xref: /petsc/src/ksp/pc/impls/mg/mg.c (revision 5544a5bc068ab743e489f2e333f3664c6a477a2a)

/*
    Defines the multigrid preconditioner interface.
*/
#include <petsc/private/pcmgimpl.h> /*I "petscksp.h" I*/
#include <petsc/private/kspimpl.h>
#include <petscdm.h>
PETSC_INTERN PetscErrorCode PCPreSolveChangeRHS(PC, PetscBool *);

/*
   Contains the list of registered coarse space construction routines
*/
PetscFunctionList PCMGCoarseList = NULL;

PetscErrorCode PCMGMCycle_Private(PC pc, PC_MG_Levels **mglevelsin, PetscBool transpose, PetscBool matapp, PCRichardsonConvergedReason *reason)
{
  PC_MG        *mg = (PC_MG *)pc->data;
  PC_MG_Levels *mgc, *mglevels = *mglevelsin;
  PetscInt      cycles = (mglevels->level == 1) ? 1 : mglevels->cycles;

  PetscFunctionBegin;
  if (mglevels->eventsmoothsolve) PetscCall(PetscLogEventBegin(mglevels->eventsmoothsolve, 0, 0, 0, 0));
  if (!transpose) {
    if (matapp) {
      PetscCall(KSPMatSolve(mglevels->smoothd, mglevels->B, mglevels->X)); /* pre-smooth */
      PetscCall(KSPCheckSolve(mglevels->smoothd, pc, NULL));
    } else {
      PetscCall(KSPSolve(mglevels->smoothd, mglevels->b, mglevels->x)); /* pre-smooth */
      PetscCall(KSPCheckSolve(mglevels->smoothd, pc, mglevels->x));
    }
  } else {
    PetscCheck(!matapp, PetscObjectComm((PetscObject)pc), PETSC_ERR_SUP, "Not supported");
    PetscCall(KSPSolveTranspose(mglevels->smoothu, mglevels->b, mglevels->x)); /* transpose of post-smooth */
    PetscCall(KSPCheckSolve(mglevels->smoothu, pc, mglevels->x));
  }
  if (mglevels->eventsmoothsolve) PetscCall(PetscLogEventEnd(mglevels->eventsmoothsolve, 0, 0, 0, 0));
  if (mglevels->level) { /* not the coarsest grid */
    if (mglevels->eventresidual) PetscCall(PetscLogEventBegin(mglevels->eventresidual, 0, 0, 0, 0));
    if (matapp && !mglevels->R) PetscCall(MatDuplicate(mglevels->B, MAT_DO_NOT_COPY_VALUES, &mglevels->R));
    if (!transpose) {
      if (matapp) PetscCall((*mglevels->matresidual)(mglevels->A, mglevels->B, mglevels->X, mglevels->R));
      else PetscCall((*mglevels->residual)(mglevels->A, mglevels->b, mglevels->x, mglevels->r));
    } else {
      if (matapp) PetscCall((*mglevels->matresidualtranspose)(mglevels->A, mglevels->B, mglevels->X, mglevels->R));
      else PetscCall((*mglevels->residualtranspose)(mglevels->A, mglevels->b, mglevels->x, mglevels->r));
    }
    if (mglevels->eventresidual) PetscCall(PetscLogEventEnd(mglevels->eventresidual, 0, 0, 0, 0));

    /* if on finest level and have convergence criteria set */
    if (mglevels->level == mglevels->levels - 1 && mg->ttol && reason) {
      PetscReal rnorm;
      PetscCall(VecNorm(mglevels->r, NORM_2, &rnorm));
      if (rnorm <= mg->ttol) {
        if (rnorm < mg->abstol) {
          *reason = PCRICHARDSON_CONVERGED_ATOL;
          PetscCall(PetscInfo(pc, "Linear solver has converged. Residual norm %g is less than absolute tolerance %g\n", (double)rnorm, (double)mg->abstol));
        } else {
          *reason = PCRICHARDSON_CONVERGED_RTOL;
          PetscCall(PetscInfo(pc, "Linear solver has converged. Residual norm %g is less than relative tolerance times initial residual norm %g\n", (double)rnorm, (double)mg->ttol));
        }
        PetscFunctionReturn(PETSC_SUCCESS);
      }
    }

    mgc = *(mglevelsin - 1);
    if (mglevels->eventinterprestrict) PetscCall(PetscLogEventBegin(mglevels->eventinterprestrict, 0, 0, 0, 0));
    if (!transpose) {
      if (matapp) PetscCall(MatMatRestrict(mglevels->restrct, mglevels->R, &mgc->B));
      else PetscCall(MatRestrict(mglevels->restrct, mglevels->r, mgc->b));
    } else {
      if (matapp) PetscCall(MatMatRestrict(mglevels->interpolate, mglevels->R, &mgc->B));
      else PetscCall(MatRestrict(mglevels->interpolate, mglevels->r, mgc->b));
    }
    if (mglevels->eventinterprestrict) PetscCall(PetscLogEventEnd(mglevels->eventinterprestrict, 0, 0, 0, 0));
    if (matapp) {
      if (!mgc->X) {
        PetscCall(MatDuplicate(mgc->B, MAT_DO_NOT_COPY_VALUES, &mgc->X));
      } else {
        PetscCall(MatZeroEntries(mgc->X));
      }
    } else {
      PetscCall(VecZeroEntries(mgc->x));
    }
    while (cycles--) PetscCall(PCMGMCycle_Private(pc, mglevelsin - 1, transpose, matapp, reason));
    if (mglevels->eventinterprestrict) PetscCall(PetscLogEventBegin(mglevels->eventinterprestrict, 0, 0, 0, 0));
    if (!transpose) {
      if (matapp) PetscCall(MatMatInterpolateAdd(mglevels->interpolate, mgc->X, mglevels->X, &mglevels->X));
      else PetscCall(MatInterpolateAdd(mglevels->interpolate, mgc->x, mglevels->x, mglevels->x));
    } else {
      PetscCall(MatInterpolateAdd(mglevels->restrct, mgc->x, mglevels->x, mglevels->x));
    }
    if (mglevels->eventinterprestrict) PetscCall(PetscLogEventEnd(mglevels->eventinterprestrict, 0, 0, 0, 0));
    if (mglevels->eventsmoothsolve) PetscCall(PetscLogEventBegin(mglevels->eventsmoothsolve, 0, 0, 0, 0));
    if (!transpose) {
      if (matapp) {
        PetscCall(KSPMatSolve(mglevels->smoothu, mglevels->B, mglevels->X)); /* post smooth */
        PetscCall(KSPCheckSolve(mglevels->smoothu, pc, NULL));
      } else {
        PetscCall(KSPSolve(mglevels->smoothu, mglevels->b, mglevels->x)); /* post smooth */
        PetscCall(KSPCheckSolve(mglevels->smoothu, pc, mglevels->x));
      }
    } else {
      PetscCheck(!matapp, PetscObjectComm((PetscObject)pc), PETSC_ERR_SUP, "Not supported");
      PetscCall(KSPSolveTranspose(mglevels->smoothd, mglevels->b, mglevels->x)); /* post smooth */
      PetscCall(KSPCheckSolve(mglevels->smoothd, pc, mglevels->x));
    }
    if (mglevels->cr) {
      PetscCheck(!matapp, PetscObjectComm((PetscObject)pc), PETSC_ERR_SUP, "Not supported");
      /* TODO Turn on copy and turn off noisy if we have an exact solution
      PetscCall(VecCopy(mglevels->x, mglevels->crx));
      PetscCall(VecCopy(mglevels->b, mglevels->crb)); */
      PetscCall(KSPSetNoisy_Private(mglevels->crx));
      PetscCall(KSPSolve(mglevels->cr, mglevels->crb, mglevels->crx)); /* compatible relaxation */
      PetscCall(KSPCheckSolve(mglevels->cr, pc, mglevels->crx));
    }
    if (mglevels->eventsmoothsolve) PetscCall(PetscLogEventEnd(mglevels->eventsmoothsolve, 0, 0, 0, 0));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode PCApplyRichardson_MG(PC pc, Vec b, Vec x, Vec w, PetscReal rtol, PetscReal abstol, PetscReal dtol, PetscInt its, PetscBool zeroguess, PetscInt *outits, PCRichardsonConvergedReason *reason)
{
  PC_MG         *mg       = (PC_MG *)pc->data;
  PC_MG_Levels **mglevels = mg->levels;
  PC             tpc;
  PetscBool      changeu, changed;
  PetscInt       levels = mglevels[0]->levels, i;

  PetscFunctionBegin;
  /* When the DM is supplying the matrix then it will not exist until here */
  for (i = 0; i < levels; i++) {
    if (!mglevels[i]->A) {
      PetscCall(KSPGetOperators(mglevels[i]->smoothu, &mglevels[i]->A, NULL));
      PetscCall(PetscObjectReference((PetscObject)mglevels[i]->A));
    }
  }

  PetscCall(KSPGetPC(mglevels[levels - 1]->smoothd, &tpc));
  PetscCall(PCPreSolveChangeRHS(tpc, &changed));
  PetscCall(KSPGetPC(mglevels[levels - 1]->smoothu, &tpc));
  PetscCall(PCPreSolveChangeRHS(tpc, &changeu));
  if (!changed && !changeu) {
    PetscCall(VecDestroy(&mglevels[levels - 1]->b));
    mglevels[levels - 1]->b = b;
  } else { /* if the smoother changes the rhs during PreSolve, we cannot use the input vector */
    if (!mglevels[levels - 1]->b) {
      Vec *vec;

      PetscCall(KSPCreateVecs(mglevels[levels - 1]->smoothd, 1, &vec, 0, NULL));
      mglevels[levels - 1]->b = *vec;
      PetscCall(PetscFree(vec));
    }
    PetscCall(VecCopy(b, mglevels[levels - 1]->b));
  }
  mglevels[levels - 1]->x = x;

  mg->rtol   = rtol;
  mg->abstol = abstol;
  mg->dtol   = dtol;
  if (rtol) {
    /* compute initial residual norm for relative convergence test */
    PetscReal rnorm;
    if (zeroguess) {
      PetscCall(VecNorm(b, NORM_2, &rnorm));
    } else {
      PetscCall((*mglevels[levels - 1]->residual)(mglevels[levels - 1]->A, b, x, w));
      PetscCall(VecNorm(w, NORM_2, &rnorm));
    }
    mg->ttol = PetscMax(rtol * rnorm, abstol);
  } else if (abstol) mg->ttol = abstol;
  else mg->ttol = 0.0;

  /* since smoother is applied to full system, not just residual we need to make sure that smoothers don't
     stop prematurely due to small residual */
  for (i = 1; i < levels; i++) {
    PetscCall(KSPSetTolerances(mglevels[i]->smoothu, 0, PETSC_CURRENT, PETSC_CURRENT, PETSC_CURRENT));
    if (mglevels[i]->smoothu != mglevels[i]->smoothd) {
      /* For Richardson the initial guess is nonzero since it is solving in each cycle the original system not just applying as a preconditioner */
      PetscCall(KSPSetInitialGuessNonzero(mglevels[i]->smoothd, PETSC_TRUE));
      PetscCall(KSPSetTolerances(mglevels[i]->smoothd, 0, PETSC_CURRENT, PETSC_CURRENT, PETSC_CURRENT));
    }
  }

  *reason = PCRICHARDSON_NOT_SET;
  for (i = 0; i < its; i++) {
    PetscCall(PCMGMCycle_Private(pc, mglevels + levels - 1, PETSC_FALSE, PETSC_FALSE, reason));
    if (*reason) break;
  }
  if (*reason == PCRICHARDSON_NOT_SET) *reason = PCRICHARDSON_CONVERGED_ITS;
  *outits = i;
  if (!changed && !changeu) mglevels[levels - 1]->b = NULL;
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode PCReset_MG(PC pc)
{
  PC_MG         *mg       = (PC_MG *)pc->data;
  PC_MG_Levels **mglevels = mg->levels;
  PetscInt       i, n;

  PetscFunctionBegin;
  if (mglevels) {
    n = mglevels[0]->levels;
    for (i = 0; i < n - 1; i++) {
      PetscCall(VecDestroy(&mglevels[i + 1]->r));
      PetscCall(VecDestroy(&mglevels[i]->b));
      PetscCall(VecDestroy(&mglevels[i]->x));
      PetscCall(MatDestroy(&mglevels[i + 1]->R));
      PetscCall(MatDestroy(&mglevels[i]->B));
      PetscCall(MatDestroy(&mglevels[i]->X));
      PetscCall(VecDestroy(&mglevels[i]->crx));
      PetscCall(VecDestroy(&mglevels[i]->crb));
      PetscCall(MatDestroy(&mglevels[i + 1]->restrct));
      PetscCall(MatDestroy(&mglevels[i + 1]->interpolate));
      PetscCall(MatDestroy(&mglevels[i + 1]->inject));
      PetscCall(VecDestroy(&mglevels[i + 1]->rscale));
    }
    PetscCall(VecDestroy(&mglevels[n - 1]->crx));
    PetscCall(VecDestroy(&mglevels[n - 1]->crb));
    /* this is not null only if the smoother on the finest level
       changes the rhs during PreSolve */
    PetscCall(VecDestroy(&mglevels[n - 1]->b));
    PetscCall(MatDestroy(&mglevels[n - 1]->B));

    for (i = 0; i < n; i++) {
      PetscCall(MatDestroy(&mglevels[i]->coarseSpace));
      PetscCall(MatDestroy(&mglevels[i]->A));
      if (mglevels[i]->smoothd != mglevels[i]->smoothu) PetscCall(KSPReset(mglevels[i]->smoothd));
      PetscCall(KSPReset(mglevels[i]->smoothu));
      if (mglevels[i]->cr) PetscCall(KSPReset(mglevels[i]->cr));
    }
    mg->Nc = 0;
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/* Implementing CR

We only want to make corrections that ``do not change'' the coarse solution. What we mean by not changing is that if I prolong my coarse solution to the fine grid and then inject that fine solution back to the coarse grid, I get the same answer. Injection is what Brannick calls R. We want the complementary projector to Inj, which we will call S, after Brannick, so that Inj S = 0. Now the orthogonal projector onto the range of Inj^T is

  Inj^T (Inj Inj^T)^{-1} Inj

and if Inj is a VecScatter, as it is now in PETSc, we have

  Inj^T Inj

and

  S = I - Inj^T Inj

since

  Inj S = Inj - (Inj Inj^T) Inj = 0.

Brannick suggests

  A \to S^T A S  \qquad\mathrm{and}\qquad M \to S^T M S

but I do not think his :math:`S^T S = I` is correct. Our S is an orthogonal projector, so :math:`S^T S = S^2 = S`. We will use

  M^{-1} A \to S M^{-1} A S

In fact, since it is somewhat hard in PETSc to do the symmetric application, we will just apply S on the left.

  Check: || Inj P - I ||_F < tol
  Check: In general, Inj Inj^T = I
*/

typedef struct {
  PC       mg;  /* The PCMG object */
  PetscInt l;   /* The multigrid level for this solver */
  Mat      Inj; /* The injection matrix */
  Mat      S;   /* I - Inj^T Inj */
} CRContext;

static PetscErrorCode CRSetup_Private(PC pc)
{
  CRContext *ctx;
  Mat        It;

  PetscFunctionBeginUser;
  PetscCall(PCShellGetContext(pc, &ctx));
  PetscCall(PCMGGetInjection(ctx->mg, ctx->l, &It));
  PetscCheck(It, PetscObjectComm((PetscObject)pc), PETSC_ERR_ARG_WRONGSTATE, "CR requires that injection be defined for this PCMG");
  PetscCall(MatCreateTranspose(It, &ctx->Inj));
  PetscCall(MatCreateNormal(ctx->Inj, &ctx->S));
  PetscCall(MatScale(ctx->S, -1.0));
  PetscCall(MatShift(ctx->S, 1.0));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode CRApply_Private(PC pc, Vec x, Vec y)
{
  CRContext *ctx;

  PetscFunctionBeginUser;
  PetscCall(PCShellGetContext(pc, &ctx));
  PetscCall(MatMult(ctx->S, x, y));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode CRDestroy_Private(PC pc)
{
  CRContext *ctx;

  PetscFunctionBeginUser;
  PetscCall(PCShellGetContext(pc, &ctx));
  PetscCall(MatDestroy(&ctx->Inj));
  PetscCall(MatDestroy(&ctx->S));
  PetscCall(PetscFree(ctx));
  PetscCall(PCShellSetContext(pc, NULL));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode CreateCR_Private(PC pc, PetscInt l, PC *cr)
{
  CRContext *ctx;

  PetscFunctionBeginUser;
  PetscCall(PCCreate(PetscObjectComm((PetscObject)pc), cr));
  PetscCall(PetscObjectSetName((PetscObject)*cr, "S (complementary projector to injection)"));
  PetscCall(PetscCalloc1(1, &ctx));
  ctx->mg = pc;
  ctx->l  = l;
  PetscCall(PCSetType(*cr, PCSHELL));
  PetscCall(PCShellSetContext(*cr, ctx));
  PetscCall(PCShellSetApply(*cr, CRApply_Private));
  PetscCall(PCShellSetSetUp(*cr, CRSetup_Private));
  PetscCall(PCShellSetDestroy(*cr, CRDestroy_Private));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PETSC_EXTERN PetscErrorCode PetscOptionsFindPairPrefix_Private(PetscOptions, const char[], const char[], const char *[], const char *[], PetscBool *);

PetscErrorCode PCMGSetLevels_MG(PC pc, PetscInt levels, MPI_Comm *comms)
{
  PC_MG         *mg = (PC_MG *)pc->data;
  MPI_Comm       comm;
  PC_MG_Levels **mglevels = mg->levels;
  PCMGType       mgtype   = mg->am;
  PetscInt       mgctype  = (PetscInt)PC_MG_CYCLE_V;
  PetscInt       i;
  PetscMPIInt    size;
  const char    *prefix;
  PC             ipc;
  PetscInt       n;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(pc, PC_CLASSID, 1);
  PetscValidLogicalCollectiveInt(pc, levels, 2);
  if (mg->nlevels == levels) PetscFunctionReturn(PETSC_SUCCESS);
  PetscCall(PetscObjectGetComm((PetscObject)pc, &comm));
  if (mglevels) {
    mgctype = mglevels[0]->cycles;
    /* changing the number of levels so free up the previous stuff */
    PetscCall(PCReset_MG(pc));
    n = mglevels[0]->levels;
    for (i = 0; i < n; i++) {
      if (mglevels[i]->smoothd != mglevels[i]->smoothu) PetscCall(KSPDestroy(&mglevels[i]->smoothd));
      PetscCall(KSPDestroy(&mglevels[i]->smoothu));
      PetscCall(KSPDestroy(&mglevels[i]->cr));
      PetscCall(PetscFree(mglevels[i]));
    }
    PetscCall(PetscFree(mg->levels));
  }

  mg->nlevels = levels;

  PetscCall(PetscMalloc1(levels, &mglevels));

  PetscCall(PCGetOptionsPrefix(pc, &prefix));

  mg->stageApply = 0;
  for (i = 0; i < levels; i++) {
    PetscCall(PetscNew(&mglevels[i]));

    mglevels[i]->level               = i;
    mglevels[i]->levels              = levels;
    mglevels[i]->cycles              = mgctype;
    mg->default_smoothu              = 2;
    mg->default_smoothd              = 2;
    mglevels[i]->eventsmoothsetup    = 0;
    mglevels[i]->eventsmoothsolve    = 0;
    mglevels[i]->eventresidual       = 0;
    mglevels[i]->eventinterprestrict = 0;

    if (comms) comm = comms[i];
    if (comm != MPI_COMM_NULL) {
      PetscCall(KSPCreate(comm, &mglevels[i]->smoothd));
      PetscCall(KSPSetNestLevel(mglevels[i]->smoothd, pc->kspnestlevel));
      PetscCall(KSPSetErrorIfNotConverged(mglevels[i]->smoothd, pc->erroriffailure));
      PetscCall(PetscObjectIncrementTabLevel((PetscObject)mglevels[i]->smoothd, (PetscObject)pc, levels - i));
      PetscCall(KSPSetOptionsPrefix(mglevels[i]->smoothd, prefix));
      PetscCall(PetscObjectComposedDataSetInt((PetscObject)mglevels[i]->smoothd, PetscMGLevelId, mglevels[i]->level));
      if (i == 0 && levels > 1) { // coarse grid
        PetscCall(KSPAppendOptionsPrefix(mglevels[0]->smoothd, "mg_coarse_"));

        /* coarse solve is (redundant) LU by default; set shifttype NONZERO to avoid annoying zero-pivot in LU preconditioner */
        PetscCall(KSPSetType(mglevels[0]->smoothd, KSPPREONLY));
        PetscCall(KSPGetPC(mglevels[0]->smoothd, &ipc));
        PetscCallMPI(MPI_Comm_size(comm, &size));
        if (size > 1) {
          PetscCall(PCSetType(ipc, PCREDUNDANT));
        } else {
          PetscCall(PCSetType(ipc, PCLU));
        }
        PetscCall(PCFactorSetShiftType(ipc, MAT_SHIFT_INBLOCKS));
      } else {
        char tprefix[128];

        PetscCall(KSPSetType(mglevels[i]->smoothd, KSPCHEBYSHEV));
        PetscCall(KSPSetConvergenceTest(mglevels[i]->smoothd, KSPConvergedSkip, NULL, NULL));
        PetscCall(KSPSetNormType(mglevels[i]->smoothd, KSP_NORM_NONE));
        PetscCall(KSPGetPC(mglevels[i]->smoothd, &ipc));
        PetscCall(PCSetType(ipc, PCSOR));
        PetscCall(KSPSetTolerances(mglevels[i]->smoothd, PETSC_CURRENT, PETSC_CURRENT, PETSC_CURRENT, mg->default_smoothd));

        if (i == levels - 1 && levels > 1) { // replace 'mg_finegrid_' with 'mg_levels_X_'
          PetscBool set;
          PetscCall(PetscOptionsFindPairPrefix_Private(((PetscObject)mglevels[i]->smoothd)->options, ((PetscObject)mglevels[i]->smoothd)->prefix, "-mg_fine_", NULL, NULL, &set));
          if (set) {
            if (prefix) PetscCall(PetscSNPrintf(tprefix, 128, "%smg_fine_", prefix));
            else PetscCall(PetscSNPrintf(tprefix, 128, "mg_fine_"));
            PetscCall(KSPSetOptionsPrefix(mglevels[i]->smoothd, tprefix));
          } else {
            PetscCall(PetscSNPrintf(tprefix, 128, "mg_levels_%" PetscInt_FMT "_", i));
            PetscCall(KSPAppendOptionsPrefix(mglevels[i]->smoothd, tprefix));
          }
        } else {
          PetscCall(PetscSNPrintf(tprefix, 128, "mg_levels_%" PetscInt_FMT "_", i));
          PetscCall(KSPAppendOptionsPrefix(mglevels[i]->smoothd, tprefix));
        }
      }
    }
    mglevels[i]->smoothu = mglevels[i]->smoothd;
    mg->rtol             = 0.0;
    mg->abstol           = 0.0;
    mg->dtol             = 0.0;
    mg->ttol             = 0.0;
    mg->cyclesperpcapply = 1;
  }
  mg->levels = mglevels;
  PetscCall(PCMGSetType(pc, mgtype));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  PCMGSetLevels - Sets the number of levels to use with `PCMG`.
  Must be called before any other `PCMG` routine.

  Logically Collective

  Input Parameters:
+ pc     - the preconditioner context
. levels - the number of levels
- comms  - optional communicators for each level; this is to allow solving the coarser problems
           on smaller sets of processes. For processes that are not included in the computation
           you must pass `MPI_COMM_NULL`. Use comms = `NULL` to specify that all processes
           should participate in each level of problem.

  Level: intermediate

  Notes:
  If the number of levels is one then the multigrid uses the `-mg_levels` prefix
  for setting the level options rather than the `-mg_coarse` or `-mg_fine` prefix.

  You can free the information in comms after this routine is called.

  The array of MPI communicators must contain `MPI_COMM_NULL` for those ranks that at each level
  are not participating in the coarser solve. For example, with 2 levels and 1 and 2 ranks on
  the two levels, rank 0 in the original communicator will pass in an array of 2 communicators
  of size 2 and 1, while rank 1 in the original communicator will pass in array of 2 communicators
  the first of size 2 and the second of value `MPI_COMM_NULL` since the rank 1 does not participate
  in the coarse grid solve.

  Since each coarser level may have a new `MPI_Comm` with fewer ranks than the previous, one
  must take special care in providing the restriction and interpolation operation. We recommend
  providing these as two step operations; first perform a standard restriction or interpolation on
  the full number of ranks for that level and then use an MPI call to copy the resulting vector
  array entries (after calls to VecGetArray()) to the smaller or larger number of ranks, note in both
  cases the MPI calls must be made on the larger of the two communicators. Traditional MPI send and
  receives or `MPI_AlltoAllv()` could be used to do the reshuffling of the vector entries.

  Fortran Notes:
  Use comms = `PETSC_NULL_MPI_COMM` as the equivalent of `NULL` in the C interface. Note `PETSC_NULL_MPI_COMM`
  is not `MPI_COMM_NULL`. It is more like `PETSC_NULL_INTEGER`, `PETSC_NULL_REAL` etc.

.seealso: [](ch_ksp), `PCMGSetType()`, `PCMGGetLevels()`
@*/
PetscErrorCode PCMGSetLevels(PC pc, PetscInt levels, MPI_Comm *comms)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(pc, PC_CLASSID, 1);
  if (comms) PetscAssertPointer(comms, 3);
  PetscTryMethod(pc, "PCMGSetLevels_C", (PC, PetscInt, MPI_Comm *), (pc, levels, comms));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode PCDestroy_MG(PC pc)
{
  PC_MG         *mg       = (PC_MG *)pc->data;
  PC_MG_Levels **mglevels = mg->levels;
  PetscInt       i, n;

  PetscFunctionBegin;
  PetscCall(PCReset_MG(pc));
  if (mglevels) {
    n = mglevels[0]->levels;
    for (i = 0; i < n; i++) {
      if (mglevels[i]->smoothd != mglevels[i]->smoothu) PetscCall(KSPDestroy(&mglevels[i]->smoothd));
      PetscCall(KSPDestroy(&mglevels[i]->smoothu));
      PetscCall(KSPDestroy(&mglevels[i]->cr));
      PetscCall(PetscFree(mglevels[i]));
    }
    PetscCall(PetscFree(mg->levels));
  }
  PetscCall(PetscFree(pc->data));
  PetscCall(PetscObjectComposeFunction((PetscObject)pc, "PCGetInterpolations_C", NULL));
  PetscCall(PetscObjectComposeFunction((PetscObject)pc, "PCGetCoarseOperators_C", NULL));
  PetscCall(PetscObjectComposeFunction((PetscObject)pc, "PCMGSetGalerkin_C", NULL));
  PetscCall(PetscObjectComposeFunction((PetscObject)pc, "PCMGGetLevels_C", NULL));
  PetscCall(PetscObjectComposeFunction((PetscObject)pc, "PCMGSetLevels_C", NULL));
  PetscCall(PetscObjectComposeFunction((PetscObject)pc, "PCGetInterpolations_C", NULL));
  PetscCall(PetscObjectComposeFunction((PetscObject)pc, "PCGetCoarseOperators_C", NULL));
  PetscCall(PetscObjectComposeFunction((PetscObject)pc, "PCMGSetAdaptInterpolation_C", NULL));
  PetscCall(PetscObjectComposeFunction((PetscObject)pc, "PCMGGetAdaptInterpolation_C", NULL));
  PetscCall(PetscObjectComposeFunction((PetscObject)pc, "PCMGSetAdaptCR_C", NULL));
  PetscCall(PetscObjectComposeFunction((PetscObject)pc, "PCMGGetAdaptCR_C", NULL));
  PetscCall(PetscObjectComposeFunction((PetscObject)pc, "PCMGSetAdaptCoarseSpaceType_C", NULL));
  PetscCall(PetscObjectComposeFunction((PetscObject)pc, "PCMGGetAdaptCoarseSpaceType_C", NULL));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*
   PCApply_MG - Runs either an additive, multiplicative, Kaskadic
             or full cycle of multigrid.

  Note:
  A simple wrapper which calls PCMGMCycle(),PCMGACycle(), or PCMGFCycle().
*/
static PetscErrorCode PCApply_MG_Internal(PC pc, Vec b, Vec x, Mat B, Mat X, PetscBool transpose)
{
  PC_MG         *mg       = (PC_MG *)pc->data;
  PC_MG_Levels **mglevels = mg->levels;
  PC             tpc;
  PetscInt       levels = mglevels[0]->levels, i;
  PetscBool      changeu, changed, matapp;

  PetscFunctionBegin;
  matapp = (PetscBool)(B && X);
  if (mg->stageApply) PetscCall(PetscLogStagePush(mg->stageApply));
  /* When the DM is supplying the matrix then it will not exist until here */
  for (i = 0; i < levels; i++) {
    if (!mglevels[i]->A) {
      PetscCall(KSPGetOperators(mglevels[i]->smoothu, &mglevels[i]->A, NULL));
      PetscCall(PetscObjectReference((PetscObject)mglevels[i]->A));
    }
  }

  PetscCall(KSPGetPC(mglevels[levels - 1]->smoothd, &tpc));
  PetscCall(PCPreSolveChangeRHS(tpc, &changed));
  PetscCall(KSPGetPC(mglevels[levels - 1]->smoothu, &tpc));
  PetscCall(PCPreSolveChangeRHS(tpc, &changeu));
  if (!changeu && !changed) {
    if (matapp) {
      PetscCall(MatDestroy(&mglevels[levels - 1]->B));
      mglevels[levels - 1]->B = B;
    } else {
      PetscCall(VecDestroy(&mglevels[levels - 1]->b));
      mglevels[levels - 1]->b = b;
    }
  } else { /* if the smoother changes the rhs during PreSolve, we cannot use the input vector */
    if (matapp) {
      if (mglevels[levels - 1]->B) {
        PetscInt  N1, N2;
        PetscBool flg;

        PetscCall(MatGetSize(mglevels[levels - 1]->B, NULL, &N1));
        PetscCall(MatGetSize(B, NULL, &N2));
        PetscCall(PetscObjectTypeCompare((PetscObject)mglevels[levels - 1]->B, ((PetscObject)B)->type_name, &flg));
        if (N1 != N2 || !flg) PetscCall(MatDestroy(&mglevels[levels - 1]->B));
      }
      if (!mglevels[levels - 1]->B) {
        PetscCall(MatDuplicate(B, MAT_COPY_VALUES, &mglevels[levels - 1]->B));
      } else {
        PetscCall(MatCopy(B, mglevels[levels - 1]->B, SAME_NONZERO_PATTERN));
      }
    } else {
      if (!mglevels[levels - 1]->b) {
        Vec *vec;

        PetscCall(KSPCreateVecs(mglevels[levels - 1]->smoothd, 1, &vec, 0, NULL));
        mglevels[levels - 1]->b = *vec;
        PetscCall(PetscFree(vec));
      }
      PetscCall(VecCopy(b, mglevels[levels - 1]->b));
    }
  }
  if (matapp) {
    mglevels[levels - 1]->X = X;
  } else {
    mglevels[levels - 1]->x = x;
  }

  /* If coarser Xs are present, it means we have already block applied the PC at least once
     Reset operators if sizes/type do no match */
  if (matapp && levels > 1 && mglevels[levels - 2]->X) {
    PetscInt  Xc, Bc;
    PetscBool flg;

    PetscCall(MatGetSize(mglevels[levels - 2]->X, NULL, &Xc));
    PetscCall(MatGetSize(mglevels[levels - 1]->B, NULL, &Bc));
    PetscCall(PetscObjectTypeCompare((PetscObject)mglevels[levels - 2]->X, ((PetscObject)mglevels[levels - 1]->X)->type_name, &flg));
    if (Xc != Bc || !flg) {
      PetscCall(MatDestroy(&mglevels[levels - 1]->R));
      for (i = 0; i < levels - 1; i++) {
        PetscCall(MatDestroy(&mglevels[i]->R));
        PetscCall(MatDestroy(&mglevels[i]->B));
        PetscCall(MatDestroy(&mglevels[i]->X));
      }
    }
  }

  if (mg->am == PC_MG_MULTIPLICATIVE) {
    if (matapp) PetscCall(MatZeroEntries(X));
    else PetscCall(VecZeroEntries(x));
    for (i = 0; i < mg->cyclesperpcapply; i++) PetscCall(PCMGMCycle_Private(pc, mglevels + levels - 1, transpose, matapp, NULL));
  } else if (mg->am == PC_MG_ADDITIVE) {
    PetscCall(PCMGACycle_Private(pc, mglevels, transpose, matapp));
  } else if (mg->am == PC_MG_KASKADE) {
    PetscCall(PCMGKCycle_Private(pc, mglevels, transpose, matapp));
  } else {
    PetscCall(PCMGFCycle_Private(pc, mglevels, transpose, matapp));
  }
  if (mg->stageApply) PetscCall(PetscLogStagePop());
  if (!changeu && !changed) {
    if (matapp) {
      mglevels[levels - 1]->B = NULL;
    } else {
      mglevels[levels - 1]->b = NULL;
    }
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode PCApply_MG(PC pc, Vec b, Vec x)
{
  PetscFunctionBegin;
  PetscCall(PCApply_MG_Internal(pc, b, x, NULL, NULL, PETSC_FALSE));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode PCApplyTranspose_MG(PC pc, Vec b, Vec x)
{
  PetscFunctionBegin;
  PetscCall(PCApply_MG_Internal(pc, b, x, NULL, NULL, PETSC_TRUE));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode PCMatApply_MG(PC pc, Mat b, Mat x)
{
  PetscFunctionBegin;
  PetscCall(PCApply_MG_Internal(pc, NULL, NULL, b, x, PETSC_FALSE));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode PCSetFromOptions_MG(PC pc, PetscOptionItems *PetscOptionsObject)
{
  PetscInt            levels, cycles;
  PetscBool           flg, flg2;
  PC_MG              *mg = (PC_MG *)pc->data;
  PC_MG_Levels      **mglevels;
  PCMGType            mgtype;
  PCMGCycleType       mgctype;
  PCMGGalerkinType    gtype;
  PCMGCoarseSpaceType coarseSpaceType;

  PetscFunctionBegin;
  levels = PetscMax(mg->nlevels, 1);
  PetscOptionsHeadBegin(PetscOptionsObject, "Multigrid options");
  PetscCall(PetscOptionsInt("-pc_mg_levels", "Number of Levels", "PCMGSetLevels", levels, &levels, &flg));
  if (!flg && !mg->levels && pc->dm) {
    PetscCall(DMGetRefineLevel(pc->dm, &levels));
    levels++;
    mg->usedmfornumberoflevels = PETSC_TRUE;
  }
  PetscCall(PCMGSetLevels(pc, levels, NULL));
  mglevels = mg->levels;

  mgctype = (PCMGCycleType)mglevels[0]->cycles;
  PetscCall(PetscOptionsEnum("-pc_mg_cycle_type", "V cycle or for W-cycle", "PCMGSetCycleType", PCMGCycleTypes, (PetscEnum)mgctype, (PetscEnum *)&mgctype, &flg));
  if (flg) PetscCall(PCMGSetCycleType(pc, mgctype));
  coarseSpaceType = mg->coarseSpaceType;
  PetscCall(PetscOptionsEnum("-pc_mg_adapt_interp_coarse_space", "Type of adaptive coarse space: none, polynomial, harmonic, eigenvector, generalized_eigenvector, gdsw", "PCMGSetAdaptCoarseSpaceType", PCMGCoarseSpaceTypes, (PetscEnum)coarseSpaceType, (PetscEnum *)&coarseSpaceType, &flg));
  if (flg) PetscCall(PCMGSetAdaptCoarseSpaceType(pc, coarseSpaceType));
  PetscCall(PetscOptionsInt("-pc_mg_adapt_interp_n", "Size of the coarse space for adaptive interpolation", "PCMGSetCoarseSpace", mg->Nc, &mg->Nc, &flg));
  PetscCall(PetscOptionsBool("-pc_mg_mesp_monitor", "Monitor the multilevel eigensolver", "PCMGSetAdaptInterpolation", PETSC_FALSE, &mg->mespMonitor, &flg));
  flg2 = PETSC_FALSE;
  PetscCall(PetscOptionsBool("-pc_mg_adapt_cr", "Monitor coarse space quality using Compatible Relaxation (CR)", "PCMGSetAdaptCR", PETSC_FALSE, &flg2, &flg));
  if (flg) PetscCall(PCMGSetAdaptCR(pc, flg2));
  flg = PETSC_FALSE;
  PetscCall(PetscOptionsBool("-pc_mg_distinct_smoothup", "Create separate smoothup KSP and append the prefix _up", "PCMGSetDistinctSmoothUp", PETSC_FALSE, &flg, NULL));
  if (flg) PetscCall(PCMGSetDistinctSmoothUp(pc));
  PetscCall(PetscOptionsEnum("-pc_mg_galerkin", "Use Galerkin process to compute coarser operators", "PCMGSetGalerkin", PCMGGalerkinTypes, (PetscEnum)mg->galerkin, (PetscEnum *)&gtype, &flg));
  if (flg) PetscCall(PCMGSetGalerkin(pc, gtype));
  mgtype = mg->am;
  PetscCall(PetscOptionsEnum("-pc_mg_type", "Multigrid type", "PCMGSetType", PCMGTypes, (PetscEnum)mgtype, (PetscEnum *)&mgtype, &flg));
  if (flg) PetscCall(PCMGSetType(pc, mgtype));
  if (mg->am == PC_MG_MULTIPLICATIVE) {
    PetscCall(PetscOptionsInt("-pc_mg_multiplicative_cycles", "Number of cycles for each preconditioner step", "PCMGMultiplicativeSetCycles", mg->cyclesperpcapply, &cycles, &flg));
    if (flg) PetscCall(PCMGMultiplicativeSetCycles(pc, cycles));
  }
  flg = PETSC_FALSE;
  PetscCall(PetscOptionsBool("-pc_mg_log", "Log times for each multigrid level", "None", flg, &flg, NULL));
  if (flg) {
    PetscInt i;
    char     eventname[128];

    levels = mglevels[0]->levels;
    for (i = 0; i < levels; i++) {
      PetscCall(PetscSNPrintf(eventname, PETSC_STATIC_ARRAY_LENGTH(eventname), "MGSetup Level %" PetscInt_FMT, i));
      PetscCall(PetscLogEventRegister(eventname, ((PetscObject)pc)->classid, &mglevels[i]->eventsmoothsetup));
      PetscCall(PetscSNPrintf(eventname, PETSC_STATIC_ARRAY_LENGTH(eventname), "MGSmooth Level %" PetscInt_FMT, i));
      PetscCall(PetscLogEventRegister(eventname, ((PetscObject)pc)->classid, &mglevels[i]->eventsmoothsolve));
      if (i) {
        PetscCall(PetscSNPrintf(eventname, PETSC_STATIC_ARRAY_LENGTH(eventname), "MGResid Level %" PetscInt_FMT, i));
        PetscCall(PetscLogEventRegister(eventname, ((PetscObject)pc)->classid, &mglevels[i]->eventresidual));
        PetscCall(PetscSNPrintf(eventname, PETSC_STATIC_ARRAY_LENGTH(eventname), "MGInterp Level %" PetscInt_FMT, i));
        PetscCall(PetscLogEventRegister(eventname, ((PetscObject)pc)->classid, &mglevels[i]->eventinterprestrict));
      }
    }

    if (PetscDefined(USE_LOG)) {
      const char sname[] = "MG Apply";

      PetscCall(PetscLogStageGetId(sname, &mg->stageApply));
      if (mg->stageApply < 0) PetscCall(PetscLogStageRegister(sname, &mg->stageApply));
    }
  }
  PetscOptionsHeadEnd();
  /* Check option consistency */
  PetscCall(PCMGGetGalerkin(pc, &gtype));
  PetscCall(PCMGGetAdaptInterpolation(pc, &flg));
  PetscCheck(!flg || !(gtype >= PC_MG_GALERKIN_NONE), PetscObjectComm((PetscObject)pc), PETSC_ERR_ARG_INCOMP, "Must use Galerkin coarse operators when adapting the interpolator");
  PetscFunctionReturn(PETSC_SUCCESS);
}

const char *const PCMGTypes[]            = {"MULTIPLICATIVE", "ADDITIVE", "FULL", "KASKADE", "PCMGType", "PC_MG", NULL};
const char *const PCMGCycleTypes[]       = {"invalid", "v", "w", "PCMGCycleType", "PC_MG_CYCLE", NULL};
const char *const PCMGGalerkinTypes[]    = {"both", "pmat", "mat", "none", "external", "PCMGGalerkinType", "PC_MG_GALERKIN", NULL};
const char *const PCMGCoarseSpaceTypes[] = {"none", "polynomial", "harmonic", "eigenvector", "generalized_eigenvector", "gdsw", "PCMGCoarseSpaceType", "PCMG_ADAPT_NONE", NULL};

#include <petscdraw.h>
PetscErrorCode PCView_MG(PC pc, PetscViewer viewer)
{
  PC_MG         *mg       = (PC_MG *)pc->data;
  PC_MG_Levels **mglevels = mg->levels;
  PetscInt       levels   = mglevels ? mglevels[0]->levels : 0, i;
  PetscBool      iascii, isbinary, isdraw;

  PetscFunctionBegin;
  PetscCall(PetscObjectTypeCompare((PetscObject)viewer, PETSCVIEWERASCII, &iascii));
  PetscCall(PetscObjectTypeCompare((PetscObject)viewer, PETSCVIEWERBINARY, &isbinary));
  PetscCall(PetscObjectTypeCompare((PetscObject)viewer, PETSCVIEWERDRAW, &isdraw));
  if (iascii) {
    const char *cyclename = levels ? (mglevels[0]->cycles == PC_MG_CYCLE_V ? "v" : "w") : "unknown";
    PetscCall(PetscViewerASCIIPrintf(viewer, "  type is %s, levels=%" PetscInt_FMT " cycles=%s\n", PCMGTypes[mg->am], levels, cyclename));
    if (mg->am == PC_MG_MULTIPLICATIVE) PetscCall(PetscViewerASCIIPrintf(viewer, "    Cycles per PCApply=%" PetscInt_FMT "\n", mg->cyclesperpcapply));
    if (mg->galerkin == PC_MG_GALERKIN_BOTH) {
      PetscCall(PetscViewerASCIIPrintf(viewer, "    Using Galerkin computed coarse grid matrices\n"));
    } else if (mg->galerkin == PC_MG_GALERKIN_PMAT) {
      PetscCall(PetscViewerASCIIPrintf(viewer, "    Using Galerkin computed coarse grid matrices for pmat\n"));
    } else if (mg->galerkin == PC_MG_GALERKIN_MAT) {
      PetscCall(PetscViewerASCIIPrintf(viewer, "    Using Galerkin computed coarse grid matrices for mat\n"));
    } else if (mg->galerkin == PC_MG_GALERKIN_EXTERNAL) {
      PetscCall(PetscViewerASCIIPrintf(viewer, "    Using externally compute Galerkin coarse grid matrices\n"));
    } else {
      PetscCall(PetscViewerASCIIPrintf(viewer, "    Not using Galerkin computed coarse grid matrices\n"));
    }
    if (mg->view) PetscCall((*mg->view)(pc, viewer));
    for (i = 0; i < levels; i++) {
      if (i) {
        PetscCall(PetscViewerASCIIPrintf(viewer, "Down solver (pre-smoother) on level %" PetscInt_FMT " -------------------------------\n", i));
      } else {
        PetscCall(PetscViewerASCIIPrintf(viewer, "Coarse grid solver -- level %" PetscInt_FMT " -------------------------------\n", i));
      }
      PetscCall(PetscViewerASCIIPushTab(viewer));
      PetscCall(KSPView(mglevels[i]->smoothd, viewer));
      PetscCall(PetscViewerASCIIPopTab(viewer));
      if (i && mglevels[i]->smoothd == mglevels[i]->smoothu) {
        PetscCall(PetscViewerASCIIPrintf(viewer, "Up solver (post-smoother) same as down solver (pre-smoother)\n"));
      } else if (i) {
        PetscCall(PetscViewerASCIIPrintf(viewer, "Up solver (post-smoother) on level %" PetscInt_FMT " -------------------------------\n", i));
        PetscCall(PetscViewerASCIIPushTab(viewer));
        PetscCall(KSPView(mglevels[i]->smoothu, viewer));
        PetscCall(PetscViewerASCIIPopTab(viewer));
      }
      if (i && mglevels[i]->cr) {
        PetscCall(PetscViewerASCIIPrintf(viewer, "CR solver on level %" PetscInt_FMT " -------------------------------\n", i));
        PetscCall(PetscViewerASCIIPushTab(viewer));
        PetscCall(KSPView(mglevels[i]->cr, viewer));
        PetscCall(PetscViewerASCIIPopTab(viewer));
      }
    }
  } else if (isbinary) {
    for (i = levels - 1; i >= 0; i--) {
      PetscCall(KSPView(mglevels[i]->smoothd, viewer));
      if (i && mglevels[i]->smoothd != mglevels[i]->smoothu) PetscCall(KSPView(mglevels[i]->smoothu, viewer));
    }
  } else if (isdraw) {
    PetscDraw draw;
    PetscReal x, w, y, bottom, th;
    PetscCall(PetscViewerDrawGetDraw(viewer, 0, &draw));
    PetscCall(PetscDrawGetCurrentPoint(draw, &x, &y));
    PetscCall(PetscDrawStringGetSize(draw, NULL, &th));
    bottom = y - th;
    for (i = levels - 1; i >= 0; i--) {
      if (!mglevels[i]->smoothu || (mglevels[i]->smoothu == mglevels[i]->smoothd)) {
        PetscCall(PetscDrawPushCurrentPoint(draw, x, bottom));
        PetscCall(KSPView(mglevels[i]->smoothd, viewer));
        PetscCall(PetscDrawPopCurrentPoint(draw));
      } else {
        w = 0.5 * PetscMin(1.0 - x, x);
        PetscCall(PetscDrawPushCurrentPoint(draw, x + w, bottom));
        PetscCall(KSPView(mglevels[i]->smoothd, viewer));
        PetscCall(PetscDrawPopCurrentPoint(draw));
        PetscCall(PetscDrawPushCurrentPoint(draw, x - w, bottom));
        PetscCall(KSPView(mglevels[i]->smoothu, viewer));
        PetscCall(PetscDrawPopCurrentPoint(draw));
      }
      PetscCall(PetscDrawGetBoundingBox(draw, NULL, &bottom, NULL, NULL));
      bottom -= th;
    }
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

#include <petsc/private/kspimpl.h>

/*
    Calls setup for the KSP on each level
*/
PetscErrorCode PCSetUp_MG(PC pc)
{
  PC_MG         *mg       = (PC_MG *)pc->data;
  PC_MG_Levels **mglevels = mg->levels;
  PetscInt       i, n;
  PC             cpc;
  PetscBool      dump = PETSC_FALSE, opsset, use_amat, missinginterpolate = PETSC_FALSE;
  Mat            dA, dB;
  Vec            tvec;
  DM            *dms;
  PetscViewer    viewer = NULL;
  PetscBool      dAeqdB = PETSC_FALSE, needRestricts = PETSC_FALSE, doCR = PETSC_FALSE;
  PetscBool      adaptInterpolation = mg->adaptInterpolation;

  PetscFunctionBegin;
  PetscCheck(mglevels, PetscObjectComm((PetscObject)pc), PETSC_ERR_ARG_WRONGSTATE, "Must set MG levels with PCMGSetLevels() before setting up");
  n = mglevels[0]->levels;
  /* FIX: Move this to PCSetFromOptions_MG? */
  if (mg->usedmfornumberoflevels) {
    PetscInt levels;
    PetscCall(DMGetRefineLevel(pc->dm, &levels));
    levels++;
    if (levels > n) { /* the problem is now being solved on a finer grid */
      PetscCall(PCMGSetLevels(pc, levels, NULL));
      n = levels;
      PetscCall(PCSetFromOptions(pc)); /* it is bad to call this here, but otherwise will never be called for the new hierarchy */
      mglevels = mg->levels;
    }
  }
  PetscCall(KSPGetPC(mglevels[0]->smoothd, &cpc));

  /* If user did not provide fine grid operators OR operator was not updated since last global KSPSetOperators() */
  /* so use those from global PC */
  /* Is this what we always want? What if user wants to keep old one? */
  PetscCall(KSPGetOperatorsSet(mglevels[n - 1]->smoothd, NULL, &opsset));
  if (opsset) {
    Mat mmat;
    PetscCall(KSPGetOperators(mglevels[n - 1]->smoothd, NULL, &mmat));
    if (mmat == pc->pmat) opsset = PETSC_FALSE;
  }

  /* Create CR solvers */
  PetscCall(PCMGGetAdaptCR(pc, &doCR));
  if (doCR) {
    const char *prefix;

    PetscCall(PCGetOptionsPrefix(pc, &prefix));
    for (i = 1; i < n; ++i) {
      PC   ipc, cr;
      char crprefix[128];

      PetscCall(KSPCreate(PetscObjectComm((PetscObject)pc), &mglevels[i]->cr));
      PetscCall(KSPSetNestLevel(mglevels[i]->cr, pc->kspnestlevel));
      PetscCall(KSPSetErrorIfNotConverged(mglevels[i]->cr, PETSC_FALSE));
      PetscCall(PetscObjectIncrementTabLevel((PetscObject)mglevels[i]->cr, (PetscObject)pc, n - i));
      PetscCall(KSPSetOptionsPrefix(mglevels[i]->cr, prefix));
      PetscCall(PetscObjectComposedDataSetInt((PetscObject)mglevels[i]->cr, PetscMGLevelId, mglevels[i]->level));
      PetscCall(KSPSetType(mglevels[i]->cr, KSPCHEBYSHEV));
      PetscCall(KSPSetConvergenceTest(mglevels[i]->cr, KSPConvergedSkip, NULL, NULL));
      PetscCall(KSPSetNormType(mglevels[i]->cr, KSP_NORM_PRECONDITIONED));
      PetscCall(KSPGetPC(mglevels[i]->cr, &ipc));

      PetscCall(PCSetType(ipc, PCCOMPOSITE));
      PetscCall(PCCompositeSetType(ipc, PC_COMPOSITE_MULTIPLICATIVE));
      PetscCall(PCCompositeAddPCType(ipc, PCSOR));
      PetscCall(CreateCR_Private(pc, i, &cr));
      PetscCall(PCCompositeAddPC(ipc, cr));
      PetscCall(PCDestroy(&cr));

      PetscCall(KSPSetTolerances(mglevels[i]->cr, PETSC_CURRENT, PETSC_CURRENT, PETSC_CURRENT, mg->default_smoothd));
      PetscCall(KSPSetInitialGuessNonzero(mglevels[i]->cr, PETSC_TRUE));
      PetscCall(PetscSNPrintf(crprefix, 128, "mg_levels_%" PetscInt_FMT "_cr_", i));
      PetscCall(KSPAppendOptionsPrefix(mglevels[i]->cr, crprefix));
    }
  }

  if (!opsset) {
    PetscCall(PCGetUseAmat(pc, &use_amat));
    if (use_amat) {
      PetscCall(PetscInfo(pc, "Using outer operators to define finest grid operator \n  because PCMGGetSmoother(pc,nlevels-1,&ksp);KSPSetOperators(ksp,...); was not called.\n"));
      PetscCall(KSPSetOperators(mglevels[n - 1]->smoothd, pc->mat, pc->pmat));
    } else {
      PetscCall(PetscInfo(pc, "Using matrix (pmat) operators to define finest grid operator \n  because PCMGGetSmoother(pc,nlevels-1,&ksp);KSPSetOperators(ksp,...); was not called.\n"));
      PetscCall(KSPSetOperators(mglevels[n - 1]->smoothd, pc->pmat, pc->pmat));
    }
  }

  for (i = n - 1; i > 0; i--) {
    if (!(mglevels[i]->interpolate || mglevels[i]->restrct)) {
      missinginterpolate = PETSC_TRUE;
      break;
    }
  }

  PetscCall(KSPGetOperators(mglevels[n - 1]->smoothd, &dA, &dB));
  if (dA == dB) dAeqdB = PETSC_TRUE;
  if (mg->galerkin == PC_MG_GALERKIN_NONE || ((mg->galerkin == PC_MG_GALERKIN_PMAT || mg->galerkin == PC_MG_GALERKIN_MAT) && !dAeqdB)) {
    needRestricts = PETSC_TRUE; /* user must compute either mat, pmat, or both so must restrict x to coarser levels */
  }

  if (pc->dm && !pc->setupcalled) {
    /* finest smoother also gets DM but it is not active, independent of whether galerkin==PC_MG_GALERKIN_EXTERNAL */
    PetscCall(KSPSetDM(mglevels[n - 1]->smoothd, pc->dm));
    PetscCall(KSPSetDMActive(mglevels[n - 1]->smoothd, PETSC_FALSE));
    if (mglevels[n - 1]->smoothd != mglevels[n - 1]->smoothu) {
      PetscCall(KSPSetDM(mglevels[n - 1]->smoothu, pc->dm));
      PetscCall(KSPSetDMActive(mglevels[n - 1]->smoothu, PETSC_FALSE));
    }
    if (mglevels[n - 1]->cr) {
      PetscCall(KSPSetDM(mglevels[n - 1]->cr, pc->dm));
      PetscCall(KSPSetDMActive(mglevels[n - 1]->cr, PETSC_FALSE));
    }
  }

  /*
   Skipping if user has provided all interpolation/restriction needed (since DM might not be able to produce them (when coming from SNES/TS)
   Skipping for externally managed hierarchy (such as ML and GAMG). Cleaner logic here would be great. Wrap ML/GAMG as DMs?
  */
  if (missinginterpolate && mg->galerkin != PC_MG_GALERKIN_EXTERNAL && !pc->setupcalled) {
    /* first see if we can compute a coarse space */
    if (mg->coarseSpaceType == PCMG_ADAPT_GDSW) {
      for (i = n - 2; i > -1; i--) {
        if (!mglevels[i + 1]->restrct && !mglevels[i + 1]->interpolate) {
          PetscCall(PCMGComputeCoarseSpace_Internal(pc, i + 1, mg->coarseSpaceType, mg->Nc, NULL, &mglevels[i + 1]->coarseSpace));
          PetscCall(PCMGSetInterpolation(pc, i + 1, mglevels[i + 1]->coarseSpace));
        }
      }
    } else { /* construct the interpolation from the DMs */
      Mat p;
      Vec rscale;
      PetscCall(PetscMalloc1(n, &dms));
      dms[n - 1] = pc->dm;
      /* Separately create them so we do not get DMKSP interference between levels */
      for (i = n - 2; i > -1; i--) PetscCall(DMCoarsen(dms[i + 1], MPI_COMM_NULL, &dms[i]));
      for (i = n - 2; i > -1; i--) {
        DMKSP     kdm;
        PetscBool dmhasrestrict, dmhasinject;
        PetscCall(KSPSetDM(mglevels[i]->smoothd, dms[i]));
        if (!needRestricts) PetscCall(KSPSetDMActive(mglevels[i]->smoothd, PETSC_FALSE));
        if (mglevels[i]->smoothd != mglevels[i]->smoothu) {
          PetscCall(KSPSetDM(mglevels[i]->smoothu, dms[i]));
          if (!needRestricts) PetscCall(KSPSetDMActive(mglevels[i]->smoothu, PETSC_FALSE));
        }
        if (mglevels[i]->cr) {
          PetscCall(KSPSetDM(mglevels[i]->cr, dms[i]));
          if (!needRestricts) PetscCall(KSPSetDMActive(mglevels[i]->cr, PETSC_FALSE));
        }
        PetscCall(DMGetDMKSPWrite(dms[i], &kdm));
        /* Ugly hack so that the next KSPSetUp() will use the RHS that we set. A better fix is to change dmActive to take
         * a bitwise OR of computing the matrix, RHS, and initial iterate. */
        kdm->ops->computerhs = NULL;
        kdm->rhsctx          = NULL;
        if (!mglevels[i + 1]->interpolate) {
          PetscCall(DMCreateInterpolation(dms[i], dms[i + 1], &p, &rscale));
          PetscCall(PCMGSetInterpolation(pc, i + 1, p));
          if (rscale) PetscCall(PCMGSetRScale(pc, i + 1, rscale));
          PetscCall(VecDestroy(&rscale));
          PetscCall(MatDestroy(&p));
        }
        PetscCall(DMHasCreateRestriction(dms[i], &dmhasrestrict));
        if (dmhasrestrict && !mglevels[i + 1]->restrct) {
          PetscCall(DMCreateRestriction(dms[i], dms[i + 1], &p));
          PetscCall(PCMGSetRestriction(pc, i + 1, p));
          PetscCall(MatDestroy(&p));
        }
        PetscCall(DMHasCreateInjection(dms[i], &dmhasinject));
        if (dmhasinject && !mglevels[i + 1]->inject) {
          PetscCall(DMCreateInjection(dms[i], dms[i + 1], &p));
          PetscCall(PCMGSetInjection(pc, i + 1, p));
          PetscCall(MatDestroy(&p));
        }
      }

      for (i = n - 2; i > -1; i--) PetscCall(DMDestroy(&dms[i]));
      PetscCall(PetscFree(dms));
    }
  }

  if (mg->galerkin < PC_MG_GALERKIN_NONE) {
    Mat       A, B;
    PetscBool doA = PETSC_FALSE, doB = PETSC_FALSE;
    MatReuse  reuse = MAT_INITIAL_MATRIX;

    if (mg->galerkin == PC_MG_GALERKIN_PMAT || mg->galerkin == PC_MG_GALERKIN_BOTH) doB = PETSC_TRUE;
    if (mg->galerkin == PC_MG_GALERKIN_MAT || (mg->galerkin == PC_MG_GALERKIN_BOTH && dA != dB)) doA = PETSC_TRUE;
    if (pc->setupcalled) reuse = MAT_REUSE_MATRIX;
    for (i = n - 2; i > -1; i--) {
      PetscCheck(mglevels[i + 1]->restrct || mglevels[i + 1]->interpolate, PetscObjectComm((PetscObject)pc), PETSC_ERR_ARG_WRONGSTATE, "Must provide interpolation or restriction for each MG level except level 0");
      if (!mglevels[i + 1]->interpolate) PetscCall(PCMGSetInterpolation(pc, i + 1, mglevels[i + 1]->restrct));
      if (!mglevels[i + 1]->restrct) PetscCall(PCMGSetRestriction(pc, i + 1, mglevels[i + 1]->interpolate));
      if (reuse == MAT_REUSE_MATRIX) PetscCall(KSPGetOperators(mglevels[i]->smoothd, &A, &B));
      if (doA) PetscCall(MatGalerkin(mglevels[i + 1]->restrct, dA, mglevels[i + 1]->interpolate, reuse, 1.0, &A));
      if (doB) PetscCall(MatGalerkin(mglevels[i + 1]->restrct, dB, mglevels[i + 1]->interpolate, reuse, 1.0, &B));
      /* the management of the PetscObjectReference() and PetscObjecDereference() below is rather delicate */
      if (!doA && dAeqdB) {
        if (reuse == MAT_INITIAL_MATRIX) PetscCall(PetscObjectReference((PetscObject)B));
        A = B;
      } else if (!doA && reuse == MAT_INITIAL_MATRIX) {
        PetscCall(KSPGetOperators(mglevels[i]->smoothd, &A, NULL));
        PetscCall(PetscObjectReference((PetscObject)A));
      }
      if (!doB && dAeqdB) {
        if (reuse == MAT_INITIAL_MATRIX) PetscCall(PetscObjectReference((PetscObject)A));
        B = A;
      } else if (!doB && reuse == MAT_INITIAL_MATRIX) {
        PetscCall(KSPGetOperators(mglevels[i]->smoothd, NULL, &B));
        PetscCall(PetscObjectReference((PetscObject)B));
      }
      if (reuse == MAT_INITIAL_MATRIX) {
        PetscCall(KSPSetOperators(mglevels[i]->smoothd, A, B));
        PetscCall(PetscObjectDereference((PetscObject)A));
        PetscCall(PetscObjectDereference((PetscObject)B));
      }
      dA = A;
      dB = B;
    }
  }

  /* Adapt interpolation matrices */
  if (adaptInterpolation) {
    for (i = 0; i < n; ++i) {
      if (!mglevels[i]->coarseSpace) PetscCall(PCMGComputeCoarseSpace_Internal(pc, i, mg->coarseSpaceType, mg->Nc, !i ? NULL : mglevels[i - 1]->coarseSpace, &mglevels[i]->coarseSpace));
      if (i) PetscCall(PCMGAdaptInterpolator_Internal(pc, i, mglevels[i - 1]->smoothu, mglevels[i]->smoothu, mglevels[i - 1]->coarseSpace, mglevels[i]->coarseSpace));
    }
    for (i = n - 2; i > -1; --i) PetscCall(PCMGRecomputeLevelOperators_Internal(pc, i));
  }

  if (needRestricts && pc->dm) {
    for (i = n - 2; i >= 0; i--) {
      DM  dmfine, dmcoarse;
      Mat Restrict, Inject;
      Vec rscale;
      PetscCall(KSPGetDM(mglevels[i + 1]->smoothd, &dmfine));
      PetscCall(KSPGetDM(mglevels[i]->smoothd, &dmcoarse));
      PetscCall(PCMGGetRestriction(pc, i + 1, &Restrict));
      PetscCall(PCMGGetRScale(pc, i + 1, &rscale));
      PetscCall(PCMGGetInjection(pc, i + 1, &Inject));
      PetscCall(DMRestrict(dmfine, Restrict, rscale, Inject, dmcoarse));
    }
  }

  if (!pc->setupcalled) {
    for (i = 0; i < n; i++) PetscCall(KSPSetFromOptions(mglevels[i]->smoothd));
    for (i = 1; i < n; i++) {
      if (mglevels[i]->smoothu && (mglevels[i]->smoothu != mglevels[i]->smoothd)) PetscCall(KSPSetFromOptions(mglevels[i]->smoothu));
      if (mglevels[i]->cr) PetscCall(KSPSetFromOptions(mglevels[i]->cr));
    }
    /* insure that if either interpolation or restriction is set the other one is set */
    for (i = 1; i < n; i++) {
      PetscCall(PCMGGetInterpolation(pc, i, NULL));
      PetscCall(PCMGGetRestriction(pc, i, NULL));
    }
    for (i = 0; i < n - 1; i++) {
      if (!mglevels[i]->b) {
        Vec *vec;
        PetscCall(KSPCreateVecs(mglevels[i]->smoothd, 1, &vec, 0, NULL));
        PetscCall(PCMGSetRhs(pc, i, *vec));
        PetscCall(VecDestroy(vec));
        PetscCall(PetscFree(vec));
      }
      if (!mglevels[i]->r && i) {
        PetscCall(VecDuplicate(mglevels[i]->b, &tvec));
        PetscCall(PCMGSetR(pc, i, tvec));
        PetscCall(VecDestroy(&tvec));
      }
      if (!mglevels[i]->x) {
        PetscCall(VecDuplicate(mglevels[i]->b, &tvec));
        PetscCall(PCMGSetX(pc, i, tvec));
        PetscCall(VecDestroy(&tvec));
      }
      if (doCR) {
        PetscCall(VecDuplicate(mglevels[i]->b, &mglevels[i]->crx));
        PetscCall(VecDuplicate(mglevels[i]->b, &mglevels[i]->crb));
        PetscCall(VecZeroEntries(mglevels[i]->crb));
      }
    }
    if (n != 1 && !mglevels[n - 1]->r) {
      /* PCMGSetR() on the finest level if user did not supply it */
      Vec *vec;
      PetscCall(KSPCreateVecs(mglevels[n - 1]->smoothd, 1, &vec, 0, NULL));
      PetscCall(PCMGSetR(pc, n - 1, *vec));
      PetscCall(VecDestroy(vec));
      PetscCall(PetscFree(vec));
    }
    if (doCR) {
      PetscCall(VecDuplicate(mglevels[n - 1]->r, &mglevels[n - 1]->crx));
      PetscCall(VecDuplicate(mglevels[n - 1]->r, &mglevels[n - 1]->crb));
      PetscCall(VecZeroEntries(mglevels[n - 1]->crb));
    }
  }

  if (pc->dm) {
    /* need to tell all the coarser levels to rebuild the matrix using the DM for that level */
    for (i = 0; i < n - 1; i++) {
      if (mglevels[i]->smoothd->setupstage != KSP_SETUP_NEW) mglevels[i]->smoothd->setupstage = KSP_SETUP_NEWMATRIX;
    }
  }
  // We got here (PCSetUp_MG) because the matrix has changed, which means the smoother needs to be set up again (e.g.,
  // new diagonal for Jacobi). Setting it here allows it to be logged under PCSetUp rather than deep inside a PCApply.
  if (mglevels[n - 1]->smoothd->setupstage != KSP_SETUP_NEW) mglevels[n - 1]->smoothd->setupstage = KSP_SETUP_NEWMATRIX;

  for (i = 1; i < n; i++) {
    if (mglevels[i]->smoothu == mglevels[i]->smoothd || mg->am == PC_MG_FULL || mg->am == PC_MG_KASKADE || mg->cyclesperpcapply > 1) {
      /* if doing only down then initial guess is zero */
      PetscCall(KSPSetInitialGuessNonzero(mglevels[i]->smoothd, PETSC_TRUE));
    }
    if (mglevels[i]->cr) PetscCall(KSPSetInitialGuessNonzero(mglevels[i]->cr, PETSC_TRUE));
    if (mglevels[i]->eventsmoothsetup) PetscCall(PetscLogEventBegin(mglevels[i]->eventsmoothsetup, 0, 0, 0, 0));
    PetscCall(KSPSetUp(mglevels[i]->smoothd));
    if (mglevels[i]->smoothd->reason) pc->failedreason = PC_SUBPC_ERROR;
    if (mglevels[i]->eventsmoothsetup) PetscCall(PetscLogEventEnd(mglevels[i]->eventsmoothsetup, 0, 0, 0, 0));
    if (!mglevels[i]->residual) {
      Mat mat;
      PetscCall(KSPGetOperators(mglevels[i]->smoothd, &mat, NULL));
      PetscCall(PCMGSetResidual(pc, i, PCMGResidualDefault, mat));
    }
    if (!mglevels[i]->residualtranspose) {
      Mat mat;
      PetscCall(KSPGetOperators(mglevels[i]->smoothd, &mat, NULL));
      PetscCall(PCMGSetResidualTranspose(pc, i, PCMGResidualTransposeDefault, mat));
    }
  }
  for (i = 1; i < n; i++) {
    if (mglevels[i]->smoothu && mglevels[i]->smoothu != mglevels[i]->smoothd) {
      Mat downmat, downpmat;

      /* check if operators have been set for up, if not use down operators to set them */
      PetscCall(KSPGetOperatorsSet(mglevels[i]->smoothu, &opsset, NULL));
      if (!opsset) {
        PetscCall(KSPGetOperators(mglevels[i]->smoothd, &downmat, &downpmat));
        PetscCall(KSPSetOperators(mglevels[i]->smoothu, downmat, downpmat));
      }

      PetscCall(KSPSetInitialGuessNonzero(mglevels[i]->smoothu, PETSC_TRUE));
      if (mglevels[i]->eventsmoothsetup) PetscCall(PetscLogEventBegin(mglevels[i]->eventsmoothsetup, 0, 0, 0, 0));
      PetscCall(KSPSetUp(mglevels[i]->smoothu));
      if (mglevels[i]->smoothu->reason) pc->failedreason = PC_SUBPC_ERROR;
      if (mglevels[i]->eventsmoothsetup) PetscCall(PetscLogEventEnd(mglevels[i]->eventsmoothsetup, 0, 0, 0, 0));
    }
    if (mglevels[i]->cr) {
      Mat downmat, downpmat;

      /* check if operators have been set for up, if not use down operators to set them */
      PetscCall(KSPGetOperatorsSet(mglevels[i]->cr, &opsset, NULL));
      if (!opsset) {
        PetscCall(KSPGetOperators(mglevels[i]->smoothd, &downmat, &downpmat));
        PetscCall(KSPSetOperators(mglevels[i]->cr, downmat, downpmat));
      }

      PetscCall(KSPSetInitialGuessNonzero(mglevels[i]->cr, PETSC_TRUE));
      if (mglevels[i]->eventsmoothsetup) PetscCall(PetscLogEventBegin(mglevels[i]->eventsmoothsetup, 0, 0, 0, 0));
      PetscCall(KSPSetUp(mglevels[i]->cr));
      if (mglevels[i]->cr->reason) pc->failedreason = PC_SUBPC_ERROR;
      if (mglevels[i]->eventsmoothsetup) PetscCall(PetscLogEventEnd(mglevels[i]->eventsmoothsetup, 0, 0, 0, 0));
    }
  }

  if (mglevels[0]->eventsmoothsetup) PetscCall(PetscLogEventBegin(mglevels[0]->eventsmoothsetup, 0, 0, 0, 0));
  PetscCall(KSPSetUp(mglevels[0]->smoothd));
  if (mglevels[0]->smoothd->reason) pc->failedreason = PC_SUBPC_ERROR;
  if (mglevels[0]->eventsmoothsetup) PetscCall(PetscLogEventEnd(mglevels[0]->eventsmoothsetup, 0, 0, 0, 0));

  /*
     Dump the interpolation/restriction matrices plus the
   Jacobian/stiffness on each level. This allows MATLAB users to
   easily check if the Galerkin condition A_c = R A_f R^T is satisfied.

   Only support one or the other at the same time.
  */
#if defined(PETSC_USE_SOCKET_VIEWER)
  PetscCall(PetscOptionsGetBool(((PetscObject)pc)->options, ((PetscObject)pc)->prefix, "-pc_mg_dump_matlab", &dump, NULL));
  if (dump) viewer = PETSC_VIEWER_SOCKET_(PetscObjectComm((PetscObject)pc));
  dump = PETSC_FALSE;
#endif
  PetscCall(PetscOptionsGetBool(((PetscObject)pc)->options, ((PetscObject)pc)->prefix, "-pc_mg_dump_binary", &dump, NULL));
  if (dump) viewer = PETSC_VIEWER_BINARY_(PetscObjectComm((PetscObject)pc));

  if (viewer) {
    for (i = 1; i < n; i++) PetscCall(MatView(mglevels[i]->restrct, viewer));
    for (i = 0; i < n; i++) {
      PetscCall(KSPGetPC(mglevels[i]->smoothd, &pc));
      PetscCall(MatView(pc->mat, viewer));
    }
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode PCMGGetLevels_MG(PC pc, PetscInt *levels)
{
  PC_MG *mg = (PC_MG *)pc->data;

  PetscFunctionBegin;
  *levels = mg->nlevels;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PCMGGetLevels - Gets the number of levels to use with `PCMG`.

  Not Collective

  Input Parameter:
. pc - the preconditioner context

  Output Parameter:
. levels - the number of levels

  Level: advanced

.seealso: [](ch_ksp), `PCMG`, `PCMGSetLevels()`
@*/
PetscErrorCode PCMGGetLevels(PC pc, PetscInt *levels)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(pc, PC_CLASSID, 1);
  PetscAssertPointer(levels, 2);
  *levels = 0;
  PetscTryMethod(pc, "PCMGGetLevels_C", (PC, PetscInt *), (pc, levels));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PCMGGetGridComplexity - compute operator and grid complexity of the `PCMG` hierarchy

  Input Parameter:
. pc - the preconditioner context

  Output Parameters:
+ gc - grid complexity = sum_i(n_i) / n_0
- oc - operator complexity = sum_i(nnz_i) / nnz_0

  Level: advanced

  Note:
  This is often call the operator complexity in multigrid literature

.seealso: [](ch_ksp), `PCMG`, `PCMGGetLevels()`, `PCMGSetLevels()`
@*/
PetscErrorCode PCMGGetGridComplexity(PC pc, PetscReal *gc, PetscReal *oc)
{
  PC_MG         *mg       = (PC_MG *)pc->data;
  PC_MG_Levels **mglevels = mg->levels;
  PetscInt       lev, N;
  PetscLogDouble nnz0 = 0, sgc = 0, soc = 0, n0 = 0;
  MatInfo        info;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(pc, PC_CLASSID, 1);
  if (gc) PetscAssertPointer(gc, 2);
  if (oc) PetscAssertPointer(oc, 3);
  if (!pc->setupcalled) {
    if (gc) *gc = 0;
    if (oc) *oc = 0;
    PetscFunctionReturn(PETSC_SUCCESS);
  }
  PetscCheck(mg->nlevels > 0, PETSC_COMM_SELF, PETSC_ERR_PLIB, "MG has no levels");
  for (lev = 0; lev < mg->nlevels; lev++) {
    Mat dB;
    PetscCall(KSPGetOperators(mglevels[lev]->smoothd, NULL, &dB));
    PetscCall(MatGetInfo(dB, MAT_GLOBAL_SUM, &info)); /* global reduction */
    PetscCall(MatGetSize(dB, &N, NULL));
    sgc += N;
    soc += info.nz_used;
    if (lev == mg->nlevels - 1) {
      nnz0 = info.nz_used;
      n0   = N;
    }
  }
  PetscCheck(n0 > 0 && gc, PETSC_COMM_SELF, PETSC_ERR_PLIB, "Number for grid points on finest level is not available");
  *gc = (PetscReal)(sgc / n0);
  if (nnz0 > 0 && oc) *oc = (PetscReal)(soc / nnz0);
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PCMGSetType - Determines the form of multigrid to use, either
  multiplicative, additive, full, or the Kaskade algorithm.

  Logically Collective

  Input Parameters:
+ pc   - the preconditioner context
- form - multigrid form, one of `PC_MG_MULTIPLICATIVE`, `PC_MG_ADDITIVE`, `PC_MG_FULL`, `PC_MG_KASKADE`

  Options Database Key:
. -pc_mg_type <form> - Sets <form>, one of multiplicative, additive, full, kaskade

  Level: advanced

.seealso: [](ch_ksp), `PCMGType`, `PCMG`, `PCMGGetLevels()`, `PCMGSetLevels()`, `PCMGGetType()`, `PCMGCycleType`
@*/
PetscErrorCode PCMGSetType(PC pc, PCMGType form)
{
  PC_MG *mg = (PC_MG *)pc->data;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(pc, PC_CLASSID, 1);
  PetscValidLogicalCollectiveEnum(pc, form, 2);
  mg->am = form;
  if (form == PC_MG_MULTIPLICATIVE) pc->ops->applyrichardson = PCApplyRichardson_MG;
  else pc->ops->applyrichardson = NULL;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PCMGGetType - Finds the form of multigrid the `PCMG` is using  multiplicative, additive, full, or the Kaskade algorithm.

  Logically Collective

  Input Parameter:
. pc - the preconditioner context

  Output Parameter:
. type - one of `PC_MG_MULTIPLICATIVE`, `PC_MG_ADDITIVE`, `PC_MG_FULL`, `PC_MG_KASKADE`, `PCMGCycleType`

  Level: advanced

.seealso: [](ch_ksp), `PCMGType`, `PCMG`, `PCMGGetLevels()`, `PCMGSetLevels()`, `PCMGSetType()`
@*/
PetscErrorCode PCMGGetType(PC pc, PCMGType *type)
{
  PC_MG *mg = (PC_MG *)pc->data;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(pc, PC_CLASSID, 1);
  *type = mg->am;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PCMGSetCycleType - Sets the type cycles to use.  Use `PCMGSetCycleTypeOnLevel()` for more
  complicated cycling.

  Logically Collective

  Input Parameters:
+ pc - the multigrid context
- n  - either `PC_MG_CYCLE_V` or `PC_MG_CYCLE_W`

  Options Database Key:
. -pc_mg_cycle_type <v,w> - provide the cycle desired

  Level: advanced

.seealso: [](ch_ksp), `PCMG`, `PCMGSetCycleTypeOnLevel()`, `PCMGType`, `PCMGCycleType`
@*/
PetscErrorCode PCMGSetCycleType(PC pc, PCMGCycleType n)
{
  PC_MG         *mg       = (PC_MG *)pc->data;
  PC_MG_Levels **mglevels = mg->levels;
  PetscInt       i, levels;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(pc, PC_CLASSID, 1);
  PetscValidLogicalCollectiveEnum(pc, n, 2);
  PetscCheck(mglevels, PetscObjectComm((PetscObject)pc), PETSC_ERR_ORDER, "Must set MG levels with PCMGSetLevels() before calling");
  levels = mglevels[0]->levels;
  for (i = 0; i < levels; i++) mglevels[i]->cycles = n;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PCMGMultiplicativeSetCycles - Sets the number of cycles to use for each preconditioner step
  of multigrid when `PCMGType` is `PC_MG_MULTIPLICATIVE`

  Logically Collective

  Input Parameters:
+ pc - the multigrid context
- n  - number of cycles (default is 1)

  Options Database Key:
. -pc_mg_multiplicative_cycles n - set the number of cycles

  Level: advanced

  Note:
  This is not associated with setting a v or w cycle, that is set with `PCMGSetCycleType()`

.seealso: [](ch_ksp), `PCMGSetCycleTypeOnLevel()`, `PCMGSetCycleType()`, `PCMGCycleType`, `PCMGType`
@*/
PetscErrorCode PCMGMultiplicativeSetCycles(PC pc, PetscInt n)
{
  PC_MG *mg = (PC_MG *)pc->data;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(pc, PC_CLASSID, 1);
  PetscValidLogicalCollectiveInt(pc, n, 2);
  mg->cyclesperpcapply = n;
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode PCMGSetGalerkin_MG(PC pc, PCMGGalerkinType use)
{
  PC_MG *mg = (PC_MG *)pc->data;

  PetscFunctionBegin;
  mg->galerkin = use;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PCMGSetGalerkin - Causes the coarser grid matrices to be computed from the
  finest grid via the Galerkin process: A_i-1 = r_i * A_i * p_i

  Logically Collective

  Input Parameters:
+ pc  - the multigrid context
- use - one of `PC_MG_GALERKIN_BOTH`, `PC_MG_GALERKIN_PMAT`, `PC_MG_GALERKIN_MAT`, or `PC_MG_GALERKIN_NONE`

  Options Database Key:
. -pc_mg_galerkin <both,pmat,mat,none> - set the matrices to form via the Galerkin process

  Level: intermediate

  Note:
  Some codes that use `PCMG` such as `PCGAMG` use Galerkin internally while constructing the hierarchy and thus do not
  use the `PCMG` construction of the coarser grids.

.seealso: [](ch_ksp), `PCMG`, `PCMGGetGalerkin()`, `PCMGGalerkinType`
@*/
PetscErrorCode PCMGSetGalerkin(PC pc, PCMGGalerkinType use)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(pc, PC_CLASSID, 1);
  PetscTryMethod(pc, "PCMGSetGalerkin_C", (PC, PCMGGalerkinType), (pc, use));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PCMGGetGalerkin - Checks if Galerkin multigrid is being used, i.e. A_i-1 = r_i * A_i * p_i

  Not Collective

  Input Parameter:
. pc - the multigrid context

  Output Parameter:
. galerkin - one of `PC_MG_GALERKIN_BOTH`,`PC_MG_GALERKIN_PMAT`,`PC_MG_GALERKIN_MAT`, `PC_MG_GALERKIN_NONE`, or `PC_MG_GALERKIN_EXTERNAL`

  Level: intermediate

.seealso: [](ch_ksp), `PCMG`, `PCMGSetGalerkin()`, `PCMGGalerkinType`
@*/
PetscErrorCode PCMGGetGalerkin(PC pc, PCMGGalerkinType *galerkin)
{
  PC_MG *mg = (PC_MG *)pc->data;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(pc, PC_CLASSID, 1);
  *galerkin = mg->galerkin;
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode PCMGSetAdaptInterpolation_MG(PC pc, PetscBool adapt)
{
  PC_MG *mg = (PC_MG *)pc->data;

  PetscFunctionBegin;
  mg->adaptInterpolation = adapt;
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode PCMGGetAdaptInterpolation_MG(PC pc, PetscBool *adapt)
{
  PC_MG *mg = (PC_MG *)pc->data;

  PetscFunctionBegin;
  *adapt = mg->adaptInterpolation;
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode PCMGSetAdaptCoarseSpaceType_MG(PC pc, PCMGCoarseSpaceType ctype)
{
  PC_MG *mg = (PC_MG *)pc->data;

  PetscFunctionBegin;
  mg->adaptInterpolation = ctype != PCMG_ADAPT_NONE ? PETSC_TRUE : PETSC_FALSE;
  mg->coarseSpaceType    = ctype;
  PetscCall(PCMGSetGalerkin(pc, PC_MG_GALERKIN_BOTH));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode PCMGGetAdaptCoarseSpaceType_MG(PC pc, PCMGCoarseSpaceType *ctype)
{
  PC_MG *mg = (PC_MG *)pc->data;

  PetscFunctionBegin;
  *ctype = mg->coarseSpaceType;
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode PCMGSetAdaptCR_MG(PC pc, PetscBool cr)
{
  PC_MG *mg = (PC_MG *)pc->data;

  PetscFunctionBegin;
  mg->compatibleRelaxation = cr;
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode PCMGGetAdaptCR_MG(PC pc, PetscBool *cr)
{
  PC_MG *mg = (PC_MG *)pc->data;

  PetscFunctionBegin;
  *cr = mg->compatibleRelaxation;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PCMGSetAdaptCoarseSpaceType - Set the type of adaptive coarse space.

  Adapts or creates the interpolator based upon a vector space which should be accurately captured by the next coarser mesh, and thus accurately interpolated.

  Logically Collective

  Input Parameters:
+ pc    - the multigrid context
- ctype - the type of coarse space

  Options Database Keys:
+ -pc_mg_adapt_interp_n <int>             - The number of modes to use
- -pc_mg_adapt_interp_coarse_space <type> - The type of coarse space: none, `polynomial`, `harmonic`, `eigenvector`, `generalized_eigenvector`, `gdsw`

  Level: intermediate

  Note:
  Requires a `DM` with specific functionality be attached to the `PC`.

.seealso: [](ch_ksp), `PCMG`, `PCMGCoarseSpaceType`, `PCMGGetAdaptCoarseSpaceType()`, `PCMGSetGalerkin()`, `PCMGSetAdaptInterpolation()`, `DM`
@*/
PetscErrorCode PCMGSetAdaptCoarseSpaceType(PC pc, PCMGCoarseSpaceType ctype)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(pc, PC_CLASSID, 1);
  PetscValidLogicalCollectiveEnum(pc, ctype, 2);
  PetscTryMethod(pc, "PCMGSetAdaptCoarseSpaceType_C", (PC, PCMGCoarseSpaceType), (pc, ctype));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PCMGGetAdaptCoarseSpaceType - Get the type of adaptive coarse space.

  Not Collective

  Input Parameter:
. pc - the multigrid context

  Output Parameter:
. ctype - the type of coarse space

  Level: intermediate

.seealso: [](ch_ksp), `PCMG`, `PCMGCoarseSpaceType`, `PCMGSetAdaptCoarseSpaceType()`, `PCMGSetGalerkin()`, `PCMGSetAdaptInterpolation()`
@*/
PetscErrorCode PCMGGetAdaptCoarseSpaceType(PC pc, PCMGCoarseSpaceType *ctype)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(pc, PC_CLASSID, 1);
  PetscAssertPointer(ctype, 2);
  PetscUseMethod(pc, "PCMGGetAdaptCoarseSpaceType_C", (PC, PCMGCoarseSpaceType *), (pc, ctype));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/* MATT: REMOVE? */
/*@
  PCMGSetAdaptInterpolation - Adapt the interpolator based upon a vector space which should be accurately captured by the next coarser mesh, and thus accurately interpolated.

  Logically Collective

  Input Parameters:
+ pc    - the multigrid context
- adapt - flag for adaptation of the interpolator

  Options Database Keys:
+ -pc_mg_adapt_interp                     - Turn on adaptation
. -pc_mg_adapt_interp_n <int>             - The number of modes to use, should be divisible by dimension
- -pc_mg_adapt_interp_coarse_space <type> - The type of coarse space: polynomial, harmonic, eigenvector, generalized_eigenvector

  Level: intermediate

.seealso: [](ch_ksp), `PCMG`, `PCMGGetAdaptInterpolation()`, `PCMGSetGalerkin()`, `PCMGGetAdaptCoarseSpaceType()`, `PCMGSetAdaptCoarseSpaceType()`
@*/
PetscErrorCode PCMGSetAdaptInterpolation(PC pc, PetscBool adapt)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(pc, PC_CLASSID, 1);
  PetscTryMethod(pc, "PCMGSetAdaptInterpolation_C", (PC, PetscBool), (pc, adapt));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PCMGGetAdaptInterpolation - Get the flag to adapt the interpolator based upon a vector space which should be accurately captured by the next coarser mesh,
  and thus accurately interpolated.

  Not Collective

  Input Parameter:
. pc - the multigrid context

  Output Parameter:
. adapt - flag for adaptation of the interpolator

  Level: intermediate

.seealso: [](ch_ksp), `PCMG`, `PCMGSetAdaptInterpolation()`, `PCMGSetGalerkin()`, `PCMGGetAdaptCoarseSpaceType()`, `PCMGSetAdaptCoarseSpaceType()`
@*/
PetscErrorCode PCMGGetAdaptInterpolation(PC pc, PetscBool *adapt)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(pc, PC_CLASSID, 1);
  PetscAssertPointer(adapt, 2);
  PetscUseMethod(pc, "PCMGGetAdaptInterpolation_C", (PC, PetscBool *), (pc, adapt));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PCMGSetAdaptCR - Monitor the coarse space quality using an auxiliary solve with compatible relaxation.

  Logically Collective

  Input Parameters:
+ pc - the multigrid context
- cr - flag for compatible relaxation

  Options Database Key:
. -pc_mg_adapt_cr - Turn on compatible relaxation

  Level: intermediate

.seealso: [](ch_ksp), `PCMG`, `PCMGGetAdaptCR()`, `PCMGSetAdaptInterpolation()`, `PCMGSetGalerkin()`, `PCMGGetAdaptCoarseSpaceType()`, `PCMGSetAdaptCoarseSpaceType()`
@*/
PetscErrorCode PCMGSetAdaptCR(PC pc, PetscBool cr)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(pc, PC_CLASSID, 1);
  PetscTryMethod(pc, "PCMGSetAdaptCR_C", (PC, PetscBool), (pc, cr));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PCMGGetAdaptCR - Get the flag to monitor coarse space quality using an auxiliary solve with compatible relaxation.

  Not Collective

  Input Parameter:
. pc - the multigrid context

  Output Parameter:
. cr - flag for compatible relaxaion

  Level: intermediate

.seealso: [](ch_ksp), `PCMGSetAdaptCR()`, `PCMGGetAdaptInterpolation()`, `PCMGSetGalerkin()`, `PCMGGetAdaptCoarseSpaceType()`, `PCMGSetAdaptCoarseSpaceType()`
@*/
PetscErrorCode PCMGGetAdaptCR(PC pc, PetscBool *cr)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(pc, PC_CLASSID, 1);
  PetscAssertPointer(cr, 2);
  PetscUseMethod(pc, "PCMGGetAdaptCR_C", (PC, PetscBool *), (pc, cr));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PCMGSetNumberSmooth - Sets the number of pre and post-smoothing steps to use
  on all levels.  Use `PCMGDistinctSmoothUp()` to create separate up and down smoothers if you want different numbers of
  pre- and post-smoothing steps.

  Logically Collective

  Input Parameters:
+ pc - the multigrid context
- n  - the number of smoothing steps

  Options Database Key:
. -mg_levels_ksp_max_it <n> - Sets number of pre and post-smoothing steps

  Level: advanced

  Note:
  This does not set a value on the coarsest grid, since we assume that there is no separate smooth up on the coarsest grid.

.seealso: [](ch_ksp), `PCMG`, `PCMGSetDistinctSmoothUp()`
@*/
PetscErrorCode PCMGSetNumberSmooth(PC pc, PetscInt n)
{
  PC_MG         *mg       = (PC_MG *)pc->data;
  PC_MG_Levels **mglevels = mg->levels;
  PetscInt       i, levels;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(pc, PC_CLASSID, 1);
  PetscValidLogicalCollectiveInt(pc, n, 2);
  PetscCheck(mglevels, PetscObjectComm((PetscObject)pc), PETSC_ERR_ORDER, "Must set MG levels with PCMGSetLevels() before calling");
  levels = mglevels[0]->levels;

  for (i = 1; i < levels; i++) {
    PetscCall(KSPSetTolerances(mglevels[i]->smoothu, PETSC_CURRENT, PETSC_CURRENT, PETSC_CURRENT, n));
    PetscCall(KSPSetTolerances(mglevels[i]->smoothd, PETSC_CURRENT, PETSC_CURRENT, PETSC_CURRENT, n));
    mg->default_smoothu = n;
    mg->default_smoothd = n;
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PCMGSetDistinctSmoothUp - sets the up (post) smoother to be a separate `KSP` from the down (pre) smoother on all levels
  and adds the suffix _up to the options name

  Logically Collective

  Input Parameter:
. pc - the preconditioner context

  Options Database Key:
. -pc_mg_distinct_smoothup <bool> - use distinct smoothing objects

  Level: advanced

  Note:
  This does not set a value on the coarsest grid, since we assume that there is no separate smooth up on the coarsest grid.

.seealso: [](ch_ksp), `PCMG`, `PCMGSetNumberSmooth()`
@*/
PetscErrorCode PCMGSetDistinctSmoothUp(PC pc)
{
  PC_MG         *mg       = (PC_MG *)pc->data;
  PC_MG_Levels **mglevels = mg->levels;
  PetscInt       i, levels;
  KSP            subksp;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(pc, PC_CLASSID, 1);
  PetscCheck(mglevels, PetscObjectComm((PetscObject)pc), PETSC_ERR_ORDER, "Must set MG levels with PCMGSetLevels() before calling");
  levels = mglevels[0]->levels;

  for (i = 1; i < levels; i++) {
    const char *prefix = NULL;
    /* make sure smoother up and down are different */
    PetscCall(PCMGGetSmootherUp(pc, i, &subksp));
    PetscCall(KSPGetOptionsPrefix(mglevels[i]->smoothd, &prefix));
    PetscCall(KSPSetOptionsPrefix(subksp, prefix));
    PetscCall(KSPAppendOptionsPrefix(subksp, "up_"));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/* No new matrices are created, and the coarse operator matrices are the references to the original ones */
static PetscErrorCode PCGetInterpolations_MG(PC pc, PetscInt *num_levels, Mat *interpolations[])
{
  PC_MG         *mg       = (PC_MG *)pc->data;
  PC_MG_Levels **mglevels = mg->levels;
  Mat           *mat;
  PetscInt       l;

  PetscFunctionBegin;
  PetscCheck(mglevels, PetscObjectComm((PetscObject)pc), PETSC_ERR_ARG_WRONGSTATE, "Must set MG levels before calling");
  PetscCall(PetscMalloc1(mg->nlevels, &mat));
  for (l = 1; l < mg->nlevels; l++) {
    mat[l - 1] = mglevels[l]->interpolate;
    PetscCall(PetscObjectReference((PetscObject)mat[l - 1]));
  }
  *num_levels     = mg->nlevels;
  *interpolations = mat;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/* No new matrices are created, and the coarse operator matrices are the references to the original ones */
static PetscErrorCode PCGetCoarseOperators_MG(PC pc, PetscInt *num_levels, Mat *coarseOperators[])
{
  PC_MG         *mg       = (PC_MG *)pc->data;
  PC_MG_Levels **mglevels = mg->levels;
  PetscInt       l;
  Mat           *mat;

  PetscFunctionBegin;
  PetscCheck(mglevels, PetscObjectComm((PetscObject)pc), PETSC_ERR_ARG_WRONGSTATE, "Must set MG levels before calling");
  PetscCall(PetscMalloc1(mg->nlevels, &mat));
  for (l = 0; l < mg->nlevels - 1; l++) {
    PetscCall(KSPGetOperators(mglevels[l]->smoothd, NULL, &mat[l]));
    PetscCall(PetscObjectReference((PetscObject)mat[l]));
  }
  *num_levels      = mg->nlevels;
  *coarseOperators = mat;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  PCMGRegisterCoarseSpaceConstructor -  Adds a method to the `PCMG` package for coarse space construction.

  Not Collective, No Fortran Support

  Input Parameters:
+ name     - name of the constructor
- function - constructor routine

  Calling sequence of `function`:
+ pc        - The `PC` object
. l         - The multigrid level, 0 is the coarse level
. dm        - The `DM` for this level
. smooth    - The level smoother
. Nc        - The size of the coarse space
. initGuess - Basis for an initial guess for the space
- coarseSp  - A basis for the computed coarse space

  Level: advanced

  Developer Notes:
  How come this is not used by `PCGAMG`?

.seealso: [](ch_ksp), `PCMG`, `PCMGGetCoarseSpaceConstructor()`, `PCRegister()`
@*/
PetscErrorCode PCMGRegisterCoarseSpaceConstructor(const char name[], PetscErrorCode (*function)(PC pc, PetscInt l, DM dm, KSP smooth, PetscInt Nc, Mat initGuess, Mat *coarseSp))
{
  PetscFunctionBegin;
  PetscCall(PCInitializePackage());
  PetscCall(PetscFunctionListAdd(&PCMGCoarseList, name, function));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  PCMGGetCoarseSpaceConstructor -  Returns the given coarse space construction method.

  Not Collective, No Fortran Support

  Input Parameter:
. name - name of the constructor

  Output Parameter:
. function - constructor routine

  Level: advanced

.seealso: [](ch_ksp), `PCMG`, `PCMGRegisterCoarseSpaceConstructor()`, `PCRegister()`
@*/
PetscErrorCode PCMGGetCoarseSpaceConstructor(const char name[], PetscErrorCode (**function)(PC, PetscInt, DM, KSP, PetscInt, Mat, Mat *))
{
  PetscFunctionBegin;
  PetscCall(PetscFunctionListFind(PCMGCoarseList, name, function));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*MC
   PCMG - Use multigrid preconditioning. This preconditioner requires you provide additional
    information about the coarser grid matrices and restriction/interpolation operators.

   Options Database Keys:
+  -pc_mg_levels <nlevels>                            - number of levels including finest
.  -pc_mg_cycle_type <v,w>                            - provide the cycle desired
.  -pc_mg_type <additive,multiplicative,full,kaskade> - multiplicative is the default
.  -pc_mg_log                                         - log information about time spent on each level of the solver
.  -pc_mg_distinct_smoothup                           - configure up (after interpolation) and down (before restriction) smoothers separately (with different options prefixes)
.  -pc_mg_galerkin <both,pmat,mat,none>               - use Galerkin process to compute coarser operators, i.e. Acoarse = R A R'
.  -pc_mg_multiplicative_cycles                        - number of cycles to use as the preconditioner (defaults to 1)
.  -pc_mg_dump_matlab                                  - dumps the matrices for each level and the restriction/interpolation matrices
                                                         to a `PETSCVIEWERSOCKET` for reading from MATLAB.
-  -pc_mg_dump_binary                                  -dumps the matrices for each level and the restriction/interpolation matrices
                                                        to the binary output file called binaryoutput

   Level: intermediate

   Notes:
   The Krylov solver (if any) and preconditioner (smoother) and their parameters are controlled from the options database with the standard
   options database keywords prefixed with `-mg_levels_` to affect all the levels but the coarsest, which is controlled with `-mg_coarse_`,
   and the finest where `-mg_fine_` can override `-mg_levels_`.  One can set different preconditioners etc on specific levels with the prefix
   `-mg_levels_n_` where `n` is the level number (zero being the coarse level. For example
.vb
   -mg_levels_ksp_type gmres -mg_levels_pc_type bjacobi -mg_coarse_pc_type svd -mg_levels_2_pc_type sor
.ve
   These options also work for controlling the smoothers etc inside `PCGAMG`

   If one uses a Krylov method such `KSPGMRES` or `KSPCG` as the smoother then one must use `KSPFGMRES`, `KSPGCR`, or `KSPRICHARDSON` as the outer Krylov method

   When run with a single level the smoother options are used on that level NOT the coarse grid solver options

   When run with `KSPRICHARDSON` the convergence test changes slightly if monitor is turned on. The iteration count may change slightly. This
   is because without monitoring the residual norm is computed WITHIN each multigrid cycle on the finest level after the pre-smoothing
   (because the residual has just been computed for the multigrid algorithm and is hence available for free) while with monitoring the
   residual is computed at the end of each cycle.

.seealso: [](sec_mg), `PCCreate()`, `PCSetType()`, `PCType`, `PC`, `PCMGType`, `PCEXOTIC`, `PCGAMG`, `PCML`, `PCHYPRE`
          `PCMGSetLevels()`, `PCMGGetLevels()`, `PCMGSetType()`, `PCMGSetCycleType()`,
          `PCMGSetDistinctSmoothUp()`, `PCMGGetCoarseSolve()`, `PCMGSetResidual()`, `PCMGSetInterpolation()`,
          `PCMGSetRestriction()`, `PCMGGetSmoother()`, `PCMGGetSmootherUp()`, `PCMGGetSmootherDown()`,
          `PCMGSetCycleTypeOnLevel()`, `PCMGSetRhs()`, `PCMGSetX()`, `PCMGSetR()`,
          `PCMGSetAdaptCR()`, `PCMGGetAdaptInterpolation()`, `PCMGSetGalerkin()`, `PCMGGetAdaptCoarseSpaceType()`, `PCMGSetAdaptCoarseSpaceType()`
M*/

PETSC_EXTERN PetscErrorCode PCCreate_MG(PC pc)
{
  PC_MG *mg;

  PetscFunctionBegin;
  PetscCall(PetscNew(&mg));
  pc->data               = mg;
  mg->nlevels            = -1;
  mg->am                 = PC_MG_MULTIPLICATIVE;
  mg->galerkin           = PC_MG_GALERKIN_NONE;
  mg->adaptInterpolation = PETSC_FALSE;
  mg->Nc                 = -1;
  mg->eigenvalue         = -1;

  pc->useAmat = PETSC_TRUE;

  pc->ops->apply          = PCApply_MG;
  pc->ops->applytranspose = PCApplyTranspose_MG;
  pc->ops->matapply       = PCMatApply_MG;
  pc->ops->setup          = PCSetUp_MG;
  pc->ops->reset          = PCReset_MG;
  pc->ops->destroy        = PCDestroy_MG;
  pc->ops->setfromoptions = PCSetFromOptions_MG;
  pc->ops->view           = PCView_MG;

  PetscCall(PetscObjectComposedDataRegister(&mg->eigenvalue));
  PetscCall(PetscObjectComposeFunction((PetscObject)pc, "PCMGSetGalerkin_C", PCMGSetGalerkin_MG));
  PetscCall(PetscObjectComposeFunction((PetscObject)pc, "PCMGGetLevels_C", PCMGGetLevels_MG));
  PetscCall(PetscObjectComposeFunction((PetscObject)pc, "PCMGSetLevels_C", PCMGSetLevels_MG));
  PetscCall(PetscObjectComposeFunction((PetscObject)pc, "PCGetInterpolations_C", PCGetInterpolations_MG));
  PetscCall(PetscObjectComposeFunction((PetscObject)pc, "PCGetCoarseOperators_C", PCGetCoarseOperators_MG));
  PetscCall(PetscObjectComposeFunction((PetscObject)pc, "PCMGSetAdaptInterpolation_C", PCMGSetAdaptInterpolation_MG));
  PetscCall(PetscObjectComposeFunction((PetscObject)pc, "PCMGGetAdaptInterpolation_C", PCMGGetAdaptInterpolation_MG));
  PetscCall(PetscObjectComposeFunction((PetscObject)pc, "PCMGSetAdaptCR_C", PCMGSetAdaptCR_MG));
  PetscCall(PetscObjectComposeFunction((PetscObject)pc, "PCMGGetAdaptCR_C", PCMGGetAdaptCR_MG));
  PetscCall(PetscObjectComposeFunction((PetscObject)pc, "PCMGSetAdaptCoarseSpaceType_C", PCMGSetAdaptCoarseSpaceType_MG));
  PetscCall(PetscObjectComposeFunction((PetscObject)pc, "PCMGGetAdaptCoarseSpaceType_C", PCMGGetAdaptCoarseSpaceType_MG));
  PetscFunctionReturn(PETSC_SUCCESS);
}