Date: 2022-03-31T04:48:10+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: {
     40:   PetscInt       k;
     41:   KSP_PIPEFGMRES *pipefgmres = (KSP_PIPEFGMRES*)ksp->data;
     42:   const PetscInt max_k = pipefgmres->max_k;

     44:   KSPSetUp_GMRES(ksp);

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

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

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

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

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

     69:   return 0;
     70: }

     72: /*

     74:     KSPPIPEFGMRESCycle - Run pipefgmres, possibly with restart.  Return residual
     75:                   history if requested.

     77:     input parameters:
     78: .        pipefgmres  - structure containing parameters and work areas

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

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

     88: */
     89: static PetscErrorCode KSPPIPEFGMRESCycle(PetscInt *itcount,KSP ksp)
     90: {
     91:   KSP_PIPEFGMRES *pipefgmres = (KSP_PIPEFGMRES*)(ksp->data);
     92:   PetscReal      res_norm;
     93:   PetscReal      hapbnd,tt;
     94:   PetscScalar    *hh,*hes,*lhh,shift = pipefgmres->shift;
     95:   PetscBool      hapend = PETSC_FALSE;  /* indicates happy breakdown ending */
     96:   PetscInt       loc_it;                /* local count of # of dir. in Krylov space */
     97:   PetscInt       max_k = pipefgmres->max_k; /* max # of directions Krylov space */
     98:   PetscInt       i,j,k;
     99:   Mat            Amat,Pmat;
    100:   Vec            Q,W; /* Pipelining vectors */
    101:   Vec            *redux = pipefgmres->redux; /* workspace for single reduction */

    103:   if (itcount) *itcount = 0;

    105:   /* Assign simpler names to these vectors, allocated as pipelining workspace */
    106:   Q = VEC_Q;
    107:   W = VEC_W;

    109:   /* Allocate memory for orthogonalization work (freed in the GMRES Destroy routine)*/
    110:   /* Note that we add an extra value here to allow for a single reduction */
    111:   if (!pipefgmres->orthogwork) { PetscMalloc1(pipefgmres->max_k + 2 ,&pipefgmres->orthogwork);
    112:   }
    113:   lhh = pipefgmres->orthogwork;

    115:   /* Number of pseudo iterations since last restart is the number
    116:      of prestart directions */
    117:   loc_it = 0;

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

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

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

    132:   PetscObjectSAWsTakeAccess((PetscObject)ksp);
    133:   if (ksp->normtype != KSP_NORM_NONE) ksp->rnorm = res_norm;
    134:   else ksp->rnorm = 0;
    135:   PetscObjectSAWsGrantAccess((PetscObject)ksp);
    136:   KSPLogResidualHistory(ksp,ksp->rnorm);
    137:   KSPMonitor(ksp,ksp->its,ksp->rnorm);

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

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

    149:   /* Fill the pipeline */
    150:   KSP_PCApply(ksp,VEC_VV(loc_it),PREVEC(loc_it));
    151:   PCGetOperators(ksp->pc,&Amat,&Pmat);
    152:   KSP_MatMult(ksp,Amat,PREVEC(loc_it),ZVEC(loc_it));
    153:   VecAXPY(ZVEC(loc_it),-shift,VEC_VV(loc_it)); /* Note shift */

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

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

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

    178:     /* Do some local twiddling to allow for a single reduction */
    179:     for (i=0;i<loc_it+1;i++) {
    180:       redux[i] = VEC_VV(i);
    181:     }
    182:     redux[loc_it+1] = ZVEC(loc_it);

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

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

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

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

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

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

    223:     /* we delay applying the shift here */
    224:     for (j=0; j<=loc_it; j++) {
    225:       lhh[j]        = -lhh[j]; /* flip sign */
    226:     }

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

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

    250:     /* The recurred computation for the new direction
    251:        The division by tt is delayed to the happy breakdown check later
    252:        Note placement BEFORE the unshift
    253:        */
    254:     VecCopy(ZVEC(loc_it),VEC_VV(loc_it+1));
    255:     VecMAXPY(VEC_VV(loc_it+1),loc_it+1,lhh,&VEC_VV(0));
    256:     /* (VEC_VV(loc_it+1) is not normalized yet) */

    258:     /* The recurred computation for the preconditioned vector (u) */
    259:     VecCopy(Q,PREVEC(loc_it+1));
    260:     VecMAXPY(PREVEC(loc_it+1),loc_it+1,lhh,&PREVEC(0));
    261:     VecScale(PREVEC(loc_it+1),1.0/tt);

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

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

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

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

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

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

    304:     PetscObjectSAWsTakeAccess((PetscObject)ksp);
    305:     ksp->its++;
    306:     if (ksp->normtype != KSP_NORM_NONE) ksp->rnorm = res_norm;
    307:     else ksp->rnorm = 0;
    308:     PetscObjectSAWsGrantAccess((PetscObject)ksp);

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

    312:     /* Catch error in happy breakdown and signal convergence and break from loop */
    313:     if (hapend) {
    314:       if (!ksp->reason) {
    316:         else {
    317:           ksp->reason = KSP_DIVERGED_BREAKDOWN;
    318:           break;
    319:         }
    320:       }
    321:     }
    322:   }
    323:   /* END OF ITERATION LOOP */
    324:   KSPLogResidualHistory(ksp,ksp->rnorm);

    326:   /*
    327:      Monitor if we know that we will not return for a restart */
    328:   if (loc_it && (ksp->reason || ksp->its >= ksp->max_it)) {
    329:     KSPMonitor(ksp,ksp->its,ksp->rnorm);
    330:   }

    332:   if (itcount) *itcount = loc_it;

    334:   /*
    335:     Down here we have to solve for the "best" coefficients of the Krylov
    336:     columns, add the solution values together, and possibly unwind the
    337:     preconditioning from the solution
    338:    */

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

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

    346:   return 0;
    347: }

    349: /*
    350:     KSPSolve_PIPEFGMRES - This routine applies the PIPEFGMRES method.

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

    355:    Output Parameter:
    356: .     outits - number of iterations used

    358: */
    359: static PetscErrorCode KSPSolve_PIPEFGMRES(KSP ksp)
    360: {
    361:   PetscInt       its,itcount;
    362:   KSP_PIPEFGMRES *pipefgmres    = (KSP_PIPEFGMRES*)ksp->data;
    363:   PetscBool      guess_zero = ksp->guess_zero;

    365:   /* We have not checked these routines for use with complex numbers. The inner products
    366:      are likely not defined correctly for that case */

    369:   PetscCitationsRegister(citation,&cited);

    372:   PetscObjectSAWsTakeAccess((PetscObject)ksp);
    373:   ksp->its = 0;
    374:   PetscObjectSAWsGrantAccess((PetscObject)ksp);

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

    392: static PetscErrorCode KSPDestroy_PIPEFGMRES(KSP ksp)
    393: {
    394:   KSPReset_PIPEFGMRES(ksp);
    395:   KSPDestroy_GMRES(ksp);
    396:   return 0;
    397: }

    399: /*
    400:     KSPPIPEFGMRESBuildSoln - create the solution from the starting vector and the
    401:                       current iterates.

    403:     Input parameters:
    404:         nrs - work area of size it + 1.
    405:         vguess  - index of initial guess
    406:         vdest - index of result.  Note that vguess may == vdest (replace
    407:                 guess with the solution).
    408:         it - HH upper triangular part is a block of size (it+1) x (it+1)

    410:      This is an internal routine that knows about the PIPEFGMRES internals.
    411:  */
    412: static PetscErrorCode KSPPIPEFGMRESBuildSoln(PetscScalar *nrs,Vec vguess,Vec vdest,KSP ksp,PetscInt it)
    413: {
    414:   PetscScalar    tt;
    415:   PetscInt       k,j;
    416:   KSP_PIPEFGMRES *pipefgmres = (KSP_PIPEFGMRES*)(ksp->data);

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

    420:   if (it < 0) {                                 /* no pipefgmres steps have been performed */
    421:     VecCopy(vguess,vdest); /* VecCopy() is smart, exits immediately if vguess == vdest */
    422:     return 0;
    423:   }

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

    430:   for (k=it-1; k>=0; k--) {
    431:     tt = *RS(k);
    432:     for (j=k+1; j<=it; j++) tt -= *HH(k,j) * nrs[j];
    433:     nrs[k] = tt / *HH(k,k);
    434:   }

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

    440:   /* add solution to previous solution */
    441:   if (vdest == vguess) {
    442:     VecAXPY(vdest,1.0,VEC_TEMP);
    443:   } else {
    444:     VecWAXPY(vdest,1.0,VEC_TEMP,vguess);
    445:   }
    446:   return 0;
    447: }

    449: /*

    451:     KSPPIPEFGMRESUpdateHessenberg - Do the scalar work for the orthogonalization.
    452:                             Return new residual.

    454:     input parameters:

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

    461:     output parameters:
    462: .        res - the new residual

    464:  */
    465: /*
    466: .  it - column of the Hessenberg that is complete, PIPEFGMRES is actually computing two columns ahead of this
    467:  */
    468: static PetscErrorCode KSPPIPEFGMRESUpdateHessenberg(KSP ksp,PetscInt it,PetscBool *hapend,PetscReal *res)
    469: {
    470:   PetscScalar    *hh,*cc,*ss,*rs;
    471:   PetscInt       j;
    472:   PetscReal      hapbnd;
    473:   KSP_PIPEFGMRES *pipefgmres = (KSP_PIPEFGMRES*)(ksp->data);

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

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

    483:   /* check for the happy breakdown */
    484:   hapbnd = PetscMin(PetscAbsScalar(hh[it+1] / rs[it]),pipefgmres->haptol);
    485:   if (PetscAbsScalar(hh[it+1]) < hapbnd) {
    486:     PetscInfo(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)));
    487:     *hapend = PETSC_TRUE;
    488:   }

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

    495:   for (j=0; j<it; j++) {
    496:     PetscScalar hhj = hh[j];
    497:     hh[j]   = PetscConj(cc[j])*hhj + ss[j]*hh[j+1];
    498:     hh[j+1] =          -ss[j] *hhj + cc[j]*hh[j+1];
    499:   }

    501:   /*
    502:     compute the new plane rotation, and apply it to:
    503:      1) the right-hand-side of the Hessenberg system (RS)
    504:         note: it affects RS(it) and RS(it+1)
    505:      2) the new column of the Hessenberg matrix
    506:         note: it affects HH(it,it) which is currently pointed to
    507:         by hh and HH(it+1, it) (*(hh+1))
    508:     thus obtaining the updated value of the residual...
    509:   */

    511:   /* compute new plane rotation */

    513:   if (!*hapend) {
    514:     PetscReal delta = PetscSqrtReal(PetscSqr(PetscAbsScalar(hh[it])) + PetscSqr(PetscAbsScalar(hh[it+1])));
    515:     if (delta == 0.0) {
    516:       ksp->reason = KSP_DIVERGED_NULL;
    517:       return 0;
    518:     }

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

    523:     hh[it]   = PetscConj(cc[it])*hh[it] + ss[it]*hh[it+1];
    524:     rs[it+1] = -ss[it]*rs[it];
    525:     rs[it]   = PetscConj(cc[it])*rs[it];
    526:     *res     = PetscAbsScalar(rs[it+1]);
    527:   } else { /* happy breakdown: HH(it+1, it) = 0, therefore we don't need to apply
    528:             another rotation matrix (so RH doesn't change).  The new residual is
    529:             always the new sine term times the residual from last time (RS(it)),
    530:             but now the new sine rotation would be zero...so the residual should
    531:             be zero...so we will multiply "zero" by the last residual.  This might
    532:             not be exactly what we want to do here -could just return "zero". */

    534:     *res = 0.0;
    535:   }
    536:   return 0;
    537: }

    539: /*
    540:    KSPBuildSolution_PIPEFGMRES

    542:      Input Parameter:
    543: .     ksp - the Krylov space object
    544: .     ptr-

    546:    Output Parameter:
    547: .     result - the solution

    549:    Note: this calls KSPPIPEFGMRESBuildSoln - the same function that KSPPIPEFGMRESCycle
    550:    calls directly.

    552: */
    553: PetscErrorCode KSPBuildSolution_PIPEFGMRES(KSP ksp,Vec ptr,Vec *result)
    554: {
    555:   KSP_PIPEFGMRES *pipefgmres = (KSP_PIPEFGMRES*)ksp->data;

    557:   if (!ptr) {
    558:     if (!pipefgmres->sol_temp) {
    559:       VecDuplicate(ksp->vec_sol,&pipefgmres->sol_temp);
    560:       PetscLogObjectParent((PetscObject)ksp,(PetscObject)pipefgmres->sol_temp);
    561:     }
    562:     ptr = pipefgmres->sol_temp;
    563:   }
    564:   if (!pipefgmres->nrs) {
    565:     /* allocate the work area */
    566:     PetscMalloc1(pipefgmres->max_k,&pipefgmres->nrs);
    567:     PetscLogObjectMemory((PetscObject)ksp,pipefgmres->max_k*sizeof(PetscScalar));
    568:   }

    570:   KSPPIPEFGMRESBuildSoln(pipefgmres->nrs,ksp->vec_sol,ptr,ksp,pipefgmres->it);
    571:   if (result) *result = ptr;
    572:   return 0;
    573: }

    575: PetscErrorCode KSPSetFromOptions_PIPEFGMRES(PetscOptionItems *PetscOptionsObject,KSP ksp)
    576: {
    577:   KSP_PIPEFGMRES *pipefgmres = (KSP_PIPEFGMRES*)ksp->data;
    578:   PetscBool      flg;
    579:   PetscScalar    shift;

    581:   KSPSetFromOptions_GMRES(PetscOptionsObject,ksp);
    582:   PetscOptionsHead(PetscOptionsObject,"KSP pipelined FGMRES Options");
    583:   PetscOptionsScalar("-ksp_pipefgmres_shift","shift parameter","KSPPIPEFGMRESSetShift",pipefgmres->shift,&shift,&flg);
    584:   if (flg) KSPPIPEFGMRESSetShift(ksp,shift);
    585:   PetscOptionsTail();
    586:   return 0;
    587: }

    589: PetscErrorCode KSPView_PIPEFGMRES(KSP ksp,PetscViewer viewer)
    590: {
    591:   KSP_PIPEFGMRES *pipefgmres = (KSP_PIPEFGMRES*)ksp->data;
    592:   PetscBool      iascii,isstring;

    594:   PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&iascii);
    595:   PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERSTRING,&isstring);

    597:   if (iascii) {
    598:     PetscViewerASCIIPrintf(viewer,"  restart=%D\n",pipefgmres->max_k);
    599:     PetscViewerASCIIPrintf(viewer,"  happy breakdown tolerance %g\n",(double)pipefgmres->haptol);
    600: #if defined(PETSC_USE_COMPLEX)
    601:     PetscViewerASCIIPrintf(viewer,"  shift=%g+%gi\n",PetscRealPart(pipefgmres->shift),PetscImaginaryPart(pipefgmres->shift));
    602: #else
    603:     PetscViewerASCIIPrintf(viewer,"  shift=%g\n",pipefgmres->shift);
    604: #endif
    605:   } else if (isstring) {
    606:     PetscViewerStringSPrintf(viewer,"restart %D",pipefgmres->max_k);
    607: #if defined(PETSC_USE_COMPLEX)
    608:     PetscViewerStringSPrintf(viewer,"   shift=%g+%gi\n",PetscRealPart(pipefgmres->shift),PetscImaginaryPart(pipefgmres->shift));
    609: #else
    610:     PetscViewerStringSPrintf(viewer,"   shift=%g\n",pipefgmres->shift);
    611: #endif
    612:   }
    613:   return 0;
    614: }

    616: PetscErrorCode KSPReset_PIPEFGMRES(KSP ksp)
    617: {
    618:   KSP_PIPEFGMRES *pipefgmres = (KSP_PIPEFGMRES*)ksp->data;
    619:   PetscInt         i;

    621:   PetscFree(pipefgmres->prevecs);
    622:   PetscFree(pipefgmres->zvecs);
    623:   for (i=0; i<pipefgmres->nwork_alloc; i++) {
    624:     VecDestroyVecs(pipefgmres->mwork_alloc[i],&pipefgmres->prevecs_user_work[i]);
    625:     VecDestroyVecs(pipefgmres->mwork_alloc[i],&pipefgmres->zvecs_user_work[i]);
    626:   }
    627:   PetscFree(pipefgmres->prevecs_user_work);
    628:   PetscFree(pipefgmres->zvecs_user_work);
    629:   PetscFree(pipefgmres->redux);
    630:   KSPReset_GMRES(ksp);
    631:   return 0;
    632: }

    634: /*MC
    635:    KSPPIPEFGMRES - Implements the Pipelined Generalized Minimal Residual method.

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

    639:    Options Database Keys:
    640: +   -ksp_gmres_restart <restart> - the number of Krylov directions to orthogonalize against
    641: .   -ksp_gmres_haptol <tol> - sets the tolerance for "happy ending" (exact convergence)
    642: .   -ksp_gmres_preallocate - preallocate all the Krylov search directions initially (otherwise groups of
    643: .   -ksp_pipefgmres_shift - the shift to use (defaults to 1. See KSPPIPEFGMRESSetShift()
    644:                              vectors are allocated as needed)
    645: -   -ksp_gmres_krylov_monitor - plot the Krylov space generated

    647:    Level: intermediate

    649:    Notes:

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

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

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

    659:    Developer Notes:
    660:     This class is subclassed off of KSPGMRES.

    662:    Reference:
    663:     P. Sanan, S.M. Schnepp, and D.A. May,
    664:     "Pipelined, Flexible Krylov Subspace Methods,"
    665:     SIAM Journal on Scientific Computing 2016 38:5, C441-C470,
    666:     DOI: 10.1137/15M1049130

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

    672: PETSC_EXTERN PetscErrorCode KSPCreate_PIPEFGMRES(KSP ksp)
    673: {
    674:   KSP_PIPEFGMRES *pipefgmres;

    676:   PetscNewLog(ksp,&pipefgmres);

    678:   ksp->data                              = (void*)pipefgmres;
    679:   ksp->ops->buildsolution                = KSPBuildSolution_PIPEFGMRES;
    680:   ksp->ops->setup                        = KSPSetUp_PIPEFGMRES;
    681:   ksp->ops->solve                        = KSPSolve_PIPEFGMRES;
    682:   ksp->ops->reset                        = KSPReset_PIPEFGMRES;
    683:   ksp->ops->destroy                      = KSPDestroy_PIPEFGMRES;
    684:   ksp->ops->view                         = KSPView_PIPEFGMRES;
    685:   ksp->ops->setfromoptions               = KSPSetFromOptions_PIPEFGMRES;
    686:   ksp->ops->computeextremesingularvalues = KSPComputeExtremeSingularValues_GMRES;
    687:   ksp->ops->computeeigenvalues           = KSPComputeEigenvalues_GMRES;

    689:   KSPSetSupportedNorm(ksp,KSP_NORM_UNPRECONDITIONED,PC_RIGHT,3);
    690:   KSPSetSupportedNorm(ksp,KSP_NORM_NONE,PC_RIGHT,1);

    692:   PetscObjectComposeFunction((PetscObject)ksp,"KSPGMRESSetPreAllocateVectors_C",KSPGMRESSetPreAllocateVectors_GMRES);
    693:   PetscObjectComposeFunction((PetscObject)ksp,"KSPGMRESSetRestart_C",KSPGMRESSetRestart_GMRES);
    694:   PetscObjectComposeFunction((PetscObject)ksp,"KSPGMRESGetRestart_C",KSPGMRESGetRestart_GMRES);

    696:   pipefgmres->nextra_vecs    = 1;
    697:   pipefgmres->haptol         = 1.0e-30;
    698:   pipefgmres->q_preallocate  = 0;
    699:   pipefgmres->delta_allocate = PIPEFGMRES_DELTA_DIRECTIONS;
    700:   pipefgmres->orthog         = NULL;
    701:   pipefgmres->nrs            = NULL;
    702:   pipefgmres->sol_temp       = NULL;
    703:   pipefgmres->max_k          = PIPEFGMRES_DEFAULT_MAXK;
    704:   pipefgmres->Rsvd           = NULL;
    705:   pipefgmres->orthogwork     = NULL;
    706:   pipefgmres->cgstype        = KSP_GMRES_CGS_REFINE_NEVER;
    707:   pipefgmres->shift          = 1.0;
    708:   return 0;
    709: }

    711: static PetscErrorCode KSPPIPEFGMRESGetNewVectors(KSP ksp,PetscInt it)
    712: {
    713:   KSP_PIPEFGMRES *pipefgmres = (KSP_PIPEFGMRES*)ksp->data;
    714:   PetscInt       nwork   = pipefgmres->nwork_alloc; /* number of work vector chunks allocated */
    715:   PetscInt       nalloc;                            /* number to allocate */
    716:   PetscInt       k;

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

    721:   /* Adjust the number to allocate to make sure that we don't exceed the
    722:      number of available slots (pipefgmres->vecs_allocated)*/
    723:   if (it + VEC_OFFSET + nalloc >= pipefgmres->vecs_allocated) {
    724:     nalloc = pipefgmres->vecs_allocated - it - VEC_OFFSET;
    725:   }
    726:   if (!nalloc) return 0;

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

    730:   /* work vectors */
    731:   KSPCreateVecs(ksp,nalloc,&pipefgmres->user_work[nwork],0,NULL);
    732:   PetscLogObjectParents(ksp,nalloc,pipefgmres->user_work[nwork]);
    733:   for (k=0; k < nalloc; k++) {
    734:     pipefgmres->vecs[it+VEC_OFFSET+k] = pipefgmres->user_work[nwork][k];
    735:   }
    736:   /* specify size of chunk allocated */
    737:   pipefgmres->mwork_alloc[nwork] = nalloc;

    739:   /* preconditioned vectors (note we don't use VEC_OFFSET) */
    740:   KSPCreateVecs(ksp,nalloc,&pipefgmres->prevecs_user_work[nwork],0,NULL);
    741:   PetscLogObjectParents(ksp,nalloc,pipefgmres->prevecs_user_work[nwork]);
    742:   for (k=0; k < nalloc; k++) {
    743:     pipefgmres->prevecs[it+k] = pipefgmres->prevecs_user_work[nwork][k];
    744:   }

    746:   KSPCreateVecs(ksp,nalloc,&pipefgmres->zvecs_user_work[nwork],0,NULL);
    747:   PetscLogObjectParents(ksp,nalloc,pipefgmres->zvecs_user_work[nwork]);
    748:   for (k=0; k < nalloc; k++) {
    749:     pipefgmres->zvecs[it+k] = pipefgmres->zvecs_user_work[nwork][k];
    750:   }

    752:   /* increment the number of work vector chunks */
    753:   pipefgmres->nwork_alloc++;
    754:   return 0;
    755: }

    757: /*@
    758:   KSPPIPEFGMRESSetShift - Set the shift parameter for the flexible, pipelined GMRES solver.

    760:   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).

    762: Logically Collective on ksp

    764: Input Parameters:
    765: +  ksp - the Krylov space context
    766: -  shift - the shift

    768: Level: intermediate

    770: Options Database:
    771: . -ksp_pipefgmres_shift <shift> - set the shift parameter

    773: .seealso: KSPComputeEigenvalues()
    774: @*/
    775: PetscErrorCode KSPPIPEFGMRESSetShift(KSP ksp,PetscScalar shift)
    776: {
    777:   KSP_PIPEFGMRES *pipefgmres = (KSP_PIPEFGMRES*)ksp->data;

    781:   pipefgmres->shift = shift;
    782:   return 0;
    783: }