Actual source code: pipefgmres.c
petsc-3.10.5 2019-03-28
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      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.

     91:  */

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

    109:   if (itcount) *itcount = 0;

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

    115:   /* Allocate memory for orthogonalization work (freed in the GMRES Destroy routine)*/
    116:   /* Note that we add an extra value here to allow for a single reduction */
    117:   if (!pipefgmres->orthogwork) { PetscMalloc1(pipefgmres->max_k + 2 ,&pipefgmres->orthogwork);
    118:   }
    119:   lhh = pipefgmres->orthogwork;

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

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

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

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

    138:   ksp->rnorm = res_norm;
    139:   KSPLogResidualHistory(ksp,res_norm);
    140:   KSPMonitor(ksp,ksp->its,res_norm);

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    307:     PetscObjectSAWsTakeAccess((PetscObject)ksp);
    308:     ksp->its++;
    309:     ksp->rnorm = res_norm;
    310:     PetscObjectSAWsGrantAccess((PetscObject)ksp);

    312:     (*ksp->converged)(ksp,ksp->its,res_norm,&ksp->reason,ksp->cnvP);

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

    328:   /*
    329:      Monitor if we know that we will not return for a restart */
    330:   if (loc_it && (ksp->reason || ksp->its >= ksp->max_it)) {
    331:     KSPMonitor(ksp,ksp->its,res_norm);
    332:   }

    334:   if (itcount) *itcount = loc_it;

    336:   /*
    337:     Down here we have to solve for the "best" coefficients of the Krylov
    338:     columns, add the solution values together, and possibly unwind the
    339:     preconditioning from the solution
    340:    */

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

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

    348:   return(0);
    349: }

    351: /*
    352:     KSPSolve_PIPEFGMRES - This routine applies the PIPEFGMRES method.

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

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

    361: */
    362: static PetscErrorCode KSPSolve_PIPEFGMRES(KSP ksp)
    363: {
    365:   PetscInt       its,itcount;
    366:   KSP_PIPEFGMRES *pipefgmres    = (KSP_PIPEFGMRES*)ksp->data;
    367:   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 (defined(PETSC_USE_COMPLEX) && !defined(PETSC_SKIP_COMPLEX))
    374:   SETERRQ(PETSC_COMM_WORLD,PETSC_ERR_SUP,"PIPEFGMRES has not been implemented for use with complex scalars");
    375: #endif

    377:   PetscCitationsRegister(citation,&cited);

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

    384:   itcount     = 0;
    385:   ksp->reason = KSP_CONVERGED_ITERATING;
    386:   while (!ksp->reason) {
    387:     KSPInitialResidual(ksp,ksp->vec_sol,VEC_TEMP,VEC_TEMP_MATOP,VEC_VV(0),ksp->vec_rhs);
    388:     KSPPIPEFGMRESCycle(&its,ksp);
    389:     itcount += its;
    390:     if (itcount >= ksp->max_it) {
    391:       if (!ksp->reason) ksp->reason = KSP_DIVERGED_ITS;
    392:       break;
    393:     }
    394:     ksp->guess_zero = PETSC_FALSE; /* every future call to KSPInitialResidual() will have nonzero guess */
    395:   }
    396:   ksp->guess_zero = guess_zero; /* restore if user provided nonzero initial guess */
    397:   return(0);
    398: }

    400: static PetscErrorCode KSPDestroy_PIPEFGMRES(KSP ksp)
    401: {

    405:   KSPReset_PIPEFGMRES(ksp);
    406:   KSPDestroy_GMRES(ksp);
    407:   return(0);
    408: }

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

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

    421:      This is an internal routine that knows about the PIPEFGMRES internals.
    422:  */
    423: static PetscErrorCode KSPPIPEFGMRESBuildSoln(PetscScalar *nrs,Vec vguess,Vec vdest,KSP ksp,PetscInt it)
    424: {
    425:   PetscScalar    tt;
    427:   PetscInt       k,j;
    428:   KSP_PIPEFGMRES *pipefgmres = (KSP_PIPEFGMRES*)(ksp->data);

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

    433:   if (it < 0) {                                 /* no pipefgmres steps have been performed */
    434:     VecCopy(vguess,vdest); /* VecCopy() is smart, exits immediately if vguess == vdest */
    435:     return(0);
    436:   }

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

    443:   for (k=it-1; k>=0; k--) {
    444:     tt = *RS(k);
    445:     for (j=k+1; j<=it; j++) tt -= *HH(k,j) * nrs[j];
    446:     nrs[k] = tt / *HH(k,k);
    447:   }

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

    453:   /* add solution to previous solution */
    454:   if (vdest == vguess) {
    455:     VecAXPY(vdest,1.0,VEC_TEMP);
    456:   } else {
    457:     VecWAXPY(vdest,1.0,VEC_TEMP,vguess);
    458:   }
    459:   return(0);
    460: }

    462: /*

    464:     KSPPIPEFGMRESUpdateHessenberg - Do the scalar work for the orthogonalization.
    465:                             Return new residual.

    467:     input parameters:

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

    474:     output parameters:
    475: .        res - the new residual

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

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

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

    498:   /* check for the happy breakdown */
    499:   hapbnd = PetscMin(PetscAbsScalar(hh[it+1] / rs[it]),pipefgmres->haptol);
    500:   if (PetscAbsScalar(hh[it+1]) < hapbnd) {
    501:     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)));
    502:     *hapend = PETSC_TRUE;
    503:   }

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

    510:   for (j=0; j<it; j++) {
    511:     PetscScalar hhj = hh[j];
    512:     hh[j]   = PetscConj(cc[j])*hhj + ss[j]*hh[j+1];
    513:     hh[j+1] =          -ss[j] *hhj + cc[j]*hh[j+1];
    514:   }

    516:   /*
    517:     compute the new plane rotation, and apply it to:
    518:      1) the right-hand-side of the Hessenberg system (RS)
    519:         note: it affects RS(it) and RS(it+1)
    520:      2) the new column of the Hessenberg matrix
    521:         note: it affects HH(it,it) which is currently pointed to
    522:         by hh and HH(it+1, it) (*(hh+1))
    523:     thus obtaining the updated value of the residual...
    524:   */

    526:   /* compute new plane rotation */

    528:   if (!*hapend) {
    529:     PetscReal delta = PetscSqrtReal(PetscSqr(PetscAbsScalar(hh[it])) + PetscSqr(PetscAbsScalar(hh[it+1])));
    530:     if (delta == 0.0) {
    531:       ksp->reason = KSP_DIVERGED_NULL;
    532:       return(0);
    533:     }

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

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

    549:     *res = 0.0;
    550:   }
    551:   return(0);
    552: }

    554: /*
    555:    KSPBuildSolution_PIPEFGMRES

    557:      Input Parameter:
    558: .     ksp - the Krylov space object
    559: .     ptr-

    561:    Output Parameter:
    562: .     result - the solution

    564:    Note: this calls KSPPIPEFGMRESBuildSoln - the same function that KSPPIPEFGMRESCycle
    565:    calls directly.

    567: */
    568: PetscErrorCode KSPBuildSolution_PIPEFGMRES(KSP ksp,Vec ptr,Vec *result)
    569: {
    570:   KSP_PIPEFGMRES *pipefgmres = (KSP_PIPEFGMRES*)ksp->data;

    574:   if (!ptr) {
    575:     if (!pipefgmres->sol_temp) {
    576:       VecDuplicate(ksp->vec_sol,&pipefgmres->sol_temp);
    577:       PetscLogObjectParent((PetscObject)ksp,(PetscObject)pipefgmres->sol_temp);
    578:     }
    579:     ptr = pipefgmres->sol_temp;
    580:   }
    581:   if (!pipefgmres->nrs) {
    582:     /* allocate the work area */
    583:     PetscMalloc1(pipefgmres->max_k,&pipefgmres->nrs);
    584:     PetscLogObjectMemory((PetscObject)ksp,pipefgmres->max_k*sizeof(PetscScalar));
    585:   }

    587:   KSPPIPEFGMRESBuildSoln(pipefgmres->nrs,ksp->vec_sol,ptr,ksp,pipefgmres->it);
    588:   if (result) *result = ptr;
    589:   return(0);
    590: }

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

    600:   KSPSetFromOptions_GMRES(PetscOptionsObject,ksp);
    601:   PetscOptionsHead(PetscOptionsObject,"KSP pipelined FGMRES Options");
    602:   PetscOptionsScalar("-ksp_pipefgmres_shift","shift parameter","KSPPIPEFGMRESSetShift",pipefgmres->shift,&shift,&flg);
    603:   if (flg) { KSPPIPEFGMRESSetShift(ksp,shift); }
    604:   PetscOptionsTail();
    605:   return(0);
    606: }

    608: PetscErrorCode KSPView_PIPEFGMRES(KSP ksp,PetscViewer viewer)
    609: {
    610:   KSP_PIPEFGMRES *pipefgmres = (KSP_PIPEFGMRES*)ksp->data;
    612:   PetscBool      iascii,isstring;

    615:   PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&iascii);
    616:   PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERSTRING,&isstring);

    618:   if (iascii) {
    619:     PetscViewerASCIIPrintf(viewer,"  restart=%D\n",pipefgmres->max_k);
    620:     PetscViewerASCIIPrintf(viewer,"  happy breakdown tolerance %g\n",(double)pipefgmres->haptol);
    621: #if defined(PETSC_USE_COMPLEX)
    622:     PetscViewerASCIIPrintf(viewer,"  shift=%g+%gi\n",PetscRealPart(pipefgmres->shift),PetscImaginaryPart(pipefgmres->shift));
    623: #else
    624:     PetscViewerASCIIPrintf(viewer,"  shift=%g\n",pipefgmres->shift);
    625: #endif
    626:   } else if (isstring) {
    627:     PetscViewerStringSPrintf(viewer,"restart %D",pipefgmres->max_k);
    628: #if defined(PETSC_USE_COMPLEX)
    629:     PetscViewerStringSPrintf(viewer,"   shift=%g+%gi\n",PetscRealPart(pipefgmres->shift),PetscImaginaryPart(pipefgmres->shift));
    630: #else
    631:     PetscViewerStringSPrintf(viewer,"   shift=%g\n",pipefgmres->shift);
    632: #endif
    633:   }
    634:   return(0);
    635: }

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

    644:   PetscFree(pipefgmres->prevecs);
    645:   PetscFree(pipefgmres->zvecs);
    646:   for (i=0; i<pipefgmres->nwork_alloc; i++) {
    647:     VecDestroyVecs(pipefgmres->mwork_alloc[i],&pipefgmres->prevecs_user_work[i]);
    648:     VecDestroyVecs(pipefgmres->mwork_alloc[i],&pipefgmres->zvecs_user_work[i]);
    649:   }
    650:   PetscFree(pipefgmres->prevecs_user_work);
    651:   PetscFree(pipefgmres->zvecs_user_work);
    652:   PetscFree(pipefgmres->redux);
    653:   KSPReset_GMRES(ksp);
    654:   return(0);
    655: }

    657: /*MC
    658:    KSPPIPEFGMRES - Implements the Pipelined Generalized Minimal Residual method.

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

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

    671:    Level: intermediate

    673:    Notes:

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

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

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

    683:    Developer Notes:
    684:     This class is subclassed off of KSPGMRES.

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

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

    696: PETSC_EXTERN PetscErrorCode KSPCreate_PIPEFGMRES(KSP ksp)
    697: {
    698:   KSP_PIPEFGMRES *pipefgmres;

    702:   PetscNewLog(ksp,&pipefgmres);

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

    715:   KSPSetSupportedNorm(ksp,KSP_NORM_UNPRECONDITIONED,PC_RIGHT,3);

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

    721:   pipefgmres->nextra_vecs    = 1;
    722:   pipefgmres->haptol         = 1.0e-30;
    723:   pipefgmres->q_preallocate  = 0;
    724:   pipefgmres->delta_allocate = PIPEFGMRES_DELTA_DIRECTIONS;
    725:   pipefgmres->orthog         = 0;
    726:   pipefgmres->nrs            = 0;
    727:   pipefgmres->sol_temp       = 0;
    728:   pipefgmres->max_k          = PIPEFGMRES_DEFAULT_MAXK;
    729:   pipefgmres->Rsvd           = 0;
    730:   pipefgmres->orthogwork     = 0;
    731:   pipefgmres->cgstype        = KSP_GMRES_CGS_REFINE_NEVER;
    732:   pipefgmres->shift          = 1.0;
    733:   return(0);
    734: }

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

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

    748:   /* Adjust the number to allocate to make sure that we don't exceed the
    749:      number of available slots (pipefgmres->vecs_allocated)*/
    750:   if (it + VEC_OFFSET + nalloc >= pipefgmres->vecs_allocated) {
    751:     nalloc = pipefgmres->vecs_allocated - it - VEC_OFFSET;
    752:   }
    753:   if (!nalloc) return(0);

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

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

    766:   /* preconditioned vectors (note we don't use VEC_OFFSET) */
    767:   KSPCreateVecs(ksp,nalloc,&pipefgmres->prevecs_user_work[nwork],0,NULL);
    768:   PetscLogObjectParents(ksp,nalloc,pipefgmres->prevecs_user_work[nwork]);
    769:   for (k=0; k < nalloc; k++) {
    770:     pipefgmres->prevecs[it+k] = pipefgmres->prevecs_user_work[nwork][k];
    771:   }

    773:   KSPCreateVecs(ksp,nalloc,&pipefgmres->zvecs_user_work[nwork],0,NULL);
    774:   PetscLogObjectParents(ksp,nalloc,pipefgmres->zvecs_user_work[nwork]);
    775:   for (k=0; k < nalloc; k++) {
    776:     pipefgmres->zvecs[it+k] = pipefgmres->zvecs_user_work[nwork][k];
    777:   }

    779:   /* increment the number of work vector chunks */
    780:   pipefgmres->nwork_alloc++;
    781:   return(0);
    782: }
    783: /*@
    784:   KSPPIPEFGMRESSetShift - Set the shift parameter for the flexible, pipelined GMRES solver.

    786:   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).

    788: Logically Collective on KSP

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

    794: Level: intermediate

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

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

    808:   pipefgmres->shift = shift;
    809:   return(0);
    810: }