xref: /petsc/src/tao/leastsquares/impls/pounders/pounders.c (revision c2558b51628bcbdfd117134d53247b2b6d3b5b7c)
1 #include <../src/tao/leastsquares/impls/pounders/pounders.h>
2 
3 static PetscErrorCode pounders_h(Tao subtao, Vec v, Mat H, Mat Hpre, void *ctx)
4 {
5   PetscFunctionBegin;
6   PetscFunctionReturn(0);
7 }
8 
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;
13   PetscErrorCode ierr;
14 
15   PetscFunctionBegin;
16   /* g = A*x  (add b later)*/
17   ierr = MatMult(mfqP->subH,x,g);CHKERRQ(ierr);
18 
19   /* f = 1/2 * x'*(Ax) + b'*x  */
20   ierr = VecDot(x,g,&d1);CHKERRQ(ierr);
21   ierr = VecDot(mfqP->subb,x,&d2);CHKERRQ(ierr);
22   *f = 0.5 *d1 + d2;
23 
24   /* now  g = g + b */
25   ierr = VecAXPY(g, 1.0, mfqP->subb);CHKERRQ(ierr);
26   PetscFunctionReturn(0);
27 }
28 
29 static PetscErrorCode pounders_feval(Tao tao, Vec x, Vec F, PetscReal *fsum)
30 {
31   PetscErrorCode ierr;
32   TAO_POUNDERS   *mfqP = (TAO_POUNDERS*)tao->data;
33   PetscInt       i,row,col;
34   PetscReal      fr,fc;
35 
36   PetscFunctionBegin;
37   ierr = TaoComputeResidual(tao,x,F);CHKERRQ(ierr);
38   if (tao->res_weights_v) {
39     ierr = VecPointwiseMult(mfqP->workfvec,tao->res_weights_v,F);CHKERRQ(ierr);
40     ierr = VecDot(mfqP->workfvec,mfqP->workfvec,fsum);CHKERRQ(ierr);
41   } else if (tao->res_weights_w) {
42     *fsum=0;
43     for (i=0;i<tao->res_weights_n;i++) {
44       row=tao->res_weights_rows[i];
45       col=tao->res_weights_cols[i];
46       ierr = VecGetValues(F,1,&row,&fr);CHKERRQ(ierr);
47       ierr = VecGetValues(F,1,&col,&fc);CHKERRQ(ierr);
48       *fsum += tao->res_weights_w[i]*fc*fr;
49     }
50   } else {
51     ierr = VecDot(F,F,fsum);CHKERRQ(ierr);
52   }
53   ierr = PetscInfo1(tao,"Least-squares residual norm: %20.19e\n",(double)*fsum);CHKERRQ(ierr);
54   if (PetscIsInfOrNanReal(*fsum)) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_USER, "User provided compute function generated Inf or NaN");
55   PetscFunctionReturn(0);
56 }
57 
58 static PetscErrorCode gqtwrap(Tao tao,PetscReal *gnorm, PetscReal *qmin)
59 {
60   PetscErrorCode ierr;
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;
68 
69   PetscFunctionBegin;
70   if (!mfqP->usegqt) {
71     PetscReal maxval;
72     PetscInt  i,j;
73 
74     ierr = VecSetValues(mfqP->subb,mfqP->n,mfqP->indices,mfqP->Gres,INSERT_VALUES);CHKERRQ(ierr);
75     ierr = VecAssemblyBegin(mfqP->subb);CHKERRQ(ierr);
76     ierr = VecAssemblyEnd(mfqP->subb);CHKERRQ(ierr);
77 
78     ierr = VecSet(mfqP->subx,0.0);CHKERRQ(ierr);
79 
80     ierr = VecSet(mfqP->subndel,-1.0);CHKERRQ(ierr);
81     ierr = VecSet(mfqP->subpdel,+1.0);CHKERRQ(ierr);
82 
83     /* Complete the lower triangle of the Hessian matrix */
84     for (i=0;i<mfqP->n;i++) {
85       for (j=i+1;j<mfqP->n;j++) {
86         mfqP->Hres[j+mfqP->n*i] = mfqP->Hres[mfqP->n*j+i];
87       }
88     }
89     ierr = MatSetValues(mfqP->subH,mfqP->n,mfqP->indices,mfqP->n,mfqP->indices,mfqP->Hres,INSERT_VALUES);CHKERRQ(ierr);
90     ierr = MatAssemblyBegin(mfqP->subH,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
91     ierr = MatAssemblyEnd(mfqP->subH,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
92 
93     ierr = TaoResetStatistics(mfqP->subtao);CHKERRQ(ierr);
94     /* ierr = TaoSetTolerances(mfqP->subtao,*gnorm,*gnorm,PETSC_DEFAULT);CHKERRQ(ierr); */
95     /* enforce bound constraints -- experimental */
96     if (tao->XU && tao->XL) {
97       ierr = VecCopy(tao->XU,mfqP->subxu);CHKERRQ(ierr);
98       ierr = VecAXPY(mfqP->subxu,-1.0,tao->solution);CHKERRQ(ierr);
99       ierr = VecScale(mfqP->subxu,1.0/mfqP->delta);CHKERRQ(ierr);
100       ierr = VecCopy(tao->XL,mfqP->subxl);CHKERRQ(ierr);
101       ierr = VecAXPY(mfqP->subxl,-1.0,tao->solution);CHKERRQ(ierr);
102       ierr = VecScale(mfqP->subxl,1.0/mfqP->delta);CHKERRQ(ierr);
103 
104       ierr = VecPointwiseMin(mfqP->subxu,mfqP->subxu,mfqP->subpdel);CHKERRQ(ierr);
105       ierr = VecPointwiseMax(mfqP->subxl,mfqP->subxl,mfqP->subndel);CHKERRQ(ierr);
106     } else {
107       ierr = VecCopy(mfqP->subpdel,mfqP->subxu);CHKERRQ(ierr);
108       ierr = VecCopy(mfqP->subndel,mfqP->subxl);CHKERRQ(ierr);
109     }
110     /* Make sure xu > xl */
111     ierr = VecCopy(mfqP->subxl,mfqP->subpdel);CHKERRQ(ierr);
112     ierr = VecAXPY(mfqP->subpdel,-1.0,mfqP->subxu);CHKERRQ(ierr);
113     ierr = VecMax(mfqP->subpdel,NULL,&maxval);CHKERRQ(ierr);
114     if (maxval > 1e-10) SETERRQ(PetscObjectComm((PetscObject)tao),PETSC_ERR_ARG_OUTOFRANGE,"upper bound < lower bound in subproblem");
115     /* Make sure xu > tao->solution > xl */
116     ierr = VecCopy(mfqP->subxl,mfqP->subpdel);CHKERRQ(ierr);
117     ierr = VecAXPY(mfqP->subpdel,-1.0,mfqP->subx);CHKERRQ(ierr);
118     ierr = VecMax(mfqP->subpdel,NULL,&maxval);CHKERRQ(ierr);
119     if (maxval > 1e-10) SETERRQ(PetscObjectComm((PetscObject)tao),PETSC_ERR_ARG_OUTOFRANGE,"initial guess < lower bound in subproblem");
120 
121     ierr = VecCopy(mfqP->subx,mfqP->subpdel);CHKERRQ(ierr);
122     ierr = VecAXPY(mfqP->subpdel,-1.0,mfqP->subxu);CHKERRQ(ierr);
123     ierr = VecMax(mfqP->subpdel,NULL,&maxval);CHKERRQ(ierr);
124     if (maxval > 1e-10) SETERRQ(PetscObjectComm((PetscObject)tao),PETSC_ERR_ARG_OUTOFRANGE,"initial guess > upper bound in subproblem");
125 
126     ierr = TaoSolve(mfqP->subtao);CHKERRQ(ierr);
127     ierr = TaoGetSolutionStatus(mfqP->subtao,NULL,qmin,NULL,NULL,NULL,NULL);CHKERRQ(ierr);
128 
129     /* test bounds post-solution*/
130     ierr = VecCopy(mfqP->subxl,mfqP->subpdel);CHKERRQ(ierr);
131     ierr = VecAXPY(mfqP->subpdel,-1.0,mfqP->subx);CHKERRQ(ierr);
132     ierr = VecMax(mfqP->subpdel,NULL,&maxval);CHKERRQ(ierr);
133     if (maxval > 1e-5) {
134       ierr = PetscInfo(tao,"subproblem solution < lower bound\n");CHKERRQ(ierr);
135       tao->reason = TAO_DIVERGED_TR_REDUCTION;
136     }
137 
138     ierr = VecCopy(mfqP->subx,mfqP->subpdel);CHKERRQ(ierr);
139     ierr = VecAXPY(mfqP->subpdel,-1.0,mfqP->subxu);CHKERRQ(ierr);
140     ierr = VecMax(mfqP->subpdel,NULL,&maxval);CHKERRQ(ierr);
141     if (maxval > 1e-5) {
142       ierr = PetscInfo(tao,"subproblem solution > upper bound\n");CHKERRQ(ierr);
143       tao->reason = TAO_DIVERGED_TR_REDUCTION;
144     }
145   } else {
146     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);
147   }
148   *qmin *= -1;
149   PetscFunctionReturn(0);
150 }
151 
152 static PetscErrorCode pounders_update_res(Tao tao)
153 {
154   TAO_POUNDERS   *mfqP = (TAO_POUNDERS*)tao->data;
155   PetscInt       i,row,col;
156   PetscBLASInt   blasn=mfqP->n,blasn2=blasn*blasn,blasm=mfqP->m,ione=1;
157   PetscReal      zero=0.0,one=1.0,wii,factor;
158   PetscErrorCode ierr;
159 
160   PetscFunctionBegin;
161   for (i=0;i<mfqP->n;i++) {
162     mfqP->Gres[i]=0;
163   }
164   for (i=0;i<mfqP->n*mfqP->n;i++) {
165     mfqP->Hres[i]=0;
166   }
167 
168   /* Compute Gres= sum_ij[wij * (cjgi + cigj)] */
169   if (tao->res_weights_v) {
170     /* Vector(diagonal) weights: gres = sum_i(wii*ci*gi) */
171     for (i=0;i<mfqP->m;i++) {
172       ierr = VecGetValues(tao->res_weights_v,1,&i,&factor);CHKERRQ(ierr);
173       factor=factor*mfqP->C[i];
174       PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&blasn,&factor,&mfqP->Fdiff[blasn*i],&ione,mfqP->Gres,&ione));
175     }
176 
177     /* compute Hres = sum_ij [wij * (*ci*Hj + cj*Hi + gi gj' + gj gi') ] */
178     /* vector(diagonal weights) Hres = sum_i(wii*(ci*Hi + gi * gi' )*/
179     for (i=0;i<mfqP->m;i++) {
180       ierr = VecGetValues(tao->res_weights_v,1,&i,&wii);CHKERRQ(ierr);
181       if (tao->niter>1) {
182         factor=wii*mfqP->C[i];
183         /* add wii * ci * Hi */
184         PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&blasn2,&factor,&mfqP->H[i],&blasm,mfqP->Hres,&ione));
185       }
186       /* add wii * gi * gi' */
187       PetscStackCallBLAS("BLASgemm",BLASgemm_("N","T",&blasn,&blasn,&ione,&wii,&mfqP->Fdiff[blasn*i],&blasn,&mfqP->Fdiff[blasn*i],&blasn,&one,mfqP->Hres,&blasn));
188     }
189   } else if (tao->res_weights_w) {
190     /* General case: .5 * Gres= sum_ij[wij * (cjgi + cigj)] */
191     for (i=0;i<tao->res_weights_n;i++) {
192       row=tao->res_weights_rows[i];
193       col=tao->res_weights_cols[i];
194 
195       factor = tao->res_weights_w[i]*mfqP->C[col]/2.0;
196       PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&blasn,&factor,&mfqP->Fdiff[blasn*row],&ione,mfqP->Gres,&ione));
197       factor = tao->res_weights_w[i]*mfqP->C[row]/2.0;
198       PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&blasn,&factor,&mfqP->Fdiff[blasn*col],&ione,mfqP->Gres,&ione));
199     }
200 
201     /* compute Hres = sum_ij [wij * (*ci*Hj + cj*Hi + gi gj' + gj gi') ] */
202     /* .5 * sum_ij [wij * (*ci*Hj + cj*Hi + gi gj' + gj gi') ] */
203     for (i=0;i<tao->res_weights_n;i++) {
204       row=tao->res_weights_rows[i];
205       col=tao->res_weights_cols[i];
206       factor=tao->res_weights_w[i]/2.0;
207       /* add wij * gi gj' + wij * gj gi' */
208       PetscStackCallBLAS("BLASgemm",BLASgemm_("N","T",&blasn,&blasn,&ione,&factor,&mfqP->Fdiff[blasn*row],&blasn,&mfqP->Fdiff[blasn*col],&blasn,&one,mfqP->Hres,&blasn));
209       PetscStackCallBLAS("BLASgemm",BLASgemm_("N","T",&blasn,&blasn,&ione,&factor,&mfqP->Fdiff[blasn*col],&blasn,&mfqP->Fdiff[blasn*row],&blasn,&one,mfqP->Hres,&blasn));
210     }
211     if (tao->niter > 1) {
212       for (i=0;i<tao->res_weights_n;i++) {
213         row=tao->res_weights_rows[i];
214         col=tao->res_weights_cols[i];
215 
216         /* add  wij*cj*Hi */
217         factor = tao->res_weights_w[i]*mfqP->C[col]/2.0;
218         PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&blasn2,&factor,&mfqP->H[row],&blasm,mfqP->Hres,&ione));
219 
220         /* add wij*ci*Hj */
221         factor = tao->res_weights_w[i]*mfqP->C[row]/2.0;
222         PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&blasn2,&factor,&mfqP->H[col],&blasm,mfqP->Hres,&ione));
223       }
224     }
225   } else {
226     /* Default: Gres= sum_i[cigi] = G*c' */
227     ierr = PetscInfo(tao,"Identity weights\n");CHKERRQ(ierr);
228     PetscStackCallBLAS("BLASgemv",BLASgemv_("N",&blasn,&blasm,&one,mfqP->Fdiff,&blasn,mfqP->C,&ione,&zero,mfqP->Gres,&ione));
229 
230     /* compute Hres = sum_ij [wij * (*ci*Hj + cj*Hi + gi gj' + gj gi') ] */
231     /*  Hres = G*G' + 0.5 sum {F(xkin,i)*H(:,:,i)}  */
232     PetscStackCallBLAS("BLASgemm",BLASgemm_("N","T",&blasn,&blasn,&blasm,&one,mfqP->Fdiff, &blasn,mfqP->Fdiff, &blasn,&zero,mfqP->Hres,&blasn));
233 
234     /* sum(F(xkin,i)*H(:,:,i)) */
235     if (tao->niter>1) {
236       for (i=0;i<mfqP->m;i++) {
237         factor = mfqP->C[i];
238         PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&blasn2,&factor,&mfqP->H[i],&blasm,mfqP->Hres,&ione));
239       }
240     }
241   }
242   PetscFunctionReturn(0);
243 }
244 
245 static PetscErrorCode phi2eval(PetscReal *x, PetscInt n, PetscReal *phi)
246 {
247 /* 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] */
248   PetscInt  i,j,k;
249   PetscReal sqrt2 = PetscSqrtReal(2.0);
250 
251   PetscFunctionBegin;
252   j=0;
253   for (i=0;i<n;i++) {
254     phi[j] = 0.5 * x[i]*x[i];
255     j++;
256     for (k=i+1;k<n;k++) {
257       phi[j]  = x[i]*x[k]/sqrt2;
258       j++;
259     }
260   }
261   PetscFunctionReturn(0);
262 }
263 
264 static PetscErrorCode getquadpounders(TAO_POUNDERS *mfqP)
265 {
266 /* Computes the parameters of the quadratic Q(x) = c + g'*x + 0.5*x*G*x'
267    that satisfies the interpolation conditions Q(X[:,j]) = f(j)
268    for j=1,...,m and with a Hessian matrix of least Frobenius norm */
269 
270     /* NB --we are ignoring c */
271   PetscInt     i,j,k,num,np = mfqP->nmodelpoints;
272   PetscReal    one = 1.0,zero=0.0,negone=-1.0;
273   PetscBLASInt blasnpmax = mfqP->npmax;
274   PetscBLASInt blasnplus1 = mfqP->n+1;
275   PetscBLASInt blasnp = np;
276   PetscBLASInt blasint = mfqP->n*(mfqP->n+1) / 2;
277   PetscBLASInt blasint2 = np - mfqP->n-1;
278   PetscBLASInt info,ione=1;
279   PetscReal    sqrt2 = PetscSqrtReal(2.0);
280 
281   PetscFunctionBegin;
282   for (i=0;i<mfqP->n*mfqP->m;i++) {
283     mfqP->Gdel[i] = 0;
284   }
285   for (i=0;i<mfqP->n*mfqP->n*mfqP->m;i++) {
286     mfqP->Hdel[i] = 0;
287   }
288 
289     /* factor M */
290   PetscStackCallBLAS("LAPACKgetrf",LAPACKgetrf_(&blasnplus1,&blasnp,mfqP->M,&blasnplus1,mfqP->npmaxiwork,&info));
291   if (info != 0) SETERRQ1(PETSC_COMM_SELF,1,"LAPACK routine getrf returned with value %d\n",info);
292 
293   if (np == mfqP->n+1) {
294     for (i=0;i<mfqP->npmax-mfqP->n-1;i++) {
295       mfqP->omega[i]=0.0;
296     }
297     for (i=0;i<mfqP->n*(mfqP->n+1)/2;i++) {
298       mfqP->beta[i]=0.0;
299     }
300   } else {
301     /* Let Ltmp = (L'*L) */
302     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));
303 
304     /* factor Ltmp */
305     PetscStackCallBLAS("LAPACKpotrf",LAPACKpotrf_("L",&blasint2,mfqP->L_tmp,&blasint,&info));
306     if (info != 0) SETERRQ1(PETSC_COMM_SELF,1,"LAPACK routine potrf returned with value %d\n",info);
307   }
308 
309   for (k=0;k<mfqP->m;k++) {
310     if (np != mfqP->n+1) {
311       /* Solve L'*L*Omega = Z' * RESk*/
312       PetscStackCallBLAS("BLASgemv",BLASgemv_("T",&blasnp,&blasint2,&one,mfqP->Z,&blasnpmax,&mfqP->RES[mfqP->npmax*k],&ione,&zero,mfqP->omega,&ione));
313       PetscStackCallBLAS("LAPACKpotrs",LAPACKpotrs_("L",&blasint2,&ione,mfqP->L_tmp,&blasint,mfqP->omega,&blasint2,&info));
314       if (info != 0) SETERRQ1(PETSC_COMM_SELF,1,"LAPACK routine potrs returned with value %d\n",info);
315 
316       /* Beta = L*Omega */
317       PetscStackCallBLAS("BLASgemv",BLASgemv_("N",&blasint,&blasint2,&one,&mfqP->L[(mfqP->n+1)*blasint],&blasint,mfqP->omega,&ione,&zero,mfqP->beta,&ione));
318     }
319 
320     /* solve M'*Alpha = RESk - N'*Beta */
321     PetscStackCallBLAS("BLASgemv",BLASgemv_("T",&blasint,&blasnp,&negone,mfqP->N,&blasint,mfqP->beta,&ione,&one,&mfqP->RES[mfqP->npmax*k],&ione));
322     PetscStackCallBLAS("LAPACKgetrs",LAPACKgetrs_("T",&blasnplus1,&ione,mfqP->M,&blasnplus1,mfqP->npmaxiwork,&mfqP->RES[mfqP->npmax*k],&blasnplus1,&info));
323     if (info != 0) SETERRQ1(PETSC_COMM_SELF,1,"LAPACK routine getrs returned with value %d\n",info);
324 
325     /* Gdel(:,k) = Alpha(2:n+1) */
326     for (i=0;i<mfqP->n;i++) {
327       mfqP->Gdel[i + mfqP->n*k] = mfqP->RES[mfqP->npmax*k + i+1];
328     }
329 
330     /* Set Hdels */
331     num=0;
332     for (i=0;i<mfqP->n;i++) {
333       /* H[i,i,k] = Beta(num) */
334       mfqP->Hdel[(i*mfqP->n + i)*mfqP->m + k] = mfqP->beta[num];
335       num++;
336       for (j=i+1;j<mfqP->n;j++) {
337         /* H[i,j,k] = H[j,i,k] = Beta(num)/sqrt(2) */
338         mfqP->Hdel[(j*mfqP->n + i)*mfqP->m + k] = mfqP->beta[num]/sqrt2;
339         mfqP->Hdel[(i*mfqP->n + j)*mfqP->m + k] = mfqP->beta[num]/sqrt2;
340         num++;
341       }
342     }
343   }
344   PetscFunctionReturn(0);
345 }
346 
347 static PetscErrorCode morepoints(TAO_POUNDERS *mfqP)
348 {
349   /* Assumes mfqP->model_indices[0]  is minimum index
350    Finishes adding points to mfqP->model_indices (up to npmax)
351    Computes L,Z,M,N
352    np is actual number of points in model (should equal npmax?) */
353   PetscInt        point,i,j,offset;
354   PetscInt        reject;
355   PetscBLASInt    blasn=mfqP->n,blasnpmax=mfqP->npmax,blasnplus1=mfqP->n+1,info,blasnmax=mfqP->nmax,blasint,blasint2,blasnp,blasmaxmn;
356   const PetscReal *x;
357   PetscReal       normd;
358   PetscErrorCode  ierr;
359 
360   PetscFunctionBegin;
361   /* Initialize M,N */
362   for (i=0;i<mfqP->n+1;i++) {
363     ierr = VecGetArrayRead(mfqP->Xhist[mfqP->model_indices[i]],&x);CHKERRQ(ierr);
364     mfqP->M[(mfqP->n+1)*i] = 1.0;
365     for (j=0;j<mfqP->n;j++) {
366       mfqP->M[j+1+((mfqP->n+1)*i)] = (x[j]  - mfqP->xmin[j]) / mfqP->delta;
367     }
368     ierr = VecRestoreArrayRead(mfqP->Xhist[mfqP->model_indices[i]],&x);CHKERRQ(ierr);
369     ierr = phi2eval(&mfqP->M[1+((mfqP->n+1)*i)],mfqP->n,&mfqP->N[mfqP->n*(mfqP->n+1)/2 * i]);CHKERRQ(ierr);
370   }
371 
372   /* Now we add points until we have npmax starting with the most recent ones */
373   point = mfqP->nHist-1;
374   mfqP->nmodelpoints = mfqP->n+1;
375   while (mfqP->nmodelpoints < mfqP->npmax && point>=0) {
376     /* Reject any points already in the model */
377     reject = 0;
378     for (j=0;j<mfqP->n+1;j++) {
379       if (point == mfqP->model_indices[j]) {
380         reject = 1;
381         break;
382       }
383     }
384 
385     /* Reject if norm(d) >c2 */
386     if (!reject) {
387       ierr = VecCopy(mfqP->Xhist[point],mfqP->workxvec);CHKERRQ(ierr);
388       ierr = VecAXPY(mfqP->workxvec,-1.0,mfqP->Xhist[mfqP->minindex]);CHKERRQ(ierr);
389       ierr = VecNorm(mfqP->workxvec,NORM_2,&normd);CHKERRQ(ierr);
390       normd /= mfqP->delta;
391       if (normd > mfqP->c2) {
392         reject =1;
393       }
394     }
395     if (reject){
396       point--;
397       continue;
398     }
399 
400     ierr = VecGetArrayRead(mfqP->Xhist[point],&x);CHKERRQ(ierr);
401     mfqP->M[(mfqP->n+1)*mfqP->nmodelpoints] = 1.0;
402     for (j=0;j<mfqP->n;j++) {
403       mfqP->M[j+1+((mfqP->n+1)*mfqP->nmodelpoints)] = (x[j]  - mfqP->xmin[j]) / mfqP->delta;
404     }
405     ierr = VecRestoreArrayRead(mfqP->Xhist[point],&x);CHKERRQ(ierr);
406     ierr = phi2eval(&mfqP->M[1+(mfqP->n+1)*mfqP->nmodelpoints],mfqP->n,&mfqP->N[mfqP->n*(mfqP->n+1)/2 * (mfqP->nmodelpoints)]);CHKERRQ(ierr);
407 
408     /* Update QR factorization */
409     /* Copy M' to Q_tmp */
410     for (i=0;i<mfqP->n+1;i++) {
411       for (j=0;j<mfqP->npmax;j++) {
412         mfqP->Q_tmp[j+mfqP->npmax*i] = mfqP->M[i+(mfqP->n+1)*j];
413       }
414     }
415     blasnp = mfqP->nmodelpoints+1;
416     /* Q_tmp,R = qr(M') */
417     blasmaxmn=PetscMax(mfqP->m,mfqP->n+1);
418     PetscStackCallBLAS("LAPACKgeqrf",LAPACKgeqrf_(&blasnp,&blasnplus1,mfqP->Q_tmp,&blasnpmax,mfqP->tau_tmp,mfqP->mwork,&blasmaxmn,&info));
419     if (info != 0) SETERRQ1(PETSC_COMM_SELF,1,"LAPACK routine geqrf returned with value %d\n",info);
420 
421     /* Reject if min(svd(N*Q(:,n+2:np+1)) <= theta2 */
422     /* L = N*Qtmp */
423     blasint2 = mfqP->n * (mfqP->n+1) / 2;
424     /* Copy N to L_tmp */
425     for (i=0;i<mfqP->n*(mfqP->n+1)/2 * mfqP->npmax;i++) {
426       mfqP->L_tmp[i]= mfqP->N[i];
427     }
428     /* Copy L_save to L_tmp */
429 
430     /* L_tmp = N*Qtmp' */
431 #if defined(PETSC_MISSING_LAPACK_ORMQR)
432     SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"ORMQR - Lapack routine is unavailable.");
433 #else
434     PetscStackCallBLAS("LAPACKormqr",LAPACKormqr_("R","N",&blasint2,&blasnp,&blasnplus1,mfqP->Q_tmp,&blasnpmax,mfqP->tau_tmp,mfqP->L_tmp,&blasint2,mfqP->npmaxwork,&blasnmax,&info));
435     if (info != 0) SETERRQ1(PETSC_COMM_SELF,1,"LAPACK routine ormqr returned with value %d\n",info);
436 #endif
437 
438     /* Copy L_tmp to L_save */
439     for (i=0;i<mfqP->npmax * mfqP->n*(mfqP->n+1)/2;i++) {
440       mfqP->L_save[i] = mfqP->L_tmp[i];
441     }
442 
443     /* Get svd for L_tmp(:,n+2:np+1) (L_tmp is modified in process) */
444     blasint = mfqP->nmodelpoints - mfqP->n;
445     ierr = PetscFPTrapPush(PETSC_FP_TRAP_OFF);CHKERRQ(ierr);
446     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));
447     ierr = PetscFPTrapPop();CHKERRQ(ierr);
448     if (info != 0) SETERRQ1(PETSC_COMM_SELF,1,"LAPACK routine gesvd returned with value %d\n",info);
449 
450     if (mfqP->beta[PetscMin(blasint,blasint2)-1] > mfqP->theta2) {
451       /* accept point */
452       mfqP->model_indices[mfqP->nmodelpoints] = point;
453       /* Copy Q_tmp to Q */
454       for (i=0;i<mfqP->npmax* mfqP->npmax;i++) {
455         mfqP->Q[i] = mfqP->Q_tmp[i];
456       }
457       for (i=0;i<mfqP->npmax;i++){
458         mfqP->tau[i] = mfqP->tau_tmp[i];
459       }
460       mfqP->nmodelpoints++;
461       blasnp = mfqP->nmodelpoints;
462 
463       /* Copy L_save to L */
464       for (i=0;i<mfqP->npmax * mfqP->n*(mfqP->n+1)/2;i++) {
465         mfqP->L[i] = mfqP->L_save[i];
466       }
467     }
468     point--;
469   }
470 
471   blasnp = mfqP->nmodelpoints;
472   /* Copy Q(:,n+2:np) to Z */
473   /* First set Q_tmp to I */
474   for (i=0;i<mfqP->npmax*mfqP->npmax;i++) {
475     mfqP->Q_tmp[i] = 0.0;
476   }
477   for (i=0;i<mfqP->npmax;i++) {
478     mfqP->Q_tmp[i + mfqP->npmax*i] = 1.0;
479   }
480 
481   /* Q_tmp = I * Q */
482 #if defined(PETSC_MISSING_LAPACK_ORMQR)
483   SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"ORMQR - Lapack routine is unavailable.");
484 #else
485   PetscStackCallBLAS("LAPACKormqr",LAPACKormqr_("R","N",&blasnp,&blasnp,&blasnplus1,mfqP->Q,&blasnpmax,mfqP->tau,mfqP->Q_tmp,&blasnpmax,mfqP->npmaxwork,&blasnmax,&info));
486   if (info != 0) SETERRQ1(PETSC_COMM_SELF,1,"LAPACK routine ormqr returned with value %d\n",info);
487 #endif
488 
489   /* Copy Q_tmp(:,n+2:np) to Z) */
490   offset = mfqP->npmax * (mfqP->n+1);
491   for (i=offset;i<mfqP->npmax*mfqP->npmax;i++) {
492     mfqP->Z[i-offset] = mfqP->Q_tmp[i];
493   }
494 
495   if (mfqP->nmodelpoints == mfqP->n + 1) {
496     /* Set L to I_{n+1} */
497     for (i=0;i<mfqP->npmax * mfqP->n*(mfqP->n+1)/2;i++) {
498       mfqP->L[i] = 0.0;
499     }
500     for (i=0;i<mfqP->n;i++) {
501       mfqP->L[(mfqP->n*(mfqP->n+1)/2)*i + i] = 1.0;
502     }
503   }
504   PetscFunctionReturn(0);
505 }
506 
507 /* Only call from modelimprove, addpoint() needs ->Q_tmp and ->work to be set */
508 static PetscErrorCode addpoint(Tao tao, TAO_POUNDERS *mfqP, PetscInt index)
509 {
510   PetscErrorCode ierr;
511 
512   PetscFunctionBegin;
513   /* Create new vector in history: X[newidx] = X[mfqP->index] + delta*X[index]*/
514   ierr = VecDuplicate(mfqP->Xhist[0],&mfqP->Xhist[mfqP->nHist]);CHKERRQ(ierr);
515   ierr = VecSetValues(mfqP->Xhist[mfqP->nHist],mfqP->n,mfqP->indices,&mfqP->Q_tmp[index*mfqP->npmax],INSERT_VALUES);CHKERRQ(ierr);
516   ierr = VecAssemblyBegin(mfqP->Xhist[mfqP->nHist]);CHKERRQ(ierr);
517   ierr = VecAssemblyEnd(mfqP->Xhist[mfqP->nHist]);CHKERRQ(ierr);
518   ierr = VecAYPX(mfqP->Xhist[mfqP->nHist],mfqP->delta,mfqP->Xhist[mfqP->minindex]);CHKERRQ(ierr);
519 
520   /* Project into feasible region */
521   if (tao->XU && tao->XL) {
522     ierr = VecMedian(mfqP->Xhist[mfqP->nHist], tao->XL, tao->XU, mfqP->Xhist[mfqP->nHist]);CHKERRQ(ierr);
523   }
524 
525   /* Compute value of new vector */
526   ierr = VecDuplicate(mfqP->Fhist[0],&mfqP->Fhist[mfqP->nHist]);CHKERRQ(ierr);
527   CHKMEMQ;
528   ierr = pounders_feval(tao,mfqP->Xhist[mfqP->nHist],mfqP->Fhist[mfqP->nHist],&mfqP->Fres[mfqP->nHist]);CHKERRQ(ierr);
529 
530   /* Add new vector to model */
531   mfqP->model_indices[mfqP->nmodelpoints] = mfqP->nHist;
532   mfqP->nmodelpoints++;
533   mfqP->nHist++;
534   PetscFunctionReturn(0);
535 }
536 
537 static PetscErrorCode modelimprove(Tao tao, TAO_POUNDERS *mfqP, PetscInt addallpoints)
538 {
539   /* modeld = Q(:,np+1:n)' */
540   PetscErrorCode ierr;
541   PetscInt       i,j,minindex=0;
542   PetscReal      dp,half=0.5,one=1.0,minvalue=PETSC_INFINITY;
543   PetscBLASInt   blasn=mfqP->n,  blasnpmax = mfqP->npmax, blask,info;
544   PetscBLASInt   blas1=1,blasnmax = mfqP->nmax;
545 
546   blask = mfqP->nmodelpoints;
547   /* Qtmp = I(n x n) */
548   for (i=0;i<mfqP->n;i++) {
549     for (j=0;j<mfqP->n;j++) {
550       mfqP->Q_tmp[i + mfqP->npmax*j] = 0.0;
551     }
552   }
553   for (j=0;j<mfqP->n;j++) {
554     mfqP->Q_tmp[j + mfqP->npmax*j] = 1.0;
555   }
556 
557   /* Qtmp = Q * I */
558 #if defined(PETSC_MISSING_LAPACK_ORMQR)
559   SETERRQ(PetscObjectComm((PetscObject)tao),PETSC_ERR_SUP,"ORMQR - Lapack routine is unavailable.");
560 #else
561   PetscStackCallBLAS("LAPACKormqr",LAPACKormqr_("R","N",&blasn,&blasn,&blask,mfqP->Q,&blasnpmax,mfqP->tau, mfqP->Q_tmp, &blasnpmax, mfqP->npmaxwork,&blasnmax, &info));
562 #endif
563 
564   for (i=mfqP->nmodelpoints;i<mfqP->n;i++) {
565     dp = BLASdot_(&blasn,&mfqP->Q_tmp[i*mfqP->npmax],&blas1,mfqP->Gres,&blas1);
566     if (dp>0.0) { /* Model says use the other direction! */
567       for (j=0;j<mfqP->n;j++) {
568         mfqP->Q_tmp[i*mfqP->npmax+j] *= -1;
569       }
570     }
571     /* mfqP->work[i] = Cres+Modeld(i,:)*(Gres+.5*Hres*Modeld(i,:)') */
572     for (j=0;j<mfqP->n;j++) {
573       mfqP->work2[j] = mfqP->Gres[j];
574     }
575     PetscStackCallBLAS("BLASgemv",BLASgemv_("N",&blasn,&blasn,&half,mfqP->Hres,&blasn,&mfqP->Q_tmp[i*mfqP->npmax], &blas1, &one, mfqP->work2,&blas1));
576     mfqP->work[i] = BLASdot_(&blasn,&mfqP->Q_tmp[i*mfqP->npmax],&blas1,mfqP->work2,&blas1);
577     if (i==mfqP->nmodelpoints || mfqP->work[i] < minvalue) {
578       minindex=i;
579       minvalue = mfqP->work[i];
580     }
581     if (addallpoints != 0) {
582       ierr = addpoint(tao,mfqP,i);CHKERRQ(ierr);
583     }
584   }
585   if (!addallpoints) {
586     ierr = addpoint(tao,mfqP,minindex);CHKERRQ(ierr);
587   }
588   PetscFunctionReturn(0);
589 }
590 
591 static PetscErrorCode affpoints(TAO_POUNDERS *mfqP, PetscReal *xmin,PetscReal c)
592 {
593   PetscInt        i,j;
594   PetscBLASInt    blasm=mfqP->m,blasj,blask,blasn=mfqP->n,ione=1,info;
595   PetscBLASInt    blasnpmax = mfqP->npmax,blasmaxmn;
596   PetscReal       proj,normd;
597   const PetscReal *x;
598   PetscErrorCode  ierr;
599 
600   PetscFunctionBegin;
601   for (i=mfqP->nHist-1;i>=0;i--) {
602     ierr = VecGetArrayRead(mfqP->Xhist[i],&x);CHKERRQ(ierr);
603     for (j=0;j<mfqP->n;j++) {
604       mfqP->work[j] = (x[j] - xmin[j])/mfqP->delta;
605     }
606     ierr = VecRestoreArrayRead(mfqP->Xhist[i],&x);CHKERRQ(ierr);
607     PetscStackCallBLAS("BLAScopy",BLAScopy_(&blasn,mfqP->work,&ione,mfqP->work2,&ione));
608     normd = BLASnrm2_(&blasn,mfqP->work,&ione);
609     if (normd <= c) {
610       blasj=PetscMax((mfqP->n - mfqP->nmodelpoints),0);
611       if (!mfqP->q_is_I) {
612         /* project D onto null */
613         blask=(mfqP->nmodelpoints);
614 #if defined(PETSC_MISSING_LAPACK_ORMQR)
615         SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"ORMQR - Lapack routine is unavailable.");
616 #else
617         PetscStackCallBLAS("LAPACKormqr",LAPACKormqr_("R","N",&ione,&blasn,&blask,mfqP->Q,&blasnpmax,mfqP->tau,mfqP->work2,&ione,mfqP->mwork,&blasm,&info));
618         if (info < 0) SETERRQ1(PETSC_COMM_SELF,1,"ormqr returned value %d\n",info);
619 #endif
620       }
621       proj = BLASnrm2_(&blasj,&mfqP->work2[mfqP->nmodelpoints],&ione);
622 
623       if (proj >= mfqP->theta1) { /* add this index to model */
624         mfqP->model_indices[mfqP->nmodelpoints]=i;
625         mfqP->nmodelpoints++;
626         PetscStackCallBLAS("BLAScopy",BLAScopy_(&blasn,mfqP->work,&ione,&mfqP->Q_tmp[mfqP->npmax*(mfqP->nmodelpoints-1)],&ione));
627         blask=mfqP->npmax*(mfqP->nmodelpoints);
628         PetscStackCallBLAS("BLAScopy",BLAScopy_(&blask,mfqP->Q_tmp,&ione,mfqP->Q,&ione));
629         blask = mfqP->nmodelpoints;
630         blasmaxmn = PetscMax(mfqP->m,mfqP->n);
631         PetscStackCallBLAS("LAPACKgeqrf",LAPACKgeqrf_(&blasn,&blask,mfqP->Q,&blasnpmax,mfqP->tau,mfqP->mwork,&blasmaxmn,&info));
632         if (info < 0) SETERRQ1(PETSC_COMM_SELF,1,"geqrf returned value %d\n",info);
633         mfqP->q_is_I = 0;
634       }
635       if (mfqP->nmodelpoints == mfqP->n)  {
636         break;
637       }
638     }
639   }
640 
641   PetscFunctionReturn(0);
642 }
643 
644 static PetscErrorCode TaoSolve_POUNDERS(Tao tao)
645 {
646   TAO_POUNDERS       *mfqP = (TAO_POUNDERS *)tao->data;
647   PetscInt           i,ii,j,k,l;
648   PetscReal          step=1.0;
649   PetscInt           low,high;
650   PetscReal          minnorm;
651   PetscReal          *x,*f;
652   const PetscReal    *xmint,*fmin;
653   PetscReal          cres,deltaold;
654   PetscReal          gnorm;
655   PetscBLASInt       info,ione=1,iblas;
656   PetscBool          valid,same;
657   PetscReal          mdec, rho, normxsp;
658   PetscReal          one=1.0,zero=0.0,ratio;
659   PetscBLASInt       blasm,blasn,blasncopy,blasnpmax;
660   PetscErrorCode     ierr;
661   static PetscBool   set = PETSC_FALSE;
662 
663   /* n = # of parameters
664      m = dimension (components) of function  */
665   PetscFunctionBegin;
666   ierr = PetscCitationsRegister("@article{UNEDF0,\n"
667                                 "title = {Nuclear energy density optimization},\n"
668                                 "author = {Kortelainen, M.  and Lesinski, T.  and Mor\'e, J.  and Nazarewicz, W.\n"
669                                 "          and Sarich, J.  and Schunck, N.  and Stoitsov, M. V. and Wild, S. },\n"
670                                 "journal = {Phys. Rev. C},\n"
671                                 "volume = {82},\n"
672                                 "number = {2},\n"
673                                 "pages = {024313},\n"
674                                 "numpages = {18},\n"
675                                 "year = {2010},\n"
676                                 "month = {Aug},\n"
677                                 "doi = {10.1103/PhysRevC.82.024313}\n}\n",&set);CHKERRQ(ierr);
678   tao->niter=0;
679   if (tao->XL && tao->XU) {
680     /* Check x0 <= XU */
681     PetscReal val;
682 
683     ierr = VecCopy(tao->solution,mfqP->Xhist[0]);CHKERRQ(ierr);
684     ierr = VecAXPY(mfqP->Xhist[0],-1.0,tao->XU);CHKERRQ(ierr);
685     ierr = VecMax(mfqP->Xhist[0],NULL,&val);CHKERRQ(ierr);
686     if (val > 1e-10) SETERRQ(PetscObjectComm((PetscObject)tao),PETSC_ERR_ARG_OUTOFRANGE,"X0 > upper bound");
687 
688     /* Check x0 >= xl */
689     ierr = VecCopy(tao->XL,mfqP->Xhist[0]);CHKERRQ(ierr);
690     ierr = VecAXPY(mfqP->Xhist[0],-1.0,tao->solution);CHKERRQ(ierr);
691     ierr = VecMax(mfqP->Xhist[0],NULL,&val);CHKERRQ(ierr);
692     if (val > 1e-10) SETERRQ(PetscObjectComm((PetscObject)tao),PETSC_ERR_ARG_OUTOFRANGE,"X0 < lower bound");
693 
694     /* Check x0 + delta < XU  -- should be able to get around this eventually */
695 
696     ierr = VecSet(mfqP->Xhist[0],mfqP->delta);CHKERRQ(ierr);
697     ierr = VecAXPY(mfqP->Xhist[0],1.0,tao->solution);CHKERRQ(ierr);
698     ierr = VecAXPY(mfqP->Xhist[0],-1.0,tao->XU);CHKERRQ(ierr);
699     ierr = VecMax(mfqP->Xhist[0],NULL,&val);CHKERRQ(ierr);
700     if (val > 1e-10) SETERRQ(PetscObjectComm((PetscObject)tao),PETSC_ERR_ARG_OUTOFRANGE,"X0 + delta > upper bound");
701   }
702 
703   blasm = mfqP->m; blasn=mfqP->n; blasnpmax = mfqP->npmax;
704   for (i=0;i<mfqP->n*mfqP->n*mfqP->m;++i) mfqP->H[i]=0;
705 
706   ierr = VecCopy(tao->solution,mfqP->Xhist[0]);CHKERRQ(ierr);
707 
708   /* This provides enough information to approximate the gradient of the objective */
709   /* using a forward difference scheme. */
710 
711   ierr = PetscInfo1(tao,"Initialize simplex; delta = %10.9e\n",(double)mfqP->delta);CHKERRQ(ierr);
712   ierr = pounders_feval(tao,mfqP->Xhist[0],mfqP->Fhist[0],&mfqP->Fres[0]);CHKERRQ(ierr);
713   mfqP->minindex = 0;
714   minnorm = mfqP->Fres[0];
715 
716   ierr = VecGetOwnershipRange(mfqP->Xhist[0],&low,&high);CHKERRQ(ierr);
717   for (i=1;i<mfqP->n+1;++i) {
718     ierr = VecCopy(mfqP->Xhist[0],mfqP->Xhist[i]);CHKERRQ(ierr);
719 
720     if (i-1 >= low && i-1 < high) {
721       ierr = VecGetArray(mfqP->Xhist[i],&x);CHKERRQ(ierr);
722       x[i-1-low] += mfqP->delta;
723       ierr = VecRestoreArray(mfqP->Xhist[i],&x);CHKERRQ(ierr);
724     }
725     CHKMEMQ;
726     ierr = pounders_feval(tao,mfqP->Xhist[i],mfqP->Fhist[i],&mfqP->Fres[i]);CHKERRQ(ierr);
727     if (mfqP->Fres[i] < minnorm) {
728       mfqP->minindex = i;
729       minnorm = mfqP->Fres[i];
730     }
731   }
732   ierr = VecCopy(mfqP->Xhist[mfqP->minindex],tao->solution);CHKERRQ(ierr);
733   ierr = VecCopy(mfqP->Fhist[mfqP->minindex],tao->ls_res);CHKERRQ(ierr);
734   ierr = PetscInfo1(tao,"Finalize simplex; minnorm = %10.9e\n",(double)minnorm);CHKERRQ(ierr);
735 
736   /* Gather mpi vecs to one big local vec */
737 
738   /* Begin serial code */
739 
740   /* Disp[i] = Xi-xmin, i=1,..,mfqP->minindex-1,mfqP->minindex+1,..,n */
741   /* Fdiff[i] = (Fi-Fmin)', i=1,..,mfqP->minindex-1,mfqP->minindex+1,..,n */
742   /* (Column oriented for blas calls) */
743   ii=0;
744 
745   ierr = PetscInfo1(tao,"Build matrix: %D\n",(PetscInt)mfqP->size);CHKERRQ(ierr);
746   if (1 == mfqP->size) {
747     ierr = VecGetArrayRead(mfqP->Xhist[mfqP->minindex],&xmint);CHKERRQ(ierr);
748     for (i=0;i<mfqP->n;i++) mfqP->xmin[i] = xmint[i];
749     ierr = VecRestoreArrayRead(mfqP->Xhist[mfqP->minindex],&xmint);CHKERRQ(ierr);
750     ierr = VecGetArrayRead(mfqP->Fhist[mfqP->minindex],&fmin);CHKERRQ(ierr);
751     for (i=0;i<mfqP->n+1;i++) {
752       if (i == mfqP->minindex) continue;
753 
754       ierr = VecGetArray(mfqP->Xhist[i],&x);CHKERRQ(ierr);
755       for (j=0;j<mfqP->n;j++) {
756         mfqP->Disp[ii+mfqP->npmax*j] = (x[j] - mfqP->xmin[j])/mfqP->delta;
757       }
758       ierr = VecRestoreArray(mfqP->Xhist[i],&x);CHKERRQ(ierr);
759 
760       ierr = VecGetArray(mfqP->Fhist[i],&f);CHKERRQ(ierr);
761       for (j=0;j<mfqP->m;j++) {
762         mfqP->Fdiff[ii+mfqP->n*j] = f[j] - fmin[j];
763       }
764       ierr = VecRestoreArray(mfqP->Fhist[i],&f);CHKERRQ(ierr);
765 
766       mfqP->model_indices[ii++] = i;
767     }
768     for (j=0;j<mfqP->m;j++) {
769       mfqP->C[j] = fmin[j];
770     }
771     ierr = VecRestoreArrayRead(mfqP->Fhist[mfqP->minindex],&fmin);CHKERRQ(ierr);
772   } else {
773     ierr = VecSet(mfqP->localxmin,0);CHKERRQ(ierr);
774     ierr = VecScatterBegin(mfqP->scatterx,mfqP->Xhist[mfqP->minindex],mfqP->localxmin,INSERT_VALUES,SCATTER_FORWARD);CHKERRQ(ierr);
775     ierr = VecScatterEnd(mfqP->scatterx,mfqP->Xhist[mfqP->minindex],mfqP->localxmin,INSERT_VALUES,SCATTER_FORWARD);CHKERRQ(ierr);
776 
777     ierr = VecGetArrayRead(mfqP->localxmin,&xmint);CHKERRQ(ierr);
778     for (i=0;i<mfqP->n;i++) mfqP->xmin[i] = xmint[i];
779     ierr = VecRestoreArrayRead(mfqP->localxmin,&xmint);CHKERRQ(ierr);
780 
781     ierr = VecScatterBegin(mfqP->scatterf,mfqP->Fhist[mfqP->minindex],mfqP->localfmin,INSERT_VALUES,SCATTER_FORWARD);CHKERRQ(ierr);
782     ierr = VecScatterEnd(mfqP->scatterf,mfqP->Fhist[mfqP->minindex],mfqP->localfmin,INSERT_VALUES,SCATTER_FORWARD);CHKERRQ(ierr);
783     ierr = VecGetArrayRead(mfqP->localfmin,&fmin);CHKERRQ(ierr);
784     for (i=0;i<mfqP->n+1;i++) {
785       if (i == mfqP->minindex) continue;
786 
787       ierr = VecScatterBegin(mfqP->scatterx,mfqP->Xhist[ii],mfqP->localx,INSERT_VALUES, SCATTER_FORWARD);CHKERRQ(ierr);
788       ierr = VecScatterEnd(mfqP->scatterx,mfqP->Xhist[ii],mfqP->localx,INSERT_VALUES, SCATTER_FORWARD);CHKERRQ(ierr);
789       ierr = VecGetArray(mfqP->localx,&x);CHKERRQ(ierr);
790       for (j=0;j<mfqP->n;j++) {
791         mfqP->Disp[ii+mfqP->npmax*j] = (x[j] - mfqP->xmin[j])/mfqP->delta;
792       }
793       ierr = VecRestoreArray(mfqP->localx,&x);CHKERRQ(ierr);
794 
795       ierr = VecScatterBegin(mfqP->scatterf,mfqP->Fhist[ii],mfqP->localf,INSERT_VALUES, SCATTER_FORWARD);CHKERRQ(ierr);
796       ierr = VecScatterEnd(mfqP->scatterf,mfqP->Fhist[ii],mfqP->localf,INSERT_VALUES, SCATTER_FORWARD);CHKERRQ(ierr);
797       ierr = VecGetArray(mfqP->localf,&f);CHKERRQ(ierr);
798       for (j=0;j<mfqP->m;j++) {
799         mfqP->Fdiff[ii+mfqP->n*j] = f[j] - fmin[j];
800       }
801       ierr = VecRestoreArray(mfqP->localf,&f);CHKERRQ(ierr);
802 
803       mfqP->model_indices[ii++] = i;
804     }
805     for (j=0;j<mfqP->m;j++) {
806       mfqP->C[j] = fmin[j];
807     }
808     ierr = VecRestoreArrayRead(mfqP->localfmin,&fmin);CHKERRQ(ierr);
809   }
810 
811   /* Determine the initial quadratic models */
812   /* G = D(ModelIn,:) \ (F(ModelIn,1:m)-repmat(F(xkin,1:m),n,1)); */
813   /* D (nxn) Fdiff (nxm)  => G (nxm) */
814   blasncopy = blasn;
815   PetscStackCallBLAS("LAPACKgesv",LAPACKgesv_(&blasn,&blasm,mfqP->Disp,&blasnpmax,mfqP->iwork,mfqP->Fdiff,&blasncopy,&info));
816   ierr = PetscInfo1(tao,"Linear solve return: %D\n",(PetscInt)info);CHKERRQ(ierr);
817 
818   cres = minnorm;
819   ierr = pounders_update_res(tao);CHKERRQ(ierr);
820 
821   valid = PETSC_TRUE;
822 
823   ierr = VecSetValues(tao->gradient,mfqP->n,mfqP->indices,mfqP->Gres,INSERT_VALUES);CHKERRQ(ierr);
824   ierr = VecAssemblyBegin(tao->gradient);CHKERRQ(ierr);
825   ierr = VecAssemblyEnd(tao->gradient);CHKERRQ(ierr);
826   ierr = VecNorm(tao->gradient,NORM_2,&gnorm);CHKERRQ(ierr);
827   gnorm *= mfqP->delta;
828   ierr = VecCopy(mfqP->Xhist[mfqP->minindex],tao->solution);CHKERRQ(ierr);
829 
830   tao->reason = TAO_CONTINUE_ITERATING;
831   ierr = TaoLogConvergenceHistory(tao,minnorm,gnorm,0.0,tao->ksp_its);CHKERRQ(ierr);
832   ierr = TaoMonitor(tao,tao->niter,minnorm,gnorm,0.0,step);CHKERRQ(ierr);
833   ierr = (*tao->ops->convergencetest)(tao,tao->cnvP);CHKERRQ(ierr);
834 
835   mfqP->nHist = mfqP->n+1;
836   mfqP->nmodelpoints = mfqP->n+1;
837   ierr = PetscInfo1(tao,"Initial gradient: %20.19e\n",(double)gnorm);CHKERRQ(ierr);
838 
839   while (tao->reason == TAO_CONTINUE_ITERATING) {
840     PetscReal gnm = 1e-4;
841     /* Call general purpose update function */
842     if (tao->ops->update) {
843       ierr = (*tao->ops->update)(tao, tao->niter, tao->user_update);CHKERRQ(ierr);
844     }
845     tao->niter++;
846     /* Solve the subproblem min{Q(s): ||s|| <= 1.0} */
847     ierr = gqtwrap(tao,&gnm,&mdec);CHKERRQ(ierr);
848     /* Evaluate the function at the new point */
849 
850     for (i=0;i<mfqP->n;i++) {
851         mfqP->work[i] = mfqP->Xsubproblem[i]*mfqP->delta + mfqP->xmin[i];
852     }
853     ierr = VecDuplicate(tao->solution,&mfqP->Xhist[mfqP->nHist]);CHKERRQ(ierr);
854     ierr = VecDuplicate(tao->ls_res,&mfqP->Fhist[mfqP->nHist]);CHKERRQ(ierr);
855     ierr = VecSetValues(mfqP->Xhist[mfqP->nHist],mfqP->n,mfqP->indices,mfqP->work,INSERT_VALUES);CHKERRQ(ierr);
856     ierr = VecAssemblyBegin(mfqP->Xhist[mfqP->nHist]);CHKERRQ(ierr);
857     ierr = VecAssemblyEnd(mfqP->Xhist[mfqP->nHist]);CHKERRQ(ierr);
858 
859     ierr = pounders_feval(tao,mfqP->Xhist[mfqP->nHist],mfqP->Fhist[mfqP->nHist],&mfqP->Fres[mfqP->nHist]);CHKERRQ(ierr);
860 
861     rho = (mfqP->Fres[mfqP->minindex] - mfqP->Fres[mfqP->nHist]) / mdec;
862     mfqP->nHist++;
863 
864     /* Update the center */
865     if ((rho >= mfqP->eta1) || (rho > mfqP->eta0 && valid==PETSC_TRUE)) {
866       /* Update model to reflect new base point */
867       for (i=0;i<mfqP->n;i++) {
868         mfqP->work[i] = (mfqP->work[i] - mfqP->xmin[i])/mfqP->delta;
869       }
870       for (j=0;j<mfqP->m;j++) {
871         /* C(j) = C(j) + work*G(:,j) + .5*work*H(:,:,j)*work';
872          G(:,j) = G(:,j) + H(:,:,j)*work' */
873         for (k=0;k<mfqP->n;k++) {
874           mfqP->work2[k]=0.0;
875           for (l=0;l<mfqP->n;l++) {
876             mfqP->work2[k]+=mfqP->H[j + mfqP->m*(k + l*mfqP->n)]*mfqP->work[l];
877           }
878         }
879         for (i=0;i<mfqP->n;i++) {
880           mfqP->C[j]+=mfqP->work[i]*(mfqP->Fdiff[i + mfqP->n* j] + 0.5*mfqP->work2[i]);
881           mfqP->Fdiff[i+mfqP->n*j] +=mfqP-> work2[i];
882         }
883       }
884       /* Cres += work*Gres + .5*work*Hres*work';
885        Gres += Hres*work'; */
886 
887       PetscStackCallBLAS("BLASgemv",BLASgemv_("N",&blasn,&blasn,&one,mfqP->Hres,&blasn,mfqP->work,&ione,&zero,mfqP->work2,&ione));
888       for (i=0;i<mfqP->n;i++) {
889         cres += mfqP->work[i]*(mfqP->Gres[i]  + 0.5*mfqP->work2[i]);
890         mfqP->Gres[i] += mfqP->work2[i];
891       }
892       mfqP->minindex = mfqP->nHist-1;
893       minnorm = mfqP->Fres[mfqP->minindex];
894       ierr = VecCopy(mfqP->Fhist[mfqP->minindex],tao->ls_res);CHKERRQ(ierr);
895       /* Change current center */
896       ierr = VecGetArrayRead(mfqP->Xhist[mfqP->minindex],&xmint);CHKERRQ(ierr);
897       for (i=0;i<mfqP->n;i++) {
898         mfqP->xmin[i] = xmint[i];
899       }
900       ierr = VecRestoreArrayRead(mfqP->Xhist[mfqP->minindex],&xmint);CHKERRQ(ierr);
901     }
902 
903     /* Evaluate at a model-improving point if necessary */
904     if (valid == PETSC_FALSE) {
905       mfqP->q_is_I = 1;
906       mfqP->nmodelpoints = 0;
907       ierr = affpoints(mfqP,mfqP->xmin,mfqP->c1);CHKERRQ(ierr);
908       if (mfqP->nmodelpoints < mfqP->n) {
909         ierr = PetscInfo(tao,"Model not valid -- model-improving\n");CHKERRQ(ierr);
910         ierr = modelimprove(tao,mfqP,1);CHKERRQ(ierr);
911       }
912     }
913 
914     /* Update the trust region radius */
915     deltaold = mfqP->delta;
916     normxsp = 0;
917     for (i=0;i<mfqP->n;i++) {
918       normxsp += mfqP->Xsubproblem[i]*mfqP->Xsubproblem[i];
919     }
920     normxsp = PetscSqrtReal(normxsp);
921     if (rho >= mfqP->eta1 && normxsp > 0.5*mfqP->delta) {
922       mfqP->delta = PetscMin(mfqP->delta*mfqP->gamma1,mfqP->deltamax);
923     } else if (valid == PETSC_TRUE) {
924       mfqP->delta = PetscMax(mfqP->delta*mfqP->gamma0,mfqP->deltamin);
925     }
926 
927     /* Compute the next interpolation set */
928     mfqP->q_is_I = 1;
929     mfqP->nmodelpoints=0;
930     ierr = PetscInfo2(tao,"Affine Points: xmin = %20.19e, c1 = %20.19e\n",(double)*mfqP->xmin,(double)mfqP->c1);CHKERRQ(ierr);
931     ierr = affpoints(mfqP,mfqP->xmin,mfqP->c1);CHKERRQ(ierr);
932     if (mfqP->nmodelpoints == mfqP->n) {
933       valid = PETSC_TRUE;
934     } else {
935       valid = PETSC_FALSE;
936       ierr = PetscInfo2(tao,"Affine Points: xmin = %20.19e, c2 = %20.19e\n",(double)*mfqP->xmin,(double)mfqP->c2);CHKERRQ(ierr);
937       ierr = affpoints(mfqP,mfqP->xmin,mfqP->c2);CHKERRQ(ierr);
938       if (mfqP->n > mfqP->nmodelpoints) {
939         ierr = PetscInfo(tao,"Model not valid -- adding geometry points\n");CHKERRQ(ierr);
940         ierr = modelimprove(tao,mfqP,mfqP->n - mfqP->nmodelpoints);CHKERRQ(ierr);
941       }
942     }
943     for (i=mfqP->nmodelpoints;i>0;i--) {
944       mfqP->model_indices[i] = mfqP->model_indices[i-1];
945     }
946     mfqP->nmodelpoints++;
947     mfqP->model_indices[0] = mfqP->minindex;
948     ierr = morepoints(mfqP);CHKERRQ(ierr);
949     for (i=0;i<mfqP->nmodelpoints;i++) {
950       ierr = VecGetArray(mfqP->Xhist[mfqP->model_indices[i]],&x);CHKERRQ(ierr);
951       for (j=0;j<mfqP->n;j++) {
952         mfqP->Disp[i + mfqP->npmax*j] = (x[j]  - mfqP->xmin[j]) / deltaold;
953       }
954       ierr = VecRestoreArray(mfqP->Xhist[mfqP->model_indices[i]],&x);CHKERRQ(ierr);
955       ierr = VecGetArray(mfqP->Fhist[mfqP->model_indices[i]],&f);CHKERRQ(ierr);
956       for (j=0;j<mfqP->m;j++) {
957         for (k=0;k<mfqP->n;k++)  {
958           mfqP->work[k]=0.0;
959           for (l=0;l<mfqP->n;l++) {
960             mfqP->work[k] += mfqP->H[j + mfqP->m*(k + mfqP->n*l)] * mfqP->Disp[i + mfqP->npmax*l];
961           }
962         }
963         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];
964       }
965       ierr = VecRestoreArray(mfqP->Fhist[mfqP->model_indices[i]],&f);CHKERRQ(ierr);
966     }
967 
968     /* Update the quadratic model */
969     ierr = PetscInfo2(tao,"Get Quad, size: %D, points: %D\n",mfqP->n,mfqP->nmodelpoints);CHKERRQ(ierr);
970     ierr = getquadpounders(mfqP);CHKERRQ(ierr);
971     ierr = VecGetArrayRead(mfqP->Fhist[mfqP->minindex],&fmin);CHKERRQ(ierr);
972     PetscStackCallBLAS("BLAScopy",BLAScopy_(&blasm,fmin,&ione,mfqP->C,&ione));
973     /* G = G*(delta/deltaold) + Gdel */
974     ratio = mfqP->delta/deltaold;
975     iblas = blasm*blasn;
976     PetscStackCallBLAS("BLASscal",BLASscal_(&iblas,&ratio,mfqP->Fdiff,&ione));
977     PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&iblas,&one,mfqP->Gdel,&ione,mfqP->Fdiff,&ione));
978     /* H = H*(delta/deltaold)^2 + Hdel */
979     iblas = blasm*blasn*blasn;
980     ratio *= ratio;
981     PetscStackCallBLAS("BLASscal",BLASscal_(&iblas,&ratio,mfqP->H,&ione));
982     PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&iblas,&one,mfqP->Hdel,&ione,mfqP->H,&ione));
983 
984     /* Get residuals */
985     cres = mfqP->Fres[mfqP->minindex];
986     ierr = pounders_update_res(tao);CHKERRQ(ierr);
987 
988     /* Export solution and gradient residual to TAO */
989     ierr = VecCopy(mfqP->Xhist[mfqP->minindex],tao->solution);CHKERRQ(ierr);
990     ierr = VecSetValues(tao->gradient,mfqP->n,mfqP->indices,mfqP->Gres,INSERT_VALUES);CHKERRQ(ierr);
991     ierr = VecAssemblyBegin(tao->gradient);CHKERRQ(ierr);
992     ierr = VecAssemblyEnd(tao->gradient);CHKERRQ(ierr);
993     ierr = VecNorm(tao->gradient,NORM_2,&gnorm);CHKERRQ(ierr);
994     gnorm *= mfqP->delta;
995     /*  final criticality test */
996     ierr = TaoLogConvergenceHistory(tao,minnorm,gnorm,0.0,tao->ksp_its);CHKERRQ(ierr);
997     ierr = TaoMonitor(tao,tao->niter,minnorm,gnorm,0.0,step);CHKERRQ(ierr);
998     ierr = (*tao->ops->convergencetest)(tao,tao->cnvP);CHKERRQ(ierr);
999     /* test for repeated model */
1000     if (mfqP->nmodelpoints==mfqP->last_nmodelpoints) {
1001       same = PETSC_TRUE;
1002     } else {
1003       same = PETSC_FALSE;
1004     }
1005     for (i=0;i<mfqP->nmodelpoints;i++) {
1006       if (same) {
1007         if (mfqP->model_indices[i] == mfqP->last_model_indices[i]) {
1008           same = PETSC_TRUE;
1009         } else {
1010           same = PETSC_FALSE;
1011         }
1012       }
1013       mfqP->last_model_indices[i] = mfqP->model_indices[i];
1014     }
1015     mfqP->last_nmodelpoints = mfqP->nmodelpoints;
1016     if (same && mfqP->delta == deltaold) {
1017       ierr = PetscInfo(tao,"Identical model used in successive iterations\n");CHKERRQ(ierr);
1018       tao->reason = TAO_CONVERGED_STEPTOL;
1019     }
1020   }
1021   PetscFunctionReturn(0);
1022 }
1023 
1024 static PetscErrorCode TaoSetUp_POUNDERS(Tao tao)
1025 {
1026   TAO_POUNDERS   *mfqP = (TAO_POUNDERS*)tao->data;
1027   PetscInt       i,j;
1028   IS             isfloc,isfglob,isxloc,isxglob;
1029   PetscErrorCode ierr;
1030 
1031   PetscFunctionBegin;
1032   if (!tao->gradient) {ierr = VecDuplicate(tao->solution,&tao->gradient);CHKERRQ(ierr);  }
1033   if (!tao->stepdirection) {ierr = VecDuplicate(tao->solution,&tao->stepdirection);CHKERRQ(ierr);  }
1034   ierr = VecGetSize(tao->solution,&mfqP->n);CHKERRQ(ierr);
1035   ierr = VecGetSize(tao->ls_res,&mfqP->m);CHKERRQ(ierr);
1036   mfqP->c1 = PetscSqrtReal((PetscReal)mfqP->n);
1037   if (mfqP->npmax == PETSC_DEFAULT) {
1038     mfqP->npmax = 2*mfqP->n + 1;
1039   }
1040   mfqP->npmax = PetscMin((mfqP->n+1)*(mfqP->n+2)/2,mfqP->npmax);
1041   mfqP->npmax = PetscMax(mfqP->npmax, mfqP->n+2);
1042 
1043   ierr = PetscMalloc1(tao->max_funcs+100,&mfqP->Xhist);CHKERRQ(ierr);
1044   ierr = PetscMalloc1(tao->max_funcs+100,&mfqP->Fhist);CHKERRQ(ierr);
1045   for (i=0;i<mfqP->n+1;i++) {
1046     ierr = VecDuplicate(tao->solution,&mfqP->Xhist[i]);CHKERRQ(ierr);
1047     ierr = VecDuplicate(tao->ls_res,&mfqP->Fhist[i]);CHKERRQ(ierr);
1048   }
1049   ierr = VecDuplicate(tao->solution,&mfqP->workxvec);CHKERRQ(ierr);
1050   ierr = VecDuplicate(tao->ls_res,&mfqP->workfvec);CHKERRQ(ierr);
1051   mfqP->nHist = 0;
1052 
1053   ierr = PetscMalloc1(tao->max_funcs+100,&mfqP->Fres);CHKERRQ(ierr);
1054   ierr = PetscMalloc1(mfqP->npmax*mfqP->m,&mfqP->RES);CHKERRQ(ierr);
1055   ierr = PetscMalloc1(mfqP->n,&mfqP->work);CHKERRQ(ierr);
1056   ierr = PetscMalloc1(mfqP->n,&mfqP->work2);CHKERRQ(ierr);
1057   ierr = PetscMalloc1(mfqP->n,&mfqP->work3);CHKERRQ(ierr);
1058   ierr = PetscMalloc1(PetscMax(mfqP->m,mfqP->n+1),&mfqP->mwork);CHKERRQ(ierr);
1059   ierr = PetscMalloc1(mfqP->npmax - mfqP->n - 1,&mfqP->omega);CHKERRQ(ierr);
1060   ierr = PetscMalloc1(mfqP->n * (mfqP->n+1) / 2,&mfqP->beta);CHKERRQ(ierr);
1061   ierr = PetscMalloc1(mfqP->n + 1 ,&mfqP->alpha);CHKERRQ(ierr);
1062 
1063   ierr = PetscMalloc1(mfqP->n*mfqP->n*mfqP->m,&mfqP->H);CHKERRQ(ierr);
1064   ierr = PetscMalloc1(mfqP->npmax*mfqP->npmax,&mfqP->Q);CHKERRQ(ierr);
1065   ierr = PetscMalloc1(mfqP->npmax*mfqP->npmax,&mfqP->Q_tmp);CHKERRQ(ierr);
1066   ierr = PetscMalloc1(mfqP->n*(mfqP->n+1)/2*(mfqP->npmax),&mfqP->L);CHKERRQ(ierr);
1067   ierr = PetscMalloc1(mfqP->n*(mfqP->n+1)/2*(mfqP->npmax),&mfqP->L_tmp);CHKERRQ(ierr);
1068   ierr = PetscMalloc1(mfqP->n*(mfqP->n+1)/2*(mfqP->npmax),&mfqP->L_save);CHKERRQ(ierr);
1069   ierr = PetscMalloc1(mfqP->n*(mfqP->n+1)/2*(mfqP->npmax),&mfqP->N);CHKERRQ(ierr);
1070   ierr = PetscMalloc1(mfqP->npmax * (mfqP->n+1) ,&mfqP->M);CHKERRQ(ierr);
1071   ierr = PetscMalloc1(mfqP->npmax * (mfqP->npmax - mfqP->n - 1) , &mfqP->Z);CHKERRQ(ierr);
1072   ierr = PetscMalloc1(mfqP->npmax,&mfqP->tau);CHKERRQ(ierr);
1073   ierr = PetscMalloc1(mfqP->npmax,&mfqP->tau_tmp);CHKERRQ(ierr);
1074   mfqP->nmax = PetscMax(5*mfqP->npmax,mfqP->n*(mfqP->n+1)/2);
1075   ierr = PetscMalloc1(mfqP->nmax,&mfqP->npmaxwork);CHKERRQ(ierr);
1076   ierr = PetscMalloc1(mfqP->nmax,&mfqP->npmaxiwork);CHKERRQ(ierr);
1077   ierr = PetscMalloc1(mfqP->n,&mfqP->xmin);CHKERRQ(ierr);
1078   ierr = PetscMalloc1(mfqP->m,&mfqP->C);CHKERRQ(ierr);
1079   ierr = PetscMalloc1(mfqP->m*mfqP->n,&mfqP->Fdiff);CHKERRQ(ierr);
1080   ierr = PetscMalloc1(mfqP->npmax*mfqP->n,&mfqP->Disp);CHKERRQ(ierr);
1081   ierr = PetscMalloc1(mfqP->n,&mfqP->Gres);CHKERRQ(ierr);
1082   ierr = PetscMalloc1(mfqP->n*mfqP->n,&mfqP->Hres);CHKERRQ(ierr);
1083   ierr = PetscMalloc1(mfqP->n*mfqP->n,&mfqP->Gpoints);CHKERRQ(ierr);
1084   ierr = PetscMalloc1(mfqP->npmax,&mfqP->model_indices);CHKERRQ(ierr);
1085   ierr = PetscMalloc1(mfqP->npmax,&mfqP->last_model_indices);CHKERRQ(ierr);
1086   ierr = PetscMalloc1(mfqP->n,&mfqP->Xsubproblem);CHKERRQ(ierr);
1087   ierr = PetscMalloc1(mfqP->m*mfqP->n,&mfqP->Gdel);CHKERRQ(ierr);
1088   ierr = PetscMalloc1(mfqP->n*mfqP->n*mfqP->m, &mfqP->Hdel);CHKERRQ(ierr);
1089   ierr = PetscMalloc1(PetscMax(mfqP->m,mfqP->n),&mfqP->indices);CHKERRQ(ierr);
1090   ierr = PetscMalloc1(mfqP->n,&mfqP->iwork);CHKERRQ(ierr);
1091   ierr = PetscMalloc1(mfqP->m*mfqP->m,&mfqP->w);CHKERRQ(ierr);
1092   for (i=0;i<mfqP->m;i++) {
1093     for (j=0;j<mfqP->m;j++) {
1094       if (i==j) {
1095         mfqP->w[i+mfqP->m*j]=1.0;
1096       } else {
1097         mfqP->w[i+mfqP->m*j]=0.0;
1098       }
1099     }
1100   }
1101   for (i=0;i<PetscMax(mfqP->m,mfqP->n);i++) {
1102     mfqP->indices[i] = i;
1103   }
1104   ierr = MPI_Comm_size(((PetscObject)tao)->comm,&mfqP->size);CHKERRQ(ierr);
1105   if (mfqP->size > 1) {
1106     ierr = VecCreateSeq(PETSC_COMM_SELF,mfqP->n,&mfqP->localx);CHKERRQ(ierr);
1107     ierr = VecCreateSeq(PETSC_COMM_SELF,mfqP->n,&mfqP->localxmin);CHKERRQ(ierr);
1108     ierr = VecCreateSeq(PETSC_COMM_SELF,mfqP->m,&mfqP->localf);CHKERRQ(ierr);
1109     ierr = VecCreateSeq(PETSC_COMM_SELF,mfqP->m,&mfqP->localfmin);CHKERRQ(ierr);
1110     ierr = ISCreateStride(MPI_COMM_SELF,mfqP->n,0,1,&isxloc);CHKERRQ(ierr);
1111     ierr = ISCreateStride(MPI_COMM_SELF,mfqP->n,0,1,&isxglob);CHKERRQ(ierr);
1112     ierr = ISCreateStride(MPI_COMM_SELF,mfqP->m,0,1,&isfloc);CHKERRQ(ierr);
1113     ierr = ISCreateStride(MPI_COMM_SELF,mfqP->m,0,1,&isfglob);CHKERRQ(ierr);
1114 
1115 
1116     ierr = VecScatterCreate(tao->solution,isxglob,mfqP->localx,isxloc,&mfqP->scatterx);CHKERRQ(ierr);
1117     ierr = VecScatterCreate(tao->ls_res,isfglob,mfqP->localf,isfloc,&mfqP->scatterf);CHKERRQ(ierr);
1118 
1119     ierr = ISDestroy(&isxloc);CHKERRQ(ierr);
1120     ierr = ISDestroy(&isxglob);CHKERRQ(ierr);
1121     ierr = ISDestroy(&isfloc);CHKERRQ(ierr);
1122     ierr = ISDestroy(&isfglob);CHKERRQ(ierr);
1123   }
1124 
1125   if (!mfqP->usegqt) {
1126     KSP       ksp;
1127     PC        pc;
1128     ierr = VecCreateSeqWithArray(PETSC_COMM_SELF,mfqP->n,mfqP->n,mfqP->Xsubproblem,&mfqP->subx);CHKERRQ(ierr);
1129     ierr = VecCreateSeq(PETSC_COMM_SELF,mfqP->n,&mfqP->subxl);CHKERRQ(ierr);
1130     ierr = VecDuplicate(mfqP->subxl,&mfqP->subb);CHKERRQ(ierr);
1131     ierr = VecDuplicate(mfqP->subxl,&mfqP->subxu);CHKERRQ(ierr);
1132     ierr = VecDuplicate(mfqP->subxl,&mfqP->subpdel);CHKERRQ(ierr);
1133     ierr = VecDuplicate(mfqP->subxl,&mfqP->subndel);CHKERRQ(ierr);
1134     ierr = TaoCreate(PETSC_COMM_SELF,&mfqP->subtao);CHKERRQ(ierr);
1135     ierr = PetscObjectIncrementTabLevel((PetscObject)mfqP->subtao, (PetscObject)tao, 1);CHKERRQ(ierr);
1136     ierr = TaoSetType(mfqP->subtao,TAOBNTR);CHKERRQ(ierr);
1137     ierr = TaoSetOptionsPrefix(mfqP->subtao,"pounders_subsolver_");CHKERRQ(ierr);
1138     ierr = TaoSetInitialVector(mfqP->subtao,mfqP->subx);CHKERRQ(ierr);
1139     ierr = TaoSetObjectiveAndGradientRoutine(mfqP->subtao,pounders_fg,(void*)mfqP);CHKERRQ(ierr);
1140     ierr = TaoSetMaximumIterations(mfqP->subtao,mfqP->gqt_maxits);CHKERRQ(ierr);
1141     ierr = TaoSetFromOptions(mfqP->subtao);CHKERRQ(ierr);
1142     ierr = TaoGetKSP(mfqP->subtao,&ksp);CHKERRQ(ierr);
1143     if (ksp) {
1144       ierr = KSPGetPC(ksp,&pc);CHKERRQ(ierr);
1145       ierr = PCSetType(pc,PCNONE);CHKERRQ(ierr);
1146     }
1147     ierr = TaoSetVariableBounds(mfqP->subtao,mfqP->subxl,mfqP->subxu);CHKERRQ(ierr);
1148     ierr = MatCreateSeqDense(PETSC_COMM_SELF,mfqP->n,mfqP->n,mfqP->Hres,&mfqP->subH);CHKERRQ(ierr);
1149     ierr = TaoSetHessianRoutine(mfqP->subtao,mfqP->subH,mfqP->subH,pounders_h,(void*)mfqP);CHKERRQ(ierr);
1150   }
1151   PetscFunctionReturn(0);
1152 }
1153 
1154 static PetscErrorCode TaoDestroy_POUNDERS(Tao tao)
1155 {
1156   TAO_POUNDERS   *mfqP = (TAO_POUNDERS*)tao->data;
1157   PetscInt       i;
1158   PetscErrorCode ierr;
1159 
1160   PetscFunctionBegin;
1161   if (!mfqP->usegqt) {
1162     ierr = TaoDestroy(&mfqP->subtao);CHKERRQ(ierr);
1163     ierr = VecDestroy(&mfqP->subx);CHKERRQ(ierr);
1164     ierr = VecDestroy(&mfqP->subxl);CHKERRQ(ierr);
1165     ierr = VecDestroy(&mfqP->subxu);CHKERRQ(ierr);
1166     ierr = VecDestroy(&mfqP->subb);CHKERRQ(ierr);
1167     ierr = VecDestroy(&mfqP->subpdel);CHKERRQ(ierr);
1168     ierr = VecDestroy(&mfqP->subndel);CHKERRQ(ierr);
1169     ierr = MatDestroy(&mfqP->subH);CHKERRQ(ierr);
1170   }
1171   ierr = PetscFree(mfqP->Fres);CHKERRQ(ierr);
1172   ierr = PetscFree(mfqP->RES);CHKERRQ(ierr);
1173   ierr = PetscFree(mfqP->work);CHKERRQ(ierr);
1174   ierr = PetscFree(mfqP->work2);CHKERRQ(ierr);
1175   ierr = PetscFree(mfqP->work3);CHKERRQ(ierr);
1176   ierr = PetscFree(mfqP->mwork);CHKERRQ(ierr);
1177   ierr = PetscFree(mfqP->omega);CHKERRQ(ierr);
1178   ierr = PetscFree(mfqP->beta);CHKERRQ(ierr);
1179   ierr = PetscFree(mfqP->alpha);CHKERRQ(ierr);
1180   ierr = PetscFree(mfqP->H);CHKERRQ(ierr);
1181   ierr = PetscFree(mfqP->Q);CHKERRQ(ierr);
1182   ierr = PetscFree(mfqP->Q_tmp);CHKERRQ(ierr);
1183   ierr = PetscFree(mfqP->L);CHKERRQ(ierr);
1184   ierr = PetscFree(mfqP->L_tmp);CHKERRQ(ierr);
1185   ierr = PetscFree(mfqP->L_save);CHKERRQ(ierr);
1186   ierr = PetscFree(mfqP->N);CHKERRQ(ierr);
1187   ierr = PetscFree(mfqP->M);CHKERRQ(ierr);
1188   ierr = PetscFree(mfqP->Z);CHKERRQ(ierr);
1189   ierr = PetscFree(mfqP->tau);CHKERRQ(ierr);
1190   ierr = PetscFree(mfqP->tau_tmp);CHKERRQ(ierr);
1191   ierr = PetscFree(mfqP->npmaxwork);CHKERRQ(ierr);
1192   ierr = PetscFree(mfqP->npmaxiwork);CHKERRQ(ierr);
1193   ierr = PetscFree(mfqP->xmin);CHKERRQ(ierr);
1194   ierr = PetscFree(mfqP->C);CHKERRQ(ierr);
1195   ierr = PetscFree(mfqP->Fdiff);CHKERRQ(ierr);
1196   ierr = PetscFree(mfqP->Disp);CHKERRQ(ierr);
1197   ierr = PetscFree(mfqP->Gres);CHKERRQ(ierr);
1198   ierr = PetscFree(mfqP->Hres);CHKERRQ(ierr);
1199   ierr = PetscFree(mfqP->Gpoints);CHKERRQ(ierr);
1200   ierr = PetscFree(mfqP->model_indices);CHKERRQ(ierr);
1201   ierr = PetscFree(mfqP->last_model_indices);CHKERRQ(ierr);
1202   ierr = PetscFree(mfqP->Xsubproblem);CHKERRQ(ierr);
1203   ierr = PetscFree(mfqP->Gdel);CHKERRQ(ierr);
1204   ierr = PetscFree(mfqP->Hdel);CHKERRQ(ierr);
1205   ierr = PetscFree(mfqP->indices);CHKERRQ(ierr);
1206   ierr = PetscFree(mfqP->iwork);CHKERRQ(ierr);
1207   ierr = PetscFree(mfqP->w);CHKERRQ(ierr);
1208   for (i=0;i<mfqP->nHist;i++) {
1209     ierr = VecDestroy(&mfqP->Xhist[i]);CHKERRQ(ierr);
1210     ierr = VecDestroy(&mfqP->Fhist[i]);CHKERRQ(ierr);
1211   }
1212   ierr = VecDestroy(&mfqP->workxvec);CHKERRQ(ierr);
1213   ierr = VecDestroy(&mfqP->workfvec);CHKERRQ(ierr);
1214   ierr = PetscFree(mfqP->Xhist);CHKERRQ(ierr);
1215   ierr = PetscFree(mfqP->Fhist);CHKERRQ(ierr);
1216 
1217   if (mfqP->size > 1) {
1218     ierr = VecDestroy(&mfqP->localx);CHKERRQ(ierr);
1219     ierr = VecDestroy(&mfqP->localxmin);CHKERRQ(ierr);
1220     ierr = VecDestroy(&mfqP->localf);CHKERRQ(ierr);
1221     ierr = VecDestroy(&mfqP->localfmin);CHKERRQ(ierr);
1222   }
1223   ierr = PetscFree(tao->data);CHKERRQ(ierr);
1224   PetscFunctionReturn(0);
1225 }
1226 
1227 static PetscErrorCode TaoSetFromOptions_POUNDERS(PetscOptionItems *PetscOptionsObject,Tao tao)
1228 {
1229   TAO_POUNDERS   *mfqP = (TAO_POUNDERS*)tao->data;
1230   PetscErrorCode ierr;
1231 
1232   PetscFunctionBegin;
1233   ierr = PetscOptionsHead(PetscOptionsObject,"POUNDERS method for least-squares optimization");CHKERRQ(ierr);
1234   ierr = PetscOptionsReal("-tao_pounders_delta","initial delta","",mfqP->delta,&mfqP->delta0,NULL);CHKERRQ(ierr);
1235   mfqP->delta = mfqP->delta0;
1236   ierr = PetscOptionsInt("-tao_pounders_npmax","max number of points in model","",mfqP->npmax,&mfqP->npmax,NULL);CHKERRQ(ierr);
1237   ierr = PetscOptionsBool("-tao_pounders_gqt","use gqt algorithm for subproblem","",mfqP->usegqt,&mfqP->usegqt,NULL);CHKERRQ(ierr);
1238   ierr = PetscOptionsTail();CHKERRQ(ierr);
1239   PetscFunctionReturn(0);
1240 }
1241 
1242 static PetscErrorCode TaoView_POUNDERS(Tao tao, PetscViewer viewer)
1243 {
1244   TAO_POUNDERS   *mfqP = (TAO_POUNDERS *)tao->data;
1245   PetscBool      isascii;
1246   PetscErrorCode ierr;
1247 
1248   PetscFunctionBegin;
1249   ierr = PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&isascii);CHKERRQ(ierr);
1250   if (isascii) {
1251     ierr = PetscViewerASCIIPrintf(viewer, "initial delta: %g\n",(double)mfqP->delta0);CHKERRQ(ierr);
1252     ierr = PetscViewerASCIIPrintf(viewer, "final delta: %g\n",(double)mfqP->delta);CHKERRQ(ierr);
1253     ierr = PetscViewerASCIIPrintf(viewer, "model points: %D\n",mfqP->nmodelpoints);CHKERRQ(ierr);
1254     if (mfqP->usegqt) {
1255       ierr = PetscViewerASCIIPrintf(viewer, "subproblem solver: gqt\n");CHKERRQ(ierr);
1256     } else {
1257       ierr = TaoView(mfqP->subtao, viewer);CHKERRQ(ierr);
1258     }
1259   }
1260   PetscFunctionReturn(0);
1261 }
1262 /*MC
1263   TAOPOUNDERS - POUNDERS derivate-free model-based algorithm for nonlinear least squares
1264 
1265   Options Database Keys:
1266 + -tao_pounders_delta - initial step length
1267 . -tao_pounders_npmax - maximum number of points in model
1268 - -tao_pounders_gqt - use gqt algorithm for subproblem instead of TRON
1269 
1270   Level: beginner
1271 
1272 M*/
1273 
1274 PETSC_EXTERN PetscErrorCode TaoCreate_POUNDERS(Tao tao)
1275 {
1276   TAO_POUNDERS   *mfqP = (TAO_POUNDERS*)tao->data;
1277   PetscErrorCode ierr;
1278 
1279   PetscFunctionBegin;
1280   tao->ops->setup = TaoSetUp_POUNDERS;
1281   tao->ops->solve = TaoSolve_POUNDERS;
1282   tao->ops->view = TaoView_POUNDERS;
1283   tao->ops->setfromoptions = TaoSetFromOptions_POUNDERS;
1284   tao->ops->destroy = TaoDestroy_POUNDERS;
1285 
1286   ierr = PetscNewLog(tao,&mfqP);CHKERRQ(ierr);
1287   tao->data = (void*)mfqP;
1288   /* Override default settings (unless already changed) */
1289   if (!tao->max_it_changed) tao->max_it = 2000;
1290   if (!tao->max_funcs_changed) tao->max_funcs = 4000;
1291   mfqP->npmax = PETSC_DEFAULT;
1292   mfqP->delta0 = 0.1;
1293   mfqP->delta = 0.1;
1294   mfqP->deltamax=1e3;
1295   mfqP->deltamin=1e-6;
1296   mfqP->c2 = 10.0;
1297   mfqP->theta1=1e-5;
1298   mfqP->theta2=1e-4;
1299   mfqP->gamma0=0.5;
1300   mfqP->gamma1=2.0;
1301   mfqP->eta0 = 0.0;
1302   mfqP->eta1 = 0.1;
1303   mfqP->usegqt = PETSC_FALSE;
1304   mfqP->gqt_rtol = 0.001;
1305   mfqP->gqt_maxits = 50;
1306   mfqP->workxvec = 0;
1307   PetscFunctionReturn(0);
1308 }
1309