqP->Fdiff,&blasncopy,&info));
791:   PetscInfo1(tao,"gesv returned %d\n",info);

793:   cres = minnorm;
794:   pounders_update_res(tao);

796:   valid = PETSC_TRUE;

798:   VecSetValues(tao->gradient,mfqP->n,mfqP->indices,mfqP->Gres,INSERT_VALUES);
799:   VecAssemblyBegin(tao->gradient);
800:   VecAssemblyEnd(tao->gradient);
801:   VecNorm(tao->gradient,NORM_2,&gnorm);
802:   gnorm *= mfqP->delta;
803:   VecCopy(mfqP->Xhist[mfqP->minindex],tao->solution);
804:   TaoMonitor(tao, tao->niter, minnorm, gnorm, 0.0, step, &reason);
805:   mfqP->nHist = mfqP->n+1;
806:   mfqP->nmodelpoints = mfqP->n+1;

808:   while (reason == TAO_CONTINUE_ITERATING) {
809:     PetscReal gnm = 1e-4;
810:     tao->niter++;
811:     /* Solve the subproblem min{Q(s): ||s|| <= delta} */
812:     gqtwrap(tao,&gnm,&mdec);
813:     /* Evaluate the function at the new point */

815:     for (i=0;i<mfqP->n;i++) {
816:         mfqP->work[i] = mfqP->Xsubproblem[i]*mfqP->delta + mfqP->xmin[i];
817:     }
818:     VecDuplicate(tao->solution,&mfqP->Xhist[mfqP->nHist]);
819:     VecDuplicate(tao->sep_objective,&mfqP->Fhist[mfqP->nHist]);
820:     VecSetValues(mfqP->Xhist[mfqP->nHist],mfqP->n,mfqP->indices,mfqP->work,INSERT_VALUES);
821:     VecAssemblyBegin(mfqP->Xhist[mfqP->nHist]);
822:     VecAssemblyEnd(mfqP->Xhist[mfqP->nHist]);

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

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

829:     /* Update the center */
830:     if ((rho >= mfqP->eta1) || (rho > mfqP->eta0 && valid==PETSC_TRUE)) {
831:       /* Update model to reflect new base point */
832:       for (i=0;i<mfqP->n;i++) {
833:         mfqP->work[i] = (mfqP->work[i] - mfqP->xmin[i])/mfqP->delta;
834:       }
835:       for (j=0;j<mfqP->m;j++) {
836:         /* C(j) = C(j) + work*G(:,j) + .5*work*H(:,:,j)*work';
837:          G(:,j) = G(:,j) + H(:,:,j)*work' */
838:         for (k=0;k<mfqP->n;k++) {
839:           mfqP->work2[k]=0.0;
840:           for (l=0;l<mfqP->n;l++) {
841:             mfqP->work2[k]+=mfqP->H[j + mfqP->m*(k + l*mfqP->n)]*mfqP->work[l];
842:           }
843:         }
844:         for (i=0;i<mfqP->n;i++) {
845:           mfqP->C[j]+=mfqP->work[i]*(mfqP->Fdiff[i + mfqP->n* j] + 0.5*mfqP->work2[i]);
846:           mfqP->Fdiff[i+mfqP->n*j] +=mfqP-> work2[i];
847:         }
848:       }
849:       /* Cres += work*Gres + .5*work*Hres*work';
850:        Gres += Hres*work'; */

852:       PetscStackCallBLAS("BLASgemv",BLASgemv_("N",&blasn,&blasn,&one,mfqP->Hres,&blasn,mfqP->work,&ione,&zero,mfqP->work2,&ione));
853:       for (i=0;i<mfqP->n;i++) {
854:         cres += mfqP->work[i]*(mfqP->Gres[i]  + 0.5*mfqP->work2[i]);
855:         mfqP->Gres[i] += mfqP->work2[i];
856:       }
857:       mfqP->minindex = mfqP->nHist-1;
858:       minnorm = mfqP->Fres[mfqP->minindex];
859:       VecCopy(mfqP->Fhist[mfqP->minindex],tao->sep_objective);
860:       /* Change current center */
861:       VecGetArrayRead(mfqP->Xhist[mfqP->minindex],&xmint);
862:       for (i=0;i<mfqP->n;i++) {
863:         mfqP->xmin[i] = xmint[i];
864:       }
865:       VecRestoreArrayRead(mfqP->Xhist[mfqP->minindex],&xmint);
866:     }

868:     /* Evaluate at a model-improving point if necessary */
869:     if (valid == PETSC_FALSE) {
870:       mfqP->q_is_I = 1;
871:       mfqP->nmodelpoints = 0;
872:       affpoints(mfqP,mfqP->xmin,mfqP->c1);
873:       if (mfqP->nmodelpoints < mfqP->n) {
874:         PetscInfo(tao,"Model not valid -- model-improving\n");
875:         modelimprove(tao,mfqP,1);
876:       }
877:     }

879:     /* Update the trust region radius */
880:     deltaold = mfqP->delta;
881:     normxsp = 0;
882:     for (i=0;i<mfqP->n;i++) {
883:       normxsp += mfqP->Xsubproblem[i]*mfqP->Xsubproblem[i];
884:     }
885:     normxsp = PetscSqrtReal(normxsp);
886:     if (rho >= mfqP->eta1 && normxsp > 0.5*mfqP->delta) {
887:       mfqP->delta = PetscMin(mfqP->delta*mfqP->gamma1,mfqP->deltamax);
888:     } else if (valid == PETSC_TRUE) {
889:       mfqP->delta = PetscMax(mfqP->delta*mfqP->gamma0,mfqP->deltamin);
890:     }

892:     /* Compute the next interpolation set */
893:     mfqP->q_is_I = 1;
894:     mfqP->nmodelpoints=0;
895:     affpoints(mfqP,mfqP->xmin,mfqP->c1);
896:     if (mfqP->nmodelpoints == mfqP->n) {
897:       valid = PETSC_TRUE;
898:     } else {
899:       valid = PETSC_FALSE;
900:       affpoints(mfqP,mfqP->xmin,mfqP->c2);
901:       if (mfqP->n > mfqP->nmodelpoints) {
902:         PetscInfo(tao,"Model not valid -- adding geometry points\n");
903:         modelimprove(tao,mfqP,mfqP->n - mfqP->nmodelpoints);
904:       }
905:     }
906:     for (i=mfqP->nmodelpoints;i>0;i--) {
907:       mfqP->model_indices[i] = mfqP->model_indices[i-1];
908:     }
909:     mfqP->nmodelpoints++;
910:     mfqP->model_indices[0] = mfqP->minindex;
911:     morepoints(mfqP);
912:     for (i=0;i<mfqP->nmodelpoints;i++) {
913:       VecGetArray(mfqP->Xhist[mfqP->model_indices[i]],&x);
914:       for (j=0;j<mfqP->n;j++) {
915:         mfqP->Disp[i + mfqP->npmax*j] = (x[j]  - mfqP->xmin[j]) / deltaold;
916:       }
917:       VecRestoreArray(mfqP->Xhist[mfqP->model_indices[i]],&x);
918:       VecGetArray(mfqP->Fhist[mfqP->model_indices[i]],&f);
919:       for (j=0;j<mfqP->m;j++) {
920:         for (k=0;k<mfqP->n;k++)  {
921:           mfqP->work[k]=0.0;
922:           for (l=0;l<mfqP->n;l++) {
923:             mfqP->work[k] += mfqP->H[j + mfqP->m*(k + mfqP->n*l)] * mfqP->Disp[i + mfqP->npmax*l];
924:           }
925:         }
926:         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];
927:       }
928:       VecRestoreArray(mfqP->Fhist[mfqP->model_indices[i]],&f);
929:     }

931:     /* Update the quadratic model */
932:     getquadpounders(mfqP);
933:     VecGetArrayRead(mfqP->Fhist[mfqP->minindex],&fmin);
934:     PetscStackCallBLAS("BLAScopy",BLAScopy_(&blasm,fmin,&ione,mfqP->C,&ione));
935:     /* G = G*(delta/deltaold) + Gdel */
936:     ratio = mfqP->delta/deltaold;
937:     iblas = blasm*blasn;
938:     PetscStackCallBLAS("BLASscal",BLASscal_(&iblas,&ratio,mfqP->Fdiff,&ione));
939:     PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&iblas,&one,mfqP->Gdel,&ione,mfqP->Fdiff,&ione));
940:     /* H = H*(delta/deltaold)^2 + Hdel */
941:     iblas = blasm*blasn*blasn;
942:     ratio *= ratio;
943:     PetscStackCallBLAS("BLASscal",BLASscal_(&iblas,&ratio,mfqP->H,&ione));
944:     PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&iblas,&one,mfqP->Hdel,&ione,mfqP->H,&ione));

946:     /* Get residuals */
947:     cres = mfqP->Fres[mfqP->minindex];
948:     pounders_update_res(tao);

950:     /* Export solution and gradient residual to TAO */
951:     VecCopy(mfqP->Xhist[mfqP->minindex],tao->solution);
952:     VecSetValues(tao->gradient,mfqP->n,mfqP->indices,mfqP->Gres,INSERT_VALUES);
953:     VecAssemblyBegin(tao->gradient);
954:     VecAssemblyEnd(tao->gradient);
955:     VecNorm(tao->gradient,NORM_2,&gnorm);
956:     gnorm *= mfqP->delta;
957:     /*  final criticality test */
958:     TaoMonitor(tao, tao->niter, minnorm, gnorm, 0.0, step, &reason);
959:     /* test for repeated model */
960:     if (mfqP->nmodelpoints==mfqP->last_nmodelpoints) {
961:       same = PETSC_TRUE;
962:     } else {
963:       same = PETSC_FALSE;
964:     }
965:     for (i=0;i<mfqP->nmodelpoints;i++) {
966:       if (same) {
967:         if (mfqP->model_indices[i] == mfqP->last_model_indices[i]) {
968:           same = PETSC_TRUE;
969:         } else {
970:           same = PETSC_FALSE;
971:         }
972:       }
973:       mfqP->last_model_indices[i] = mfqP->model_indices[i];
974:     }
975:     mfqP->last_nmodelpoints = mfqP->nmodelpoints;
976:     if (same && mfqP->delta == deltaold) {
977:       PetscInfo(tao,"Identical model used in successive iterations\n");
978:       reason = TAO_CONVERGED_STEPTOL;
979:       tao->reason = TAO_CONVERGED_STEPTOL;
980:     }
981:   }
982:   return(0);
983: }

985: static PetscErrorCode TaoSetUp_POUNDERS(Tao tao)
986: {
987:   TAO_POUNDERS   *mfqP = (TAO_POUNDERS*)tao->data;
988:   PetscInt       i,j;
989:   IS             isfloc,isfglob,isxloc,isxglob;

993:   if (!tao->gradient) {VecDuplicate(tao->solution,&tao->gradient);  }
994:   if (!tao->stepdirection) {VecDuplicate(tao->solution,&tao->stepdirection);  }
995:   VecGetSize(tao->solution,&mfqP->n);
996:   VecGetSize(tao->sep_objective,&mfqP->m);
997:   mfqP->c1 = PetscSqrtReal((PetscReal)mfqP->n);
998:   if (mfqP->npmax == PETSC_DEFAULT) {
999:     mfqP->npmax = 2*mfqP->n + 1;
1000:   }
1001:   mfqP->npmax = PetscMin((mfqP->n+1)*(mfqP->n+2)/2,mfqP->npmax);
1002:   mfqP->npmax = PetscMax(mfqP->npmax, mfqP->n+2);

1004:   PetscMalloc1(tao->max_funcs+10,&mfqP->Xhist);
1005:   PetscMalloc1(tao->max_funcs+10,&mfqP->Fhist);
1006:   for (i=0;i<mfqP->n +1;i++) {
1007:     VecDuplicate(tao->solution,&mfqP->Xhist[i]);
1008:     VecDuplicate(tao->sep_objective,&mfqP->Fhist[i]);
1009:   }
1010:   VecDuplicate(tao->solution,&mfqP->workxvec);
1011:   VecDuplicate(tao->sep_objective,&mfqP->workfvec);
1012:   mfqP->nHist = 0;

1014:   PetscMalloc1(tao->max_funcs+10,&mfqP->Fres);
1015:   PetscMalloc1(mfqP->npmax*mfqP->m,&mfqP->RES);
1016:   PetscMalloc1(mfqP->n,&mfqP->work);
1017:   PetscMalloc1(mfqP->n,&mfqP->work2);
1018:   PetscMalloc1(mfqP->n,&mfqP->work3);
1019:   PetscMalloc1(PetscMax(mfqP->m,mfqP->n+1),&mfqP->mwork);
1020:   PetscMalloc1(mfqP->npmax - mfqP->n - 1,&mfqP->omega);
1021:   PetscMalloc1(mfqP->n * (mfqP->n+1) / 2,&mfqP->beta);
1022:   PetscMalloc1(mfqP->n + 1 ,&mfqP->alpha);

1024:   PetscMalloc1(mfqP->n*mfqP->n*mfqP->m,&mfqP->H);
1025:   PetscMalloc1(mfqP->npmax*mfqP->npmax,&mfqP->Q);
1026:   PetscMalloc1(mfqP->npmax*mfqP->npmax,&mfqP->Q_tmp);
1027:   PetscMalloc1(mfqP->n*(mfqP->n+1)/2*(mfqP->npmax),&mfqP->L);
1028:   PetscMalloc1(mfqP->n*(mfqP->n+1)/2*(mfqP->npmax),&mfqP->L_tmp);
1029:   PetscMalloc1(mfqP->n*(mfqP->n+1)/2*(mfqP->npmax),&mfqP->L_save);
1030:   PetscMalloc1(mfqP->n*(mfqP->n+1)/2*(mfqP->npmax),&mfqP->N);
1031:   PetscMalloc1(mfqP->npmax * (mfqP->n+1) ,&mfqP->M);
1032:   PetscMalloc1(mfqP->npmax * (mfqP->npmax - mfqP->n - 1) , &mfqP->Z);
1033:   PetscMalloc1(mfqP->npmax,&mfqP->tau);
1034:   PetscMalloc1(mfqP->npmax,&mfqP->tau_tmp);
1035:   mfqP->nmax = PetscMax(5*mfqP->npmax,mfqP->n*(mfqP->n+1)/2);
1036:   PetscMalloc1(mfqP->nmax,&mfqP->npmaxwork);
1037:   PetscMalloc1(mfqP->nmax,&mfqP->npmaxiwork);
1038:   PetscMalloc1(mfqP->n,&mfqP->xmin);
1039:   PetscMalloc1(mfqP->m,&mfqP->C);
1040:   PetscMalloc1(mfqP->m*mfqP->n,&mfqP->Fdiff);
1041:   PetscMalloc1(mfqP->npmax*mfqP->n,&mfqP->Disp);
1042:   PetscMalloc1(mfqP->n,&mfqP->Gres);
1043:   PetscMalloc1(mfqP->n*mfqP->n,&mfqP->Hres);
1044:   PetscMalloc1(mfqP->n*mfqP->n,&mfqP->Gpoints);
1045:   PetscMalloc1(mfqP->npmax,&mfqP->model_indices);
1046:   PetscMalloc1(mfqP->npmax,&mfqP->last_model_indices);
1047:   PetscMalloc1(mfqP->n,&mfqP->Xsubproblem);
1048:   PetscMalloc1(mfqP->m*mfqP->n,&mfqP->Gdel);
1049:   PetscMalloc1(mfqP->n*mfqP->n*mfqP->m, &mfqP->Hdel);
1050:   PetscMalloc1(PetscMax(mfqP->m,mfqP->n),&mfqP->indices);
1051:   PetscMalloc1(mfqP->n,&mfqP->iwork);
1052:   PetscMalloc1(mfqP->m*mfqP->m,&mfqP->w);
1053:   for (i=0;i<mfqP->m;i++) {
1054:     for (j=0;j<mfqP->m;j++) {
1055:       if (i==j) {
1056:         mfqP->w[i+mfqP->m*j]=1.0;
1057:       } else {
1058:         mfqP->w[i+mfqP->m*j]=0.0;
1059:       }
1060:     }
1061:   }
1062:   for (i=0;i<PetscMax(mfqP->m,mfqP->n);i++) {
1063:     mfqP->indices[i] = i;
1064:   }
1065:   MPI_Comm_size(((PetscObject)tao)->comm,&mfqP->size);
1066:   if (mfqP->size > 1) {
1067:     VecCreateSeq(PETSC_COMM_SELF,mfqP->n,&mfqP->localx);
1068:     VecCreateSeq(PETSC_COMM_SELF,mfqP->n,&mfqP->localxmin);
1069:     VecCreateSeq(PETSC_COMM_SELF,mfqP->m,&mfqP->localf);
1070:     VecCreateSeq(PETSC_COMM_SELF,mfqP->m,&mfqP->localfmin);
1071:     ISCreateStride(MPI_COMM_SELF,mfqP->n,0,1,&isxloc);
1072:     ISCreateStride(MPI_COMM_SELF,mfqP->n,0,1,&isxglob);
1073:     ISCreateStride(MPI_COMM_SELF,mfqP->m,0,1,&isfloc);
1074:     ISCreateStride(MPI_COMM_SELF,mfqP->m,0,1,&isfglob);


1077:     VecScatterCreate(tao->solution,isxglob,mfqP->localx,isxloc,&mfqP->scatterx);
1078:     VecScatterCreate(tao->sep_objective,isfglob,mfqP->localf,isfloc,&mfqP->scatterf);

1080:     ISDestroy(&isxloc);
1081:     ISDestroy(&isxglob);
1082:     ISDestroy(&isfloc);
1083:     ISDestroy(&isfglob);
1084:   }

1086:   if (!mfqP->usegqt) {
1087:     KSP       ksp;
1088:     PC        pc;
1089:     VecCreateSeqWithArray(PETSC_COMM_SELF,mfqP->n,mfqP->n,mfqP->Xsubproblem,&mfqP->subx);
1090:     VecCreateSeq(PETSC_COMM_SELF,mfqP->n,&mfqP->subxl);
1091:     VecDuplicate(mfqP->subxl,&mfqP->subb);
1092:     VecDuplicate(mfqP->subxl,&mfqP->subxu);
1093:     VecDuplicate(mfqP->subxl,&mfqP->subpdel);
1094:     VecDuplicate(mfqP->subxl,&mfqP->subndel);
1095:     TaoCreate(PETSC_COMM_SELF,&mfqP->subtao);
1096:     TaoSetType(mfqP->subtao,TAOTRON);
1097:     TaoSetOptionsPrefix(mfqP->subtao,"pounders_subsolver_");
1098:     TaoSetInitialVector(mfqP->subtao,mfqP->subx);
1099:     TaoSetObjectiveAndGradientRoutine(mfqP->subtao,pounders_fg,(void*)mfqP);
1100:     TaoSetMaximumIterations(mfqP->subtao,mfqP->gqt_maxits);
1101:     TaoSetFromOptions(mfqP->subtao);
1102:     TaoGetKSP(mfqP->subtao,&ksp);
1103:     if (ksp) {
1104:       KSPGetPC(ksp,&pc);
1105:       PCSetType(pc,PCNONE);
1106:     }
1107:     TaoSetVariableBounds(mfqP->subtao,mfqP->subxl,mfqP->subxu);
1108:     MatCreateSeqDense(PETSC_COMM_SELF,mfqP->n,mfqP->n,mfqP->Hres,&mfqP->subH);
1109:     TaoSetHessianRoutine(mfqP->subtao,mfqP->subH,mfqP->subH,pounders_h,(void*)mfqP);
1110:   }
1111:   return(0);
1112: }

1114: static PetscErrorCode TaoDestroy_POUNDERS(Tao tao)
1115: {
1116:   TAO_POUNDERS   *mfqP = (TAO_POUNDERS*)tao->data;
1117:   PetscInt       i;

1121:   if (!mfqP->usegqt) {
1122:     TaoDestroy(&mfqP->subtao);
1123:     VecDestroy(&mfqP->subx);
1124:     VecDestroy(&mfqP->subxl);
1125:     VecDestroy(&mfqP->subxu);
1126:     VecDestroy(&mfqP->subb);
1127:     VecDestroy(&mfqP->subpdel);
1128:     VecDestroy(&mfqP->subndel);
1129:     MatDestroy(&mfqP->subH);
1130:   }
1131:   PetscFree(mfqP->Fres);
1132:   PetscFree(mfqP->RES);
1133:   PetscFree(mfqP->work);
1134:   PetscFree(mfqP->work2);
1135:   PetscFree(mfqP->work3);
1136:   PetscFree(mfqP->mwork);
1137:   PetscFree(mfqP->omega);
1138:   PetscFree(mfqP->beta);
1139:   PetscFree(mfqP->alpha);
1140:   PetscFree(mfqP->H);
1141:   PetscFree(mfqP->Q);
1142:   PetscFree(mfqP->Q_tmp);
1143:   PetscFree(mfqP->L);
1144:   PetscFree(mfqP->L_tmp);
1145:   PetscFree(mfqP->L_save);
1146:   PetscFree(mfqP->N);
1147:   PetscFree(mfqP->M);
1148:   PetscFree(mfqP->Z);
1149:   PetscFree(mfqP->tau);
1150:   PetscFree(mfqP->tau_tmp);
1151:   PetscFree(mfqP->npmaxwork);
1152:   PetscFree(mfqP->npmaxiwork);
1153:   PetscFree(mfqP->xmin);
1154:   PetscFree(mfqP->C);
1155:   PetscFree(mfqP->Fdiff);
1156:   PetscFree(mfqP->Disp);
1157:   PetscFree(mfqP->Gres);
1158:   PetscFree(mfqP->Hres);
1159:   PetscFree(mfqP->Gpoints);
1160:   PetscFree(mfqP->model_indices);
1161:   PetscFree(mfqP->last_model_indices);
1162:   PetscFree(mfqP->Xsubproblem);
1163:   PetscFree(mfqP->Gdel);
1164:   PetscFree(mfqP->Hdel);
1165:   PetscFree(mfqP->indices);
1166:   PetscFree(mfqP->iwork);
1167:   PetscFree(mfqP->w);
1168:   for (i=0;i<mfqP->nHist;i++) {
1169:     VecDestroy(&mfqP->Xhist[i]);
1170:     VecDestroy(&mfqP->Fhist[i]);
1171:   }
1172:   VecDestroy(&mfqP->workxvec);
1173:   VecDestroy(&mfqP->workfvec);
1174:   PetscFree(mfqP->Xhist);
1175:   PetscFree(mfqP->Fhist);

1177:   if (mfqP->size > 1) {
1178:     VecDestroy(&mfqP->localx);
1179:     VecDestroy(&mfqP->localxmin);
1180:     VecDestroy(&mfqP->localf);
1181:     VecDestroy(&mfqP->localfmin);
1182:   }
1183:   PetscFree(tao->data);
1184:   return(0);
1185: }

1187: static PetscErrorCode TaoSetFromOptions_POUNDERS(PetscOptionItems *PetscOptionsObject,Tao tao)
1188: {
1189:   TAO_POUNDERS   *mfqP = (TAO_POUNDERS*)tao->data;

1193:   PetscOptionsHead(PetscOptionsObject,"POUNDERS method for least-squares optimization");
1194:   PetscOptionsReal("-tao_pounders_delta","initial delta","",mfqP->delta,&mfqP->delta0,NULL);
1195:   mfqP->delta = mfqP->delta0;
1196:   mfqP->npmax = PETSC_DEFAULT;
1197:   PetscOptionsInt("-tao_pounders_npmax","max number of points in model","",mfqP->npmax,&mfqP->npmax,NULL);
1198:   mfqP->usegqt = PETSC_FALSE;
1199:   PetscOptionsBool("-tao_pounders_gqt","use gqt algorithm for subproblem","",mfqP->usegqt,&mfqP->usegqt,NULL);
1200:   PetscOptionsTail();
1201:   return(0);
1202: }

1204: static PetscErrorCode TaoView_POUNDERS(Tao tao, PetscViewer viewer)
1205: {
1206:   TAO_POUNDERS   *mfqP = (TAO_POUNDERS *)tao->data;
1207:   PetscBool      isascii;
1208:   PetscInt       nits;

1212:   PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&isascii);
1213:   if (isascii) {
1214:     PetscViewerASCIIPushTab(viewer);
1215:     PetscViewerASCIIPrintf(viewer, "initial delta: %g\n",(double)mfqP->delta0);
1216:     PetscViewerASCIIPrintf(viewer, "final delta: %g\n",(double)mfqP->delta);
1217:     PetscViewerASCIIPrintf(viewer, "model points: %D\n",mfqP->nmodelpoints);
1218:     if (mfqP->usegqt) {
1219:       PetscViewerASCIIPrintf(viewer, "subproblem solver: gqt\n");
1220:     } else {
1221:       PetscViewerASCIIPrintf(viewer, "subproblem solver: %s\n",((PetscObject)mfqP->subtao)->type_name);
1222:       TaoGetTotalIterationNumber(mfqP->subtao,&nits);
1223:       PetscViewerASCIIPrintf(viewer, "total subproblem iterations: %D\n",nits);
1224:     }
1225:     PetscViewerASCIIPopTab(viewer);
1226:   }
1227:   return(0);
1228: }
1229: /*MC
1230:   TAOPOUNDERS - POUNDERS derivate-free model-based algorithm for nonlinear least squares

1232:   Options Database Keys:
1233: + -tao_pounders_delta - initial step length
1234: . -tao_pounders_npmax - maximum number of points in model
1235: - -tao_pounders_gqt - use gqt algorithm for subproblem instead of TRON

1237:   Level: beginner
1238:  
1239: M*/

1241: PETSC_EXTERN PetscErrorCode TaoCreate_POUNDERS(Tao tao)
1242: {
1243:   TAO_POUNDERS   *mfqP = (TAO_POUNDERS*)tao->data;

1247:   tao->ops->setup = TaoSetUp_POUNDERS;
1248:   tao->ops->solve = TaoSolve_POUNDERS;
1249:   tao->ops->view = TaoView_POUNDERS;
1250:   tao->ops->setfromoptions = TaoSetFromOptions_POUNDERS;
1251:   tao->ops->destroy = TaoDestroy_POUNDERS;

1253:   PetscNewLog(tao,&mfqP);
1254:   tao->data = (void*)mfqP;
1255:   /* Override default settings (unless already changed) */
1256:   if (!tao->max_it_changed) tao->max_it = 2000;
1257:   if (!tao->max_funcs_changed) tao->max_funcs = 4000;
1258:   mfqP->delta0 = 0.1;
1259:   mfqP->delta = 0.1;
1260:   mfqP->deltamax=1e3;
1261:   mfqP->deltamin=1e-6;
1262:   mfqP->c2 = 100.0;
1263:   mfqP->theta1=1e-5;
1264:   mfqP->theta2=1e-4;
1265:   mfqP->gamma0=0.5;
1266:   mfqP->gamma1=2.0;
1267:   mfqP->eta0 = 0.0;
1268:   mfqP->eta1 = 0.1;
1269:   mfqP->gqt_rtol = 0.001;
1270:   mfqP->gqt_maxits = 50;
1271:   mfqP->workxvec = 0;
1272:   return(0);
1273: }