[petsc-users] Richardson with multiplicative MG vs. full MG

[Jed Brown]

Should we add PCMGRegisterCycle() and/or provide a more flexible way to
specify cycle index?

On Mon, Aug 13, 2012 at 4:53 PM, Barry Smith <bsmith at mcs.anl.gov> wrote:

>
>    John,
>
>      You are right we do not have completely optimized forms of all
> variants of the multigrid algorithms.
>
>      You could make a modification to the routine
>
> #undef __FUNCT__
> #define __FUNCT__ "PCMGMCycle_Private"
> PetscErrorCode PCMGMCycle_Private(PC pc,PC_MG_Levels
> **mglevelsin,PCRichardsonConvergedReason *reason)
> {
>   PC_MG          *mg = (PC_MG*)pc->data;
>   PC_MG_Levels   *mgc,*mglevels = *mglevelsin;
>   PetscErrorCode ierr;
>   PetscInt       cycles = (mglevels->level == 1) ? 1 : (PetscInt)
> mglevels->cycles;
>
>   PetscFunctionBegin;
>
>   if (mglevels->eventsmoothsolve) {ierr =
> PetscLogEventBegin(mglevels->eventsmoothsolve,0,0,0,0);CHKERRQ(ierr);}
>   ierr =
> KSPSolve(mglevels->smoothd,mglevels->b,mglevels->x);CHKERRQ(ierr);  /*
> pre-smooth */
>   if (mglevels->eventsmoothsolve) {ierr =
> PetscLogEventEnd(mglevels->eventsmoothsolve,0,0,0,0);CHKERRQ(ierr);}
>   if (mglevels->level) {  /* not the coarsest grid */
>     if (mglevels->eventresidual) {ierr =
> PetscLogEventBegin(mglevels->eventresidual,0,0,0,0);CHKERRQ(ierr);}
>     ierr =
> (*mglevels->residual)(mglevels->A,mglevels->b,mglevels->x,mglevels->r);CHKERRQ(ierr);
>     if (mglevels->eventresidual) {ierr =
> PetscLogEventEnd(mglevels->eventresidual,0,0,0,0);CHKERRQ(ierr);}
>
>     /* if on finest level and have convergence criteria set */
>     if (mglevels->level == mglevels->levels-1 && mg->ttol && reason) {
>       PetscReal rnorm;
>       ierr = VecNorm(mglevels->r,NORM_2,&rnorm);CHKERRQ(ierr);
>       if (rnorm <= mg->ttol) {
>         if (rnorm < mg->abstol) {
>           *reason = PCRICHARDSON_CONVERGED_ATOL;
>           ierr = PetscInfo2(pc,"Linear solver has converged. Residual norm
> %G is less than absolute tolerance %G\n",rnorm,mg->abstol);CHKERRQ(ierr);
>         } else {
>           *reason = PCRICHARDSON_CONVERGED_RTOL;
>           ierr = PetscInfo2(pc,"Linear solver has converged. Residual norm
> %G is less than relative tolerance times initial residual norm
> %G\n",rnorm,mg->ttol);CHKERRQ(ierr);
>         }
>         PetscFunctionReturn(0);
>       }
>     }
>
>     mgc = *(mglevelsin - 1);
>     if (mglevels->eventinterprestrict) {ierr =
> PetscLogEventBegin(mglevels->eventinterprestrict,0,0,0,0);CHKERRQ(ierr);}
>     ierr = MatRestrict(mglevels->restrct,mglevels->r,mgc->b);CHKERRQ(ierr);
>     if (mglevels->eventinterprestrict) {ierr =
> PetscLogEventEnd(mglevels->eventinterprestrict,0,0,0,0);CHKERRQ(ierr);}
>     ierr = VecSet(mgc->x,0.0);CHKERRQ(ierr);
>     while (cycles--) {
>       ierr = PCMGMCycle_Private(pc,mglevelsin-1,reason);CHKERRQ(ierr);
>     }
>     if (mglevels->eventinterprestrict) {ierr =
> PetscLogEventBegin(mglevels->eventinterprestrict,0,0,0,0);CHKERRQ(ierr);}
>     ierr =
> MatInterpolateAdd(mglevels->interpolate,mgc->x,mglevels->x,mglevels->x);CHKERRQ(ierr);
>     if (mglevels->eventinterprestrict) {ierr =
> PetscLogEventEnd(mglevels->eventinterprestrict,0,0,0,0);CHKERRQ(ierr);}
>     if (mglevels->eventsmoothsolve) {ierr =
> PetscLogEventBegin(mglevels->eventsmoothsolve,0,0,0,0);CHKERRQ(ierr);}
>     ierr =
> KSPSolve(mglevels->smoothu,mglevels->b,mglevels->x);CHKERRQ(ierr);    /*
> post smooth */
>     if (mglevels->eventsmoothsolve) {ierr =
> PetscLogEventEnd(mglevels->eventsmoothsolve,0,0,0,0);CHKERRQ(ierr);}
>   }
>   PetscFunctionReturn(0);
> }
>
> to remove the unneeded computations.
>
>    Barry
>
>
>
> On Aug 13, 2012, at 3:39 PM, John Fettig <john.fettig at gmail.com> wrote:
>
> > What if you wanted to do a full cycle with no pre-smooths instead of a
> v-cycle?
> >
> > John
> >
> > On Mon, Aug 13, 2012 at 4:34 PM, Barry Smith <bsmith at mcs.anl.gov> wrote:
> >>
> >> #undef __FUNCT__
> >> #define __FUNCT__ "PCMGKCycle_Private"
> >> PetscErrorCode PCMGKCycle_Private(PC pc,PC_MG_Levels **mglevels)
> >> {
> >>  PetscErrorCode ierr;
> >>  PetscInt       i,l = mglevels[0]->levels;
> >>
> >>  PetscFunctionBegin;
> >>  /* restrict the RHS through all levels to coarsest. */
> >>  for (i=l-1; i>0; i--){
> >>    if (mglevels[i]->eventinterprestrict) {ierr =
> PetscLogEventBegin(mglevels[i]->eventinterprestrict,0,0,0,0);CHKERRQ(ierr);}
> >>    ierr =
> MatRestrict(mglevels[i]->restrct,mglevels[i]->b,mglevels[i-1]->b);CHKERRQ(ierr);
> >>    if (mglevels[i]->eventinterprestrict) {ierr =
> PetscLogEventEnd(mglevels[i]->eventinterprestrict,0,0,0,0);CHKERRQ(ierr);}
> >>  }
> >>
> >>  /* work our way up through the levels */
> >>  ierr = VecSet(mglevels[0]->x,0.0);CHKERRQ(ierr);
> >>  for (i=0; i<l-1; i++) {
> >>    if (mglevels[i]->eventsmoothsolve) {ierr =
> PetscLogEventBegin(mglevels[i]->eventsmoothsolve,0,0,0,0);CHKERRQ(ierr);}
> >>    ierr =
> KSPSolve(mglevels[i]->smoothd,mglevels[i]->b,mglevels[i]->x);CHKERRQ(ierr);
> >>    if (mglevels[i]->eventsmoothsolve) {ierr =
> PetscLogEventEnd(mglevels[i]->eventsmoothsolve,0,0,0,0);CHKERRQ(ierr);}
> >>    if (mglevels[i+1]->eventinterprestrict) {ierr =
> PetscLogEventBegin(mglevels[i+1]->eventinterprestrict,0,0,0,0);CHKERRQ(ierr);}
> >>    ierr =
> MatInterpolate(mglevels[i+1]->interpolate,mglevels[i]->x,mglevels[i+1]->x);CHKERRQ(ierr);
> >>    if (mglevels[i+1]->eventinterprestrict) {ierr =
> PetscLogEventEnd(mglevels[i+1]->eventinterprestrict,0,0,0,0);CHKERRQ(ierr);}
> >>  }
> >>  if (mglevels[l-1]->eventsmoothsolve) {ierr =
> PetscLogEventBegin(mglevels[l-1]->eventsmoothsolve,0,0,0,0);CHKERRQ(ierr);}
> >>  ierr =
> KSPSolve(mglevels[l-1]->smoothd,mglevels[l-1]->b,mglevels[l-1]->x);CHKERRQ(ierr);
> >>  if (mglevels[l-1]->eventsmoothsolve) {ierr =
> PetscLogEventEnd(mglevels[l-1]->eventsmoothsolve,0,0,0,0);CHKERRQ(ierr);}
> >>
> >>  PetscFunctionReturn(0);
> >> }
> >>
> >>
> >> On Aug 13, 2012, at 3:19 PM, Jed Brown <jedbrown at mcs.anl.gov> wrote:
> >>
> >>> Shorthand for this is -pc_mg_type kaskade.
> >>>
> >>> On Mon, Aug 13, 2012 at 1:01 PM, John Fettig <john.fettig at gmail.com>
> wrote:
> >>> Barry,
> >>>
> >>> Thank you for answering my question.  I have another one for you:  it
> >>> seems the special case of zero pre-smooths is somewhat non-trivial.
> >>> The best I can do is set the pre-smoother to Richardson with PCNONE
> >>> and zero as max_its. However, if you aren't careful in setting
> >>> KSPSetInitialGuessNonzero this can have unexpected results since the
> >>> generic KSPSolve will clobber your solution before it even tries a
> >>> convergence criteria (thus ruling out KSPPREONLY).  It also does a
> >>> couple of unnecessary residual calculations. Would it be reasonable to
> >>> put a zero-iteration special case in KSPSolve so that if you don't
> >>> want any iterations it doesn't actually do anything (no setup, no
> >>> preconditioner, no residual, no scaling, etc.)?
> >>>
> >>> Thanks,
> >>> John
> >>>
> >>> On Thu, Aug 9, 2012 at 6:37 PM, Barry Smith <bsmith at mcs.anl.gov>
> wrote:
> >>>>
> >>>>  John,
> >>>>
> >>>> On Aug 9, 2012, at 9:50 AM, John Fettig <john.fettig at gmail.com>
> wrote:
> >>>>
> >>>>> I am a little confused about what Richardson means.  If you use
> >>>>> multiplicative V-cycle multigrid with Richardson KSP (and no
> >>>>> convergence monitor), it sets the applyrichardson operator to
> >>>>> PCApplyRichardson_MG, which appears to just run V-cycles until
> >>>>> convergence.
> >>>>
> >>>>     Yes, this is correct.
> >>>>
> >>>>> As far as I can tell, it doesn't ever update according
> >>>>> to the documented
> >>>>>
> >>>>> x^{n+1} = x^{n} + scale*B(b - A x^{n})
> >>>>>
> >>>>      In exact arithmetic it is actually "implicitly" doing exactly
> this update.  It is difficult to see why this is true generally (because B
> is rather complicated for multigrid) but if you consider only two levels
> with a direct solver on the coarse grid and SSOR as the pre and post smooth
> you can write out the formulas and map back and forth between the two
> forms.  The reason for the PCApplyRichardson_ forms is because they are a
> bit more efficient than  separating out the action of B and then doing the
> update as above.
> >>>>
> >>>>
> >>>>> If on the other hand you use full MG, it does update according to the
> >>>>> above formula.  This also happens if you set a convergence monitor.
> >>>>>
> >>>>> I can see how multiplicative V-cycle with Richardson is simply using
> >>>>> multigrid as a solver.  What I don't understand is how full MG with
> >>>>> Richardson is using multigrid as a solver, because it is using the
> >>>>> update formula above in between cycles..  Shouldn't there be a
> >>>>> applyrichardson for full multigrid as well?  If not, why?
> >>>>
> >>>>    I think there could be a applyRichardson for full multigrid but it
> would be kind of complicated and would not benefit much because the amount
> of work in a full multigrid step is much higher so the savings would be a
> much lower percentage than with V cycle.
> >>>>
> >>>>   Barry
> >>>>
> >>>>>
> >>>>> Thanks,
> >>>>> John
> >>>>
> >>>
> >>
>
>
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