Date: 2022-03-04T17:39:04+00:00

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: }