Actual source code: pipefgmres.c

  1: /*
  2:   Contributed by Patrick Sanan and Sascha M. Schnepp
  3: */

  5: #include <../src/ksp/ksp/impls/gmres/pipefgmres/pipefgmresimpl.h>

  7: static PetscBool  cited = PETSC_FALSE;
  8: static const char citation[] =
  9:   "@article{SSM2016,\n"
 10:   "  author = {P. Sanan and S.M. Schnepp and D.A. May},\n"
 11:   "  title = {Pipelined, Flexible Krylov Subspace Methods},\n"
 12:   "  journal = {SIAM Journal on Scientific Computing},\n"
 13:   "  volume = {38},\n"
 14:   "  number = {5},\n"
 15:   "  pages = {C441-C470},\n"
 16:   "  year = {2016},\n"
 17:   "  doi = {10.1137/15M1049130},\n"
 18:   "  URL = {http://dx.doi.org/10.1137/15M1049130},\n"
 19:   "  eprint = {http://dx.doi.org/10.1137/15M1049130}\n"
 20:   "}\n";

 22: #define PIPEFGMRES_DELTA_DIRECTIONS 10
 23: #define PIPEFGMRES_DEFAULT_MAXK     30

 25: static PetscErrorCode KSPPIPEFGMRESGetNewVectors(KSP,PetscInt);
 26: static PetscErrorCode KSPPIPEFGMRESUpdateHessenberg(KSP,PetscInt,PetscBool*,PetscReal*);
 27: static PetscErrorCode KSPPIPEFGMRESBuildSoln(PetscScalar*,Vec,Vec,KSP,PetscInt);
 28: extern PetscErrorCode KSPReset_PIPEFGMRES(KSP);

 30: /*

 32:     KSPSetUp_PIPEFGMRES - Sets up the workspace needed by pipefgmres.

 34:     This is called once, usually automatically by KSPSolve() or KSPSetUp(),
 35:     but can be called directly by KSPSetUp().

 37: */
 38: static PetscErrorCode KSPSetUp_PIPEFGMRES(KSP ksp)
 39: {
 41:   PetscInt       k;
 42:   KSP_PIPEFGMRES *pipefgmres = (KSP_PIPEFGMRES*)ksp->data;
 43:   const PetscInt max_k = pipefgmres->max_k;

 46:   KSPSetUp_GMRES(ksp);

 48:   PetscMalloc1((VEC_OFFSET+max_k),&pipefgmres->prevecs);
 49:   PetscMalloc1((VEC_OFFSET+max_k),&pipefgmres->prevecs_user_work);
 50:   PetscLogObjectMemory((PetscObject)ksp,(VEC_OFFSET+max_k)*(2*sizeof(void*)));

 52:   KSPCreateVecs(ksp,pipefgmres->vv_allocated,&pipefgmres->prevecs_user_work[0],0,NULL);
 53:   PetscLogObjectParents(ksp,pipefgmres->vv_allocated,pipefgmres->prevecs_user_work[0]);
 54:   for (k=0; k < pipefgmres->vv_allocated; k++) {
 55:     pipefgmres->prevecs[k] = pipefgmres->prevecs_user_work[0][k];
 56:   }

 58:   PetscMalloc1((VEC_OFFSET+max_k),&pipefgmres->zvecs);
 59:   PetscMalloc1((VEC_OFFSET+max_k),&pipefgmres->zvecs_user_work);
 60:   PetscLogObjectMemory((PetscObject)ksp,(VEC_OFFSET+max_k)*(2*sizeof(void*)));

 62:   PetscMalloc1((VEC_OFFSET+max_k),&pipefgmres->redux);
 63:   PetscLogObjectMemory((PetscObject)ksp,(VEC_OFFSET+max_k)*(sizeof(void*)));

 65:   KSPCreateVecs(ksp,pipefgmres->vv_allocated,&pipefgmres->zvecs_user_work[0],0,NULL);
 66:   PetscLogObjectParents(ksp,pipefgmres->vv_allocated,pipefgmres->zvecs_user_work[0]);
 67:   for (k=0; k < pipefgmres->vv_allocated; k++) {
 68:     pipefgmres->zvecs[k] = pipefgmres->zvecs_user_work[0][k];
 69:   }

 71:   return(0);
 72: }

 74: /*

 76:     KSPPIPEFGMRESCycle - Run pipefgmres, possibly with restart.  Return residual
 77:                   history if requested.

 79:     input parameters:
 80: .        pipefgmres  - structure containing parameters and work areas

 82:     output parameters:
 83: .        itcount - number of iterations used.  If null, ignored.
 84: .        converged - 0 if not converged

 86:     Notes:
 87:     On entry, the value in vector VEC_VV(0) should be
 88:     the initial residual.

 90: */
 91: static PetscErrorCode KSPPIPEFGMRESCycle(PetscInt *itcount,KSP ksp)
 92: {
 93:   KSP_PIPEFGMRES *pipefgmres = (KSP_PIPEFGMRES*)(ksp->data);
 94:   PetscReal      res_norm;
 95:   PetscReal      hapbnd,tt;
 96:   PetscScalar    *hh,*hes,*lhh,shift = pipefgmres->shift;
 97:   PetscBool      hapend = PETSC_FALSE;  /* indicates happy breakdown ending */
 99:   PetscInt       loc_it;                /* local count of # of dir. in Krylov space */
100:   PetscInt       max_k = pipefgmres->max_k; /* max # of directions Krylov space */
101:   PetscInt       i,j,k;
102:   Mat            Amat,Pmat;
103:   Vec            Q,W; /* Pipelining vectors */
104:   Vec            *redux = pipefgmres->redux; /* workspace for single reduction */

107:   if (itcount) *itcount = 0;

109:   /* Assign simpler names to these vectors, allocated as pipelining workspace */
110:   Q = VEC_Q;
111:   W = VEC_W;

113:   /* Allocate memory for orthogonalization work (freed in the GMRES Destroy routine)*/
114:   /* Note that we add an extra value here to allow for a single reduction */
115:   if (!pipefgmres->orthogwork) { PetscMalloc1(pipefgmres->max_k + 2 ,&pipefgmres->orthogwork);
116:   }
117:   lhh = pipefgmres->orthogwork;

119:   /* Number of pseudo iterations since last restart is the number
120:      of prestart directions */
121:   loc_it = 0;

123:   /* note: (pipefgmres->it) is always set one less than (loc_it) It is used in
124:      KSPBUILDSolution_PIPEFGMRES, where it is passed to KSPPIPEFGMRESBuildSoln.
125:      Note that when KSPPIPEFGMRESBuildSoln is called from this function,
126:      (loc_it -1) is passed, so the two are equivalent */
127:   pipefgmres->it = (loc_it - 1);

129:   /* initial residual is in VEC_VV(0)  - compute its norm*/
130:   VecNorm(VEC_VV(0),NORM_2,&res_norm);

132:   /* first entry in right-hand-side of hessenberg system is just
133:      the initial residual norm */
134:   *RS(0) = res_norm;

136:   PetscObjectSAWsTakeAccess((PetscObject)ksp);
137:   if (ksp->normtype != KSP_NORM_NONE) ksp->rnorm = res_norm;
138:   else ksp->rnorm = 0;
139:   PetscObjectSAWsGrantAccess((PetscObject)ksp);
140:   KSPLogResidualHistory(ksp,ksp->rnorm);
141:   KSPMonitor(ksp,ksp->its,ksp->rnorm);

143:   /* check for the convergence - maybe the current guess is good enough */
144:   (*ksp->converged)(ksp,ksp->its,ksp->rnorm,&ksp->reason,ksp->cnvP);
145:   if (ksp->reason) {
146:     if (itcount) *itcount = 0;
147:     return(0);
148:   }

150:   /* scale VEC_VV (the initial residual) */
151:   VecScale(VEC_VV(0),1.0/res_norm);

153:   /* Fill the pipeline */
154:   KSP_PCApply(ksp,VEC_VV(loc_it),PREVEC(loc_it));
155:   PCGetOperators(ksp->pc,&Amat,&Pmat);
156:   KSP_MatMult(ksp,Amat,PREVEC(loc_it),ZVEC(loc_it));
157:   VecAXPY(ZVEC(loc_it),-shift,VEC_VV(loc_it)); /* Note shift */

159:   /* MAIN ITERATION LOOP BEGINNING*/
160:   /* keep iterating until we have converged OR generated the max number
161:      of directions OR reached the max number of iterations for the method */
162:   while (!ksp->reason && loc_it < max_k && ksp->its < ksp->max_it) {
163:     if (loc_it) {
164:       KSPLogResidualHistory(ksp,res_norm);
165:       KSPMonitor(ksp,ksp->its,res_norm);
166:     }
167:     pipefgmres->it = (loc_it - 1);

169:     /* see if more space is needed for work vectors */
170:     if (pipefgmres->vv_allocated <= loc_it + VEC_OFFSET + 1) {
171:       KSPPIPEFGMRESGetNewVectors(ksp,loc_it+1);
172:       /* (loc_it+1) is passed in as number of the first vector that should
173:          be allocated */
174:     }

176:     /* Note that these inner products are with "Z" now, so
177:        in particular, lhh[loc_it] is the 'barred' or 'shifted' value,
178:        not the value from the equivalent FGMRES run (even in exact arithmetic)
179:        That is, the H we need for the Arnoldi relation is different from the
180:        coefficients we use in the orthogonalization process,because of the shift */

182:     /* Do some local twiddling to allow for a single reduction */
183:     for (i=0;i<loc_it+1;i++) {
184:       redux[i] = VEC_VV(i);
185:     }
186:     redux[loc_it+1] = ZVEC(loc_it);

188:     /* note the extra dot product which ends up in lh[loc_it+1], which computes ||z||^2 */
189:     VecMDotBegin(ZVEC(loc_it),loc_it+2,redux,lhh);

191:     /* Start the split reduction (This actually calls the MPI_Iallreduce, otherwise, the reduction is simply delayed until the "end" call)*/
192:     PetscCommSplitReductionBegin(PetscObjectComm((PetscObject)ZVEC(loc_it)));

194:     /* The work to be overlapped with the inner products follows.
195:        This is application of the preconditioner and the operator
196:        to compute intermediate quantites which will be combined (locally)
197:        with the results of the inner products.
198:        */
199:     KSP_PCApply(ksp,ZVEC(loc_it),Q);
200:     PCGetOperators(ksp->pc,&Amat,&Pmat);
201:     KSP_MatMult(ksp,Amat,Q,W);

203:     /* Compute inner products of the new direction with previous directions,
204:        and the norm of the to-be-orthogonalized direction "Z".
205:        This information is enough to build the required entries
206:        of H. The inner product with VEC_VV(it_loc) is
207:        *different* than in the standard FGMRES and need to be dealt with specially.
208:        That is, for standard FGMRES the orthogonalization coefficients are the same
209:        as the coefficients used in the Arnoldi relation to reconstruct, but here this
210:        is not true (albeit only for the one entry of H which we "unshift" below. */

212:     /* Finish the dot product, retrieving the extra entry */
213:     VecMDotEnd(ZVEC(loc_it),loc_it+2,redux,lhh);
214:     tt = PetscRealPart(lhh[loc_it+1]);

216:     /* Hessenberg entries, and entries for (naive) classical Graham-Schmidt
217:       Note that the Hessenberg entries require a shift, as these are for the
218:       relation AU = VH, which is wrt unshifted basis vectors */
219:     hh = HH(0,loc_it); hes=HES(0,loc_it);
220:     for (j=0; j<loc_it; j++) {
221:       hh[j]  = lhh[j];
222:       hes[j] = lhh[j];
223:     }
224:     hh[loc_it]  = lhh[loc_it] + shift;
225:     hes[loc_it] = lhh[loc_it] + shift;

227:     /* we delay applying the shift here */
228:     for (j=0; j<=loc_it; j++) {
229:       lhh[j]        = -lhh[j]; /* flip sign */
230:     }

232:     /* Compute the norm of the un-normalized new direction using the rearranged formula
233:        Note that these are shifted ("barred") quantities */
234:     for (k=0;k<=loc_it;k++) tt -= ((PetscReal)(PetscAbsScalar(lhh[k]) * PetscAbsScalar(lhh[k])));
235:     /* On AVX512 this is accumulating roundoff errors for eg: tt=-2.22045e-16 */
236:     if ((tt < 0.0) && tt > -PETSC_SMALL) tt = 0.0 ;
237:     if (tt < 0.0) {
238:       /* If we detect square root breakdown in the norm, we must restart the algorithm.
239:          Here this means we simply break the current loop and reconstruct the solution
240:          using the basis we have computed thus far. Note that by breaking immediately,
241:          we do not update the iteration count, so computation done in this iteration
242:          should be disregarded.
243:          */
244:       PetscInfo2(ksp,"Restart due to square root breakdown at it = %D, tt=%g\n",ksp->its,(double)tt);
245:       break;
246:     } else {
247:       tt = PetscSqrtReal(tt);
248:     }

250:     /* new entry in hessenburg is the 2-norm of our new direction */
251:     hh[loc_it+1]  = tt;
252:     hes[loc_it+1] = tt;

254:     /* The recurred computation for the new direction
255:        The division by tt is delayed to the happy breakdown check later
256:        Note placement BEFORE the unshift
257:        */
258:     VecCopy(ZVEC(loc_it),VEC_VV(loc_it+1));
259:     VecMAXPY(VEC_VV(loc_it+1),loc_it+1,lhh,&VEC_VV(0));
260:     /* (VEC_VV(loc_it+1) is not normalized yet) */

262:     /* The recurred computation for the preconditioned vector (u) */
263:     VecCopy(Q,PREVEC(loc_it+1));
264:     VecMAXPY(PREVEC(loc_it+1),loc_it+1,lhh,&PREVEC(0));
265:     VecScale(PREVEC(loc_it+1),1.0/tt);

267:     /* Unshift an entry in the GS coefficients ("removing the bar") */
268:     lhh[loc_it]         -= shift;

270:     /* The recurred computation for z (Au)
271:        Note placement AFTER the "unshift" */
272:     VecCopy(W,ZVEC(loc_it+1));
273:     VecMAXPY(ZVEC(loc_it+1),loc_it+1,lhh,&ZVEC(0));
274:     VecScale(ZVEC(loc_it+1),1.0/tt);

276:     /* Happy Breakdown Check */
277:     hapbnd = PetscAbsScalar((tt) / *RS(loc_it));
278:     /* RS(loc_it) contains the res_norm from the last iteration  */
279:     hapbnd = PetscMin(pipefgmres->haptol,hapbnd);
280:     if (tt > hapbnd) {
281:       /* scale new direction by its norm  */
282:       VecScale(VEC_VV(loc_it+1),1.0/tt);
283:     } else {
284:       /* This happens when the solution is exactly reached. */
285:       /* So there is no new direction... */
286:       VecSet(VEC_TEMP,0.0);     /* set VEC_TEMP to 0 */
287:       hapend = PETSC_TRUE;
288:     }
289:     /* note that for pipefgmres we could get HES(loc_it+1, loc_it)  = 0 and the
290:        current solution would not be exact if HES was singular.  Note that
291:        HH non-singular implies that HES is not singular, and HES is guaranteed
292:        to be nonsingular when PREVECS are linearly independent and A is
293:        nonsingular (in GMRES, the nonsingularity of A implies the nonsingularity
294:        of HES). So we should really add a check to verify that HES is nonsingular.*/

296:     /* Note that to be thorough, in debug mode, one could call a LAPACK routine
297:        here to check that the hessenberg matrix is indeed non-singular (since
298:        FGMRES does not guarantee this) */

300:     /* Now apply rotations to new col of hessenberg (and right side of system),
301:        calculate new rotation, and get new residual norm at the same time*/
302:     KSPPIPEFGMRESUpdateHessenberg(ksp,loc_it,&hapend,&res_norm);
303:     if (ksp->reason) break;

305:     loc_it++;
306:     pipefgmres->it = (loc_it-1);   /* Add this here in case it has converged */

308:     PetscObjectSAWsTakeAccess((PetscObject)ksp);
309:     ksp->its++;
310:     if (ksp->normtype != KSP_NORM_NONE) ksp->rnorm = res_norm;
311:     else ksp->rnorm = 0;
312:     PetscObjectSAWsGrantAccess((PetscObject)ksp);

314:     (*ksp->converged)(ksp,ksp->its,ksp->rnorm,&ksp->reason,ksp->cnvP);

316:     /* Catch error in happy breakdown and signal convergence and break from loop */
317:     if (hapend) {
318:       if (!ksp->reason) {
319:         if (ksp->errorifnotconverged) SETERRQ1(PetscObjectComm((PetscObject)ksp),PETSC_ERR_NOT_CONVERGED,"You reached the happy break down, but convergence was not indicated. Residual norm = %g",(double)res_norm);
320:         else {
321:           ksp->reason = KSP_DIVERGED_BREAKDOWN;
322:           break;
323:         }
324:       }
325:     }
326:   }
327:   /* END OF ITERATION LOOP */
328:   KSPLogResidualHistory(ksp,ksp->rnorm);

330:   /*
331:      Monitor if we know that we will not return for a restart */
332:   if (loc_it && (ksp->reason || ksp->its >= ksp->max_it)) {
333:     KSPMonitor(ksp,ksp->its,ksp->rnorm);
334:   }

336:   if (itcount) *itcount = loc_it;

338:   /*
339:     Down here we have to solve for the "best" coefficients of the Krylov
340:     columns, add the solution values together, and possibly unwind the
341:     preconditioning from the solution
342:    */

344:   /* Form the solution (or the solution so far) */
345:   /* Note: must pass in (loc_it-1) for iteration count so that KSPPIPEGMRESIIBuildSoln
346:      properly navigates */

348:   KSPPIPEFGMRESBuildSoln(RS(0),ksp->vec_sol,ksp->vec_sol,ksp,loc_it-1);

350:   return(0);
351: }

353: /*
354:     KSPSolve_PIPEFGMRES - This routine applies the PIPEFGMRES method.

356:    Input Parameter:
357: .     ksp - the Krylov space object that was set to use pipefgmres

359:    Output Parameter:
360: .     outits - number of iterations used

362: */
363: static PetscErrorCode KSPSolve_PIPEFGMRES(KSP ksp)
364: {
366:   PetscInt       its,itcount;
367:   KSP_PIPEFGMRES *pipefgmres    = (KSP_PIPEFGMRES*)ksp->data;
368:   PetscBool      guess_zero = ksp->guess_zero;

371:   /* We have not checked these routines for use with complex numbers. The inner products
372:      are likely not defined correctly for that case */
373:   if (PetscDefined(USE_COMPLEX) && !PetscDefined(SKIP_COMPLEX)) SETERRQ(PETSC_COMM_WORLD,PETSC_ERR_SUP,"PIPEFGMRES has not been implemented for use with complex scalars");

375:   PetscCitationsRegister(citation,&cited);

377:   if (ksp->calc_sings && !pipefgmres->Rsvd) SETERRQ(PetscObjectComm((PetscObject)ksp),PETSC_ERR_ORDER,"Must call KSPSetComputeSingularValues() before KSPSetUp() is called");
378:   PetscObjectSAWsTakeAccess((PetscObject)ksp);
379:   ksp->its = 0;
380:   PetscObjectSAWsGrantAccess((PetscObject)ksp);

382:   itcount     = 0;
383:   ksp->reason = KSP_CONVERGED_ITERATING;
384:   while (!ksp->reason) {
385:     KSPInitialResidual(ksp,ksp->vec_sol,VEC_TEMP,VEC_TEMP_MATOP,VEC_VV(0),ksp->vec_rhs);
386:     KSPPIPEFGMRESCycle(&its,ksp);
387:     itcount += its;
388:     if (itcount >= ksp->max_it) {
389:       if (!ksp->reason) ksp->reason = KSP_DIVERGED_ITS;
390:       break;
391:     }
392:     ksp->guess_zero = PETSC_FALSE; /* every future call to KSPInitialResidual() will have nonzero guess */
393:   }
394:   ksp->guess_zero = guess_zero; /* restore if user provided nonzero initial guess */
395:   return(0);
396: }

398: static PetscErrorCode KSPDestroy_PIPEFGMRES(KSP ksp)
399: {

403:   KSPReset_PIPEFGMRES(ksp);
404:   KSPDestroy_GMRES(ksp);
405:   return(0);
406: }

408: /*
409:     KSPPIPEFGMRESBuildSoln - create the solution from the starting vector and the
410:                       current iterates.

412:     Input parameters:
413:         nrs - work area of size it + 1.
414:         vguess  - index of initial guess
415:         vdest - index of result.  Note that vguess may == vdest (replace
416:                 guess with the solution).
417:         it - HH upper triangular part is a block of size (it+1) x (it+1)

419:      This is an internal routine that knows about the PIPEFGMRES internals.
420:  */
421: static PetscErrorCode KSPPIPEFGMRESBuildSoln(PetscScalar *nrs,Vec vguess,Vec vdest,KSP ksp,PetscInt it)
422: {
423:   PetscScalar    tt;
425:   PetscInt       k,j;
426:   KSP_PIPEFGMRES *pipefgmres = (KSP_PIPEFGMRES*)(ksp->data);

429:   /* Solve for solution vector that minimizes the residual */

431:   if (it < 0) {                                 /* no pipefgmres steps have been performed */
432:     VecCopy(vguess,vdest); /* VecCopy() is smart, exits immediately if vguess == vdest */
433:     return(0);
434:   }

436:   /* solve the upper triangular system - RS is the right side and HH is
437:      the upper triangular matrix  - put soln in nrs */
438:   if (*HH(it,it) != 0.0) nrs[it] = *RS(it) / *HH(it,it);
439:   else nrs[it] = 0.0;

441:   for (k=it-1; k>=0; k--) {
442:     tt = *RS(k);
443:     for (j=k+1; j<=it; j++) tt -= *HH(k,j) * nrs[j];
444:     nrs[k] = tt / *HH(k,k);
445:   }

447:   /* Accumulate the correction to the solution of the preconditioned problem in VEC_TEMP */
448:   VecZeroEntries(VEC_TEMP);
449:   VecMAXPY(VEC_TEMP,it+1,nrs,&PREVEC(0));

451:   /* add solution to previous solution */
452:   if (vdest == vguess) {
453:     VecAXPY(vdest,1.0,VEC_TEMP);
454:   } else {
455:     VecWAXPY(vdest,1.0,VEC_TEMP,vguess);
456:   }
457:   return(0);
458: }

460: /*

462:     KSPPIPEFGMRESUpdateHessenberg - Do the scalar work for the orthogonalization.
463:                             Return new residual.

465:     input parameters:

467: .        ksp -    Krylov space object
468: .        it  -    plane rotations are applied to the (it+1)th column of the
469:                   modified hessenberg (i.e. HH(:,it))
470: .        hapend - PETSC_FALSE not happy breakdown ending.

472:     output parameters:
473: .        res - the new residual

475:  */
476: /*
477: .  it - column of the Hessenberg that is complete, PIPEFGMRES is actually computing two columns ahead of this
478:  */
479: static PetscErrorCode KSPPIPEFGMRESUpdateHessenberg(KSP ksp,PetscInt it,PetscBool *hapend,PetscReal *res)
480: {
481:   PetscScalar    *hh,*cc,*ss,*rs;
482:   PetscInt       j;
483:   PetscReal      hapbnd;
484:   KSP_PIPEFGMRES *pipefgmres = (KSP_PIPEFGMRES*)(ksp->data);

488:   hh = HH(0,it);   /* pointer to beginning of column to update */
489:   cc = CC(0);      /* beginning of cosine rotations */
490:   ss = SS(0);      /* beginning of sine rotations */
491:   rs = RS(0);      /* right hand side of least squares system */

493:   /* The Hessenberg matrix is now correct through column it, save that form for possible spectral analysis */
494:   for (j=0; j<=it+1; j++) *HES(j,it) = hh[j];

496:   /* check for the happy breakdown */
497:   hapbnd = PetscMin(PetscAbsScalar(hh[it+1] / rs[it]),pipefgmres->haptol);
498:   if (PetscAbsScalar(hh[it+1]) < hapbnd) {
499:     PetscInfo4(ksp,"Detected happy breakdown, current hapbnd = %14.12e H(%D,%D) = %14.12e\n",(double)hapbnd,it+1,it,(double)PetscAbsScalar(*HH(it+1,it)));
500:     *hapend = PETSC_TRUE;
501:   }

503:   /* Apply all the previously computed plane rotations to the new column
504:      of the Hessenberg matrix */
505:   /* Note: this uses the rotation [conj(c)  s ; -s   c], c= cos(theta), s= sin(theta),
506:      and some refs have [c   s ; -conj(s)  c] (don't be confused!) */

508:   for (j=0; j<it; j++) {
509:     PetscScalar hhj = hh[j];
510:     hh[j]   = PetscConj(cc[j])*hhj + ss[j]*hh[j+1];
511:     hh[j+1] =          -ss[j] *hhj + cc[j]*hh[j+1];
512:   }

514:   /*
515:     compute the new plane rotation, and apply it to:
516:      1) the right-hand-side of the Hessenberg system (RS)
517:         note: it affects RS(it) and RS(it+1)
518:      2) the new column of the Hessenberg matrix
519:         note: it affects HH(it,it) which is currently pointed to
520:         by hh and HH(it+1, it) (*(hh+1))
521:     thus obtaining the updated value of the residual...
522:   */

524:   /* compute new plane rotation */

526:   if (!*hapend) {
527:     PetscReal delta = PetscSqrtReal(PetscSqr(PetscAbsScalar(hh[it])) + PetscSqr(PetscAbsScalar(hh[it+1])));
528:     if (delta == 0.0) {
529:       ksp->reason = KSP_DIVERGED_NULL;
530:       return(0);
531:     }

533:     cc[it] = hh[it] / delta;    /* new cosine value */
534:     ss[it] = hh[it+1] / delta;  /* new sine value */

536:     hh[it]   = PetscConj(cc[it])*hh[it] + ss[it]*hh[it+1];
537:     rs[it+1] = -ss[it]*rs[it];
538:     rs[it]   = PetscConj(cc[it])*rs[it];
539:     *res     = PetscAbsScalar(rs[it+1]);
540:   } else { /* happy breakdown: HH(it+1, it) = 0, therefore we don't need to apply
541:             another rotation matrix (so RH doesn't change).  The new residual is
542:             always the new sine term times the residual from last time (RS(it)),
543:             but now the new sine rotation would be zero...so the residual should
544:             be zero...so we will multiply "zero" by the last residual.  This might
545:             not be exactly what we want to do here -could just return "zero". */

547:     *res = 0.0;
548:   }
549:   return(0);
550: }

552: /*
553:    KSPBuildSolution_PIPEFGMRES

555:      Input Parameter:
556: .     ksp - the Krylov space object
557: .     ptr-

559:    Output Parameter:
560: .     result - the solution

562:    Note: this calls KSPPIPEFGMRESBuildSoln - the same function that KSPPIPEFGMRESCycle
563:    calls directly.

565: */
566: PetscErrorCode KSPBuildSolution_PIPEFGMRES(KSP ksp,Vec ptr,Vec *result)
567: {
568:   KSP_PIPEFGMRES *pipefgmres = (KSP_PIPEFGMRES*)ksp->data;

572:   if (!ptr) {
573:     if (!pipefgmres->sol_temp) {
574:       VecDuplicate(ksp->vec_sol,&pipefgmres->sol_temp);
575:       PetscLogObjectParent((PetscObject)ksp,(PetscObject)pipefgmres->sol_temp);
576:     }
577:     ptr = pipefgmres->sol_temp;
578:   }
579:   if (!pipefgmres->nrs) {
580:     /* allocate the work area */
581:     PetscMalloc1(pipefgmres->max_k,&pipefgmres->nrs);
582:     PetscLogObjectMemory((PetscObject)ksp,pipefgmres->max_k*sizeof(PetscScalar));
583:   }

585:   KSPPIPEFGMRESBuildSoln(pipefgmres->nrs,ksp->vec_sol,ptr,ksp,pipefgmres->it);
586:   if (result) *result = ptr;
587:   return(0);
588: }

590: PetscErrorCode KSPSetFromOptions_PIPEFGMRES(PetscOptionItems *PetscOptionsObject,KSP ksp)
591: {
593:   KSP_PIPEFGMRES *pipefgmres = (KSP_PIPEFGMRES*)ksp->data;
594:   PetscBool      flg;
595:   PetscScalar    shift;

598:   KSPSetFromOptions_GMRES(PetscOptionsObject,ksp);
599:   PetscOptionsHead(PetscOptionsObject,"KSP pipelined FGMRES Options");
600:   PetscOptionsScalar("-ksp_pipefgmres_shift","shift parameter","KSPPIPEFGMRESSetShift",pipefgmres->shift,&shift,&flg);
601:   if (flg) { KSPPIPEFGMRESSetShift(ksp,shift); }
602:   PetscOptionsTail();
603:   return(0);
604: }

606: PetscErrorCode KSPView_PIPEFGMRES(KSP ksp,PetscViewer viewer)
607: {
608:   KSP_PIPEFGMRES *pipefgmres = (KSP_PIPEFGMRES*)ksp->data;
610:   PetscBool      iascii,isstring;

613:   PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&iascii);
614:   PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERSTRING,&isstring);

616:   if (iascii) {
617:     PetscViewerASCIIPrintf(viewer,"  restart=%D\n",pipefgmres->max_k);
618:     PetscViewerASCIIPrintf(viewer,"  happy breakdown tolerance %g\n",(double)pipefgmres->haptol);
619: #if defined(PETSC_USE_COMPLEX)
620:     PetscViewerASCIIPrintf(viewer,"  shift=%g+%gi\n",PetscRealPart(pipefgmres->shift),PetscImaginaryPart(pipefgmres->shift));
621: #else
622:     PetscViewerASCIIPrintf(viewer,"  shift=%g\n",pipefgmres->shift);
623: #endif
624:   } else if (isstring) {
625:     PetscViewerStringSPrintf(viewer,"restart %D",pipefgmres->max_k);
626: #if defined(PETSC_USE_COMPLEX)
627:     PetscViewerStringSPrintf(viewer,"   shift=%g+%gi\n",PetscRealPart(pipefgmres->shift),PetscImaginaryPart(pipefgmres->shift));
628: #else
629:     PetscViewerStringSPrintf(viewer,"   shift=%g\n",pipefgmres->shift);
630: #endif
631:   }
632:   return(0);
633: }

635: PetscErrorCode KSPReset_PIPEFGMRES(KSP ksp)
636: {
637:   KSP_PIPEFGMRES *pipefgmres = (KSP_PIPEFGMRES*)ksp->data;
638:   PetscErrorCode   ierr;
639:   PetscInt         i;

642:   PetscFree(pipefgmres->prevecs);
643:   PetscFree(pipefgmres->zvecs);
644:   for (i=0; i<pipefgmres->nwork_alloc; i++) {
645:     VecDestroyVecs(pipefgmres->mwork_alloc[i],&pipefgmres->prevecs_user_work[i]);
646:     VecDestroyVecs(pipefgmres->mwork_alloc[i],&pipefgmres->zvecs_user_work[i]);
647:   }
648:   PetscFree(pipefgmres->prevecs_user_work);
649:   PetscFree(pipefgmres->zvecs_user_work);
650:   PetscFree(pipefgmres->redux);
651:   KSPReset_GMRES(ksp);
652:   return(0);
653: }

655: /*MC
656:    KSPPIPEFGMRES - Implements the Pipelined Generalized Minimal Residual method.

658:    A flexible, 1-stage pipelined variant of GMRES.

660:    Options Database Keys:
661: +   -ksp_gmres_restart <restart> - the number of Krylov directions to orthogonalize against
662: .   -ksp_gmres_haptol <tol> - sets the tolerance for "happy ending" (exact convergence)
663: .   -ksp_gmres_preallocate - preallocate all the Krylov search directions initially (otherwise groups of
664: .   -ksp_pipefgmres_shift - the shift to use (defaults to 1. See KSPPIPEFGMRESSetShift()
665:                              vectors are allocated as needed)
666: -   -ksp_gmres_krylov_monitor - plot the Krylov space generated

668:    Level: intermediate

670:    Notes:

672:    This variant is not "explicitly normalized" like KSPPGMRES, and requires a shift parameter.

674:    A heuristic for choosing the shift parameter is the largest eigenvalue of the preconditioned operator.

676:    Only right preconditioning is supported (but this preconditioner may be nonlinear/variable/inexact, as with KSPFGMRES).
677:    MPI configuration may be necessary for reductions to make asynchronous progress, which is important for performance of pipelined methods.
678:    See the FAQ on the PETSc website for details.

680:    Developer Notes:
681:     This class is subclassed off of KSPGMRES.

683:    Reference:
684:     P. Sanan, S.M. Schnepp, and D.A. May,
685:     "Pipelined, Flexible Krylov Subspace Methods,"
686:     SIAM Journal on Scientific Computing 2016 38:5, C441-C470,
687:     DOI: 10.1137/15M1049130

689: .seealso:  KSPCreate(), KSPSetType(), KSPType (for list of available types), KSP, KSPLGMRES, KSPPIPECG, KSPPIPECR, KSPPGMRES, KSPFGMRES
690:            KSPGMRESSetRestart(), KSPGMRESSetHapTol(), KSPGMRESSetPreAllocateVectors(), KSPGMRESMonitorKrylov(), KSPPIPEFGMRESSetShift()
691: M*/

693: PETSC_EXTERN PetscErrorCode KSPCreate_PIPEFGMRES(KSP ksp)
694: {
695:   KSP_PIPEFGMRES *pipefgmres;

699:   PetscNewLog(ksp,&pipefgmres);

701:   ksp->data                              = (void*)pipefgmres;
702:   ksp->ops->buildsolution                = KSPBuildSolution_PIPEFGMRES;
703:   ksp->ops->setup                        = KSPSetUp_PIPEFGMRES;
704:   ksp->ops->solve                        = KSPSolve_PIPEFGMRES;
705:   ksp->ops->reset                        = KSPReset_PIPEFGMRES;
706:   ksp->ops->destroy                      = KSPDestroy_PIPEFGMRES;
707:   ksp->ops->view                         = KSPView_PIPEFGMRES;
708:   ksp->ops->setfromoptions               = KSPSetFromOptions_PIPEFGMRES;
709:   ksp->ops->computeextremesingularvalues = KSPComputeExtremeSingularValues_GMRES;
710:   ksp->ops->computeeigenvalues           = KSPComputeEigenvalues_GMRES;

712:   KSPSetSupportedNorm(ksp,KSP_NORM_UNPRECONDITIONED,PC_RIGHT,3);
713:   KSPSetSupportedNorm(ksp,KSP_NORM_NONE,PC_RIGHT,1);

715:   PetscObjectComposeFunction((PetscObject)ksp,"KSPGMRESSetPreAllocateVectors_C",KSPGMRESSetPreAllocateVectors_GMRES);
716:   PetscObjectComposeFunction((PetscObject)ksp,"KSPGMRESSetRestart_C",KSPGMRESSetRestart_GMRES);
717:   PetscObjectComposeFunction((PetscObject)ksp,"KSPGMRESGetRestart_C",KSPGMRESGetRestart_GMRES);

719:   pipefgmres->nextra_vecs    = 1;
720:   pipefgmres->haptol         = 1.0e-30;
721:   pipefgmres->q_preallocate  = 0;
722:   pipefgmres->delta_allocate = PIPEFGMRES_DELTA_DIRECTIONS;
723:   pipefgmres->orthog         = NULL;
724:   pipefgmres->nrs            = NULL;
725:   pipefgmres->sol_temp       = NULL;
726:   pipefgmres->max_k          = PIPEFGMRES_DEFAULT_MAXK;
727:   pipefgmres->Rsvd           = NULL;
728:   pipefgmres->orthogwork     = NULL;
729:   pipefgmres->cgstype        = KSP_GMRES_CGS_REFINE_NEVER;
730:   pipefgmres->shift          = 1.0;
731:   return(0);
732: }

734: static PetscErrorCode KSPPIPEFGMRESGetNewVectors(KSP ksp,PetscInt it)
735: {
736:   KSP_PIPEFGMRES *pipefgmres = (KSP_PIPEFGMRES*)ksp->data;
737:   PetscInt       nwork   = pipefgmres->nwork_alloc; /* number of work vector chunks allocated */
738:   PetscInt       nalloc;                            /* number to allocate */
740:   PetscInt       k;

743:   nalloc = pipefgmres->delta_allocate; /* number of vectors to allocate
744:                                       in a single chunk */

746:   /* Adjust the number to allocate to make sure that we don't exceed the
747:      number of available slots (pipefgmres->vecs_allocated)*/
748:   if (it + VEC_OFFSET + nalloc >= pipefgmres->vecs_allocated) {
749:     nalloc = pipefgmres->vecs_allocated - it - VEC_OFFSET;
750:   }
751:   if (!nalloc) return(0);

753:   pipefgmres->vv_allocated += nalloc; /* vv_allocated is the number of vectors allocated */

755:   /* work vectors */
756:   KSPCreateVecs(ksp,nalloc,&pipefgmres->user_work[nwork],0,NULL);
757:   PetscLogObjectParents(ksp,nalloc,pipefgmres->user_work[nwork]);
758:   for (k=0; k < nalloc; k++) {
759:     pipefgmres->vecs[it+VEC_OFFSET+k] = pipefgmres->user_work[nwork][k];
760:   }
761:   /* specify size of chunk allocated */
762:   pipefgmres->mwork_alloc[nwork] = nalloc;

764:   /* preconditioned vectors (note we don't use VEC_OFFSET) */
765:   KSPCreateVecs(ksp,nalloc,&pipefgmres->prevecs_user_work[nwork],0,NULL);
766:   PetscLogObjectParents(ksp,nalloc,pipefgmres->prevecs_user_work[nwork]);
767:   for (k=0; k < nalloc; k++) {
768:     pipefgmres->prevecs[it+k] = pipefgmres->prevecs_user_work[nwork][k];
769:   }

771:   KSPCreateVecs(ksp,nalloc,&pipefgmres->zvecs_user_work[nwork],0,NULL);
772:   PetscLogObjectParents(ksp,nalloc,pipefgmres->zvecs_user_work[nwork]);
773:   for (k=0; k < nalloc; k++) {
774:     pipefgmres->zvecs[it+k] = pipefgmres->zvecs_user_work[nwork][k];
775:   }

777:   /* increment the number of work vector chunks */
778:   pipefgmres->nwork_alloc++;
779:   return(0);
780: }

782: /*@
783:   KSPPIPEFGMRESSetShift - Set the shift parameter for the flexible, pipelined GMRES solver.

785:   A heuristic is to set this to be comparable to the largest eigenvalue of the preconditioned operator. This can be acheived with PETSc itself by using a few iterations of a Krylov method. See KSPComputeEigenvalues (and note the caveats there).

787: Logically Collective on ksp

789: Input Parameters:
790: +  ksp - the Krylov space context
791: -  shift - the shift

793: Level: intermediate

795: Options Database:
796: . -ksp_pipefgmres_shift <shift>

798: .seealso: KSPComputeEigenvalues()
799: @*/
800: PetscErrorCode KSPPIPEFGMRESSetShift(KSP ksp,PetscScalar shift)
801: {
802:   KSP_PIPEFGMRES *pipefgmres = (KSP_PIPEFGMRES*)ksp->data;

807:   pipefgmres->shift = shift;
808:   return(0);
809: }