On May 3, 2012, at 12:01 AM, Karthik Duraisamy wrote:
> Sorry.. I forgot to add that I tried just those options
>
> -pc_type ml -ksp_type fgmres -ksp_max_it 100 -ksp_view
>
> to get the result that I got.
The output you sent below was NOT obtained with these options.
>> type: gmres
>> type: bjacobi
Are you sure you have KSPSetFromOptions() in your code?
Barry
> I was asking for additional options. In any case, now I can try a few things
>
> ----- Original Message -----
> From: "Barry Smith" <bsmith at mcs.anl.gov>
> To: "PETSc users list" <petsc-users at mcs.anl.gov>
> Sent: Wednesday, May 2, 2012 9:56:05 PM
> Subject: Re: [petsc-users] Multigrid
>
>
> Make the run I suggested -pc_type ml -ksp_type fgmres -ksp_max_it 100
> -ksp_view
>
> Barry
>
> On May 2, 2012, at 10:21 PM, Karthik Duraisamy wrote:
>
>> Fair enough. I wiped out all references to petsc in my code and started from
>> scratch and used just the options you mentioned.. and I get good convergence
>> as first reported in the PCMG reference! I have attached the output. Now
>> that it is behaving, could you recommend some options to exercise the
>> multigrid?
>>
>> Thanks a lot for your time (and patience!)
>>
>> ----------------------
>>
>> KSP Object: 8 MPI processes
>> type: gmres
>> GMRES: restart=30, using Classical (unmodified) Gram-Schmidt
>> Orthogonalization with no iterative refinement
>> GMRES: happy breakdown tolerance 1e-30
>> maximum iterations=1, initial guess is zero
>> tolerances: relative=1e-05, absolute=1e-50, divergence=1e+10
>> left preconditioning
>> using PRECONDITIONED norm type for convergence test
>> PC Object: 8 MPI processes
>> type: bjacobi
>> block Jacobi: number of blocks = 8
>> Local solve is same for all blocks, in the following KSP and PC objects:
>> KSP Object: (sub_) 1 MPI processes
>> type: preonly
>> maximum iterations=10000, initial guess is zero
>> tolerances: relative=1e-05, absolute=1e-50, divergence=10000
>> left preconditioning
>> using NONE norm type for convergence test
>> PC Object: (sub_) 1 MPI processes
>> type: ilu
>> ILU: out-of-place factorization
>> 0 levels of fill
>> tolerance for zero pivot 1e-12
>> using diagonal shift to prevent zero pivot
>> matrix ordering: natural
>> factor fill ratio given 1, needed 1
>> Factored matrix follows:
>> Matrix Object: 1 MPI processes
>> type: seqaij
>> rows=9015, cols=9015
>> package used to perform factorization: petsc
>> total: nonzeros=517777, allocated nonzeros=517777
>> total number of mallocs used during MatSetValues calls =0
>> using I-node routines: found 3476 nodes, limit used is 5
>> linear system matrix = precond matrix:
>> Matrix Object: 1 MPI processes
>> type: seqaij
>> rows=9015, cols=9015
>> total: nonzeros=517777, allocated nonzeros=517777
>> total number of mallocs used during MatSetValues calls =0
>> using I-node routines: found 3476 nodes, limit used is 5
>> linear system matrix = precond matrix:
>> Matrix Object: 8 MPI processes
>> type: mpiaij
>> rows=75000, cols=75000
>> total: nonzeros=4427800, allocated nonzeros=4427800
>> total number of mallocs used during MatSetValues calls =0
>> using I-node (on process 0) routines: found 3476 nodes, limit used is 5
>>
>>
>>
>>
>> ----- Original Message -----
>> From: "Barry Smith" <bsmith at mcs.anl.gov>
>> To: "PETSc users list" <petsc-users at mcs.anl.gov>
>> Sent: Wednesday, May 2, 2012 10:34:17 AM
>> Subject: Re: [petsc-users] Multigrid
>>
>>
>> Run with -pc_type ml -ksp_type fgmres -ksp_max_it 100 -ksp_view and
>> NOTHING else. Don'ts set any KSP or PC options in the code! Then send us
>> ALL the output.
>>
>> Barry
>>
>> I am getting all confused with "I ran with these options and this happened,
>> I ran with these options and this happened, I ran with these options and
>> this happened". I am getting information overload with too much information
>> and yet not enough information.
>>
>>
>> On May 2, 2012, at 12:27 PM, Karthik Duraisamy wrote:
>>
>>> Hello,
>>>
>>> With PCML, I tried
>>>
>>> -mg_coarse_pc_type redundant ; -mg_coarse_ksp_type preonly ;
>>> KSPSetType(ksp, KSPGMRES);
>>>
>>> and the residual still diverges after a while. Strangely, if I use PCMG
>>> (with 0 levels), it works OK, but not with PCML.
>>>
>>> This is definitely strange. Here is the entire piece of relevant code.
>>>
>>> When initializing Petsc, I do the following: (let's call this the first bit
>>> and I do this only once)
>>>
>>> KSPSetOperators(ksp, A_, A_, SAME_NONZERO_PATTERN);
>>> KSPSetInitialGuessKnoll(ksp, PETSC_TRUE);
>>> KSPSetType(ksp, KSPFGMRES);
>>> KSPGMRESSetRestart(ksp, 100);
>>> KSPSetFromOptions(ksp);
>>>
>>>
>>> Then for each outer solver iteration: (let's call this the second bit and I
>>> do this every outer iteration)
>>>
>>> PC pc;
>>> KSPGetPC(ksp,&pc);
>>> PCSetType(pc, PCMG);
>>> KSPSetOperators(ksp, A_, A_, SAME_NONZERO_PATTERN);
>>> PetscOptionsSetValue("-ksp_initial_guess_nonzero", "true");
>>> KSPSetType(ksp, KSPGMRES);
>>> KSPGMRESSetRestart(ksp, 100);
>>> KSPSetFromOptions(ksp);
>>>
>>>
>>> I have tried various combinations of preconditioners and smoothers
>>> (including fgmres, bcgsl, etc) but the only combination that works is the
>>> above. Any of the PCML options that I have tried failed.
>>>
>>> Notes:
>>> - If I keep both of the above bits, it converges very well
>>>
>>> - If I keep both of the above bits, but replace PCMG with PCML, it diverges
>>> quickly
>>>
>>> - If I removed the first bit and kept the second bit, the residuals diverge
>>> even with PCMG
>>>
>>> - If I keep the first bit and remove the second bit, the residuals diverge
>>> eventually, but at a slow rate
>>>
>>>
>>> I am very confident with the rest of the code as I've been using PETSc
>>> (with just the first bit) for 3 years on 100s of problems.
>>>
>>> Regards,
>>> Karthik
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>> ----- Original Message -----
>>> From: "Barry Smith" <bsmith at mcs.anl.gov>
>>> To: "PETSc users list" <petsc-users at mcs.anl.gov>
>>> Sent: Wednesday, May 2, 2012 6:02:06 AM
>>> Subject: Re: [petsc-users] Multigrid
>>>
>>>
>>> This is likely helping a great deal:
>>>
>>> KSPGMRESSetRestart(ksp, 100);
>>>
>>> You can try this with ml also
>>>
>>> On May 2, 2012, at 1:33 AM, Karthik Duraisamy wrote:
>>>
>>>> Hello again,
>>>>
>>>> I played with BoomerAMG and ML (for the 2D problem with a reasonable
>>>> condition number), and so far, they haven't worked (the residual
>>>> eventually diverges). The options that I am using are (for instance)
>>>>
>>>> PC pc;
>>>> KSPGetPC(ksp,&pc);
>>>> PetscOptionsSetValue("-ksp_type", "gmres");
>>>
>>> Here you NEED fgmres since the smoother is gmres by default. You can see
>>> this with -ksp_view
>>>
>>>> KSPSetOperators(ksp, A_, A_, SAME_NONZERO_PATTERN);
>>>> PetscOptionsSetValue("-ksp_initial_guess_nonzero", "true");
>>>> PCSetType(pc, PCML);
>>>> PetscOptionsSetValue("-pc_ml_maxNlevels", "3");
>>>> PetscOptionsSetValue("-mg_coarse_ksp_type","richardson");
>>>> PetscOptionsSetValue("-mg_coarse_pc_type","sor");
>>>> PetscOptionsSetValue("-mg_coarse_pc_sor_its","8");
>>>
>>> Just use -mg_coarse_pc_type redundant -mg_coarse_ksp_type preonly instead
>>> of the above.
>>>
>>>
>>> Barry
>>>
>>>> PetscOptionsSetValue("-ksp_monitor","1");
>>>> KSPSetFromOptions(ksp);
>>>
>>>
>>>>
>>>> I have replaced KSP with bcgsl, fgmres, etc.
>>>>
>>>> As I mentioned earlier, the following works like a charm:
>>>>
>>>> PC pc;
>>>> KSPGetPC(ksp,&pc);
>>>> PCSetType(pc, PCMG);
>>>> KSPSetOperators(ksp, A_, A_, SAME_NONZERO_PATTERN);
>>>> PetscOptionsSetValue("-ksp_initial_guess_nonzero", "true");
>>>> KSPSetType(ksp, KSPGMRES);
>>>> KSPGMRESSetRestart(ksp, 100);
>>>> KSPSetFromOptions(ksp);
>>>>
>>>>
>>>> Regards,
>>>> Karthik.
>>>>
>>>>
>>>>
>>>> ----- Original Message -----
>>>> From: "Barry Smith" <bsmith at mcs.anl.gov>
>>>> To: "PETSc users list" <petsc-users at mcs.anl.gov>
>>>> Sent: Tuesday, May 1, 2012 9:14:05 PM
>>>> Subject: Re: [petsc-users] Multigrid
>>>>
>>>>
>>>> On May 1, 2012, at 6:00 PM, Karthik Duraisamy wrote:
>>>>
>>>>> So as I understand it, GMRES is used as a preconditioner and as a solver
>>>>> when I use PCMG with defaults. If this is the case, I should be able to
>>>>> recreate this set up without the PCMG. Any pointers as to how this can be
>>>>> done?
>>>>>
>>>>> Also, yes indeed, my mesh is completely unstructured, so I will have to
>>>>> use ml or boomeramg.
>>>>>
>>>>> The problems that I am attempting involve RANS of compressible turbulent
>>>>> combustion (finite volume, steady). The high condition numbers are
>>>>> because of the extreme grid stretching and stiff source terms (in the
>>>>> turbulence and combustion model).
>>>>
>>>> With a condition number like that I wouldn't even consider using double
>>>> precision, I think you would just be computing meaningless noise. With
>>>> petsc-dev you can use quad precision ./configure
>>>> --with-precision=__float128 using recent GNU compilers
>>>> (no we haven't tested the gfortran sides of things).
>>>>
>>>> Barry
>>>>
>>>>
>>>>> I have been trying a reduced problem in these 2D test cases, in which the
>>>>> condition number is only 1e7.
>>>>>
>>>>> Thanks,
>>>>> Karthik.
>>>>>
>>>>>
>>>>> ----- Original Message -----
>>>>> From: "Mark F. Adams" <mark.adams at columbia.edu>
>>>>> To: "PETSc users list" <petsc-users at mcs.anl.gov>
>>>>> Sent: Tuesday, May 1, 2012 3:50:31 PM
>>>>> Subject: Re: [petsc-users] Multigrid
>>>>>
>>>>>
>>>>> Also, note that PCMG can not create coarse grid spaces for an (MPI)AIJ
>>>>> matrix. If you use regular grids (DA?) then PETSc can construct geometric
>>>>> multigrid coarse grid spaces, although I don't know if PCMG will
>>>>> construct these for you (I don't think it will and I can see from your
>>>>> output that PCMG just used one grid). 'ml', hypre' and 'gamg' (a native
>>>>> AMG solver) will do real AMG solvers for you. All three can work on a
>>>>> similar class of problems.
>>>>>
>>>>>
>>>>> Also, you mention that you have a condition number of 1.e20. That is
>>>>> astronomical for such a small problem. How did you compute that number?
>>>>> Do you know where the ill-conditioning comes from? Is this an elliptic
>>>>> operator?
>>>>>
>>>>>
>>>>> Mark
>>>>>
>>>>>
>>>>>
>>>>>
>>>>> On May 1, 2012, at 6:31 PM, Matthew Knepley wrote:
>>>>>
>>>>>
>>>>>
>>>>> On Tue, May 1, 2012 at 6:27 PM, Karthik Duraisamy < dkarthik at
>>>>> stanford.edu > wrote:
>>>>>
>>>>>
>>>>>
>>>>> The following was output for the very first iteration whereas what I had
>>>>> attached earlier was output every iteration. I am still a bit perplexed
>>>>> because PCMG drops the residual like a rock (after the first few
>>>>> iterations whereas with no PCMG, it is very slow)
>>>>>
>>>>>
>>>>>
>>>>> Because the smoother IS the solver you were using before. Just like I
>>>>> said last time, what you are doing is
>>>>> wrapping up the same solver you used before, sticking it in another GMRES
>>>>> loop, and only looking at the
>>>>> outer loop. This has nothing to do with MG.
>>>>>
>>>>>
>>>>> Matt
>>>>>
>>>>>
>>>>> KSP Object: 8 MPI processes
>>>>> type: gmres
>>>>> GMRES: restart=100, using Classical (unmodified) Gram-Schmidt
>>>>> Orthogonalization with no iterative refinement
>>>>> GMRES: happy breakdown tolerance 1e-30
>>>>> maximum iterations=1, initial guess is zero
>>>>> using preconditioner applied to right hand side for initial guess
>>>>> tolerances: relative=0.01, absolute=1e-08, divergence=1e+10
>>>>> left preconditioning
>>>>> using DEFAULT norm type for convergence test
>>>>> PC Object: 8 MPI processes
>>>>> type: mg
>>>>> MG: type is MULTIPLICATIVE, levels=1 cycles=v
>>>>> Cycles per PCApply=1
>>>>> Not using Galerkin computed coarse grid matrices
>>>>> Coarse grid solver -- level -------------------------------
>>>>> KSP Object: (mg_levels_0_) 8 MPI processes
>>>>> type not yet set
>>>>> maximum iterations=1, initial guess is zero
>>>>> tolerances: relative=1e-05, absolute=1e-50, divergence=10000
>>>>> left preconditioning
>>>>> using DEFAULT norm type for convergence test
>>>>> PC Object: (mg_levels_0_) 8 MPI processes
>>>>> type not yet set
>>>>> linear system matrix = precond matrix:
>>>>> Matrix Object: 8 MPI processes
>>>>> type: mpiaij
>>>>> rows=75000, cols=75000
>>>>> total: nonzeros=4427800, allocated nonzeros=4427800
>>>>> total number of mallocs used during MatSetValues calls =0
>>>>> using I-node (on process 0) routines: found 3476 nodes, limit used is 5
>>>>>
>>>>>
>>>>> ----- Original Message -----
>>>>> From: "Matthew Knepley" < knepley at gmail.com >
>>>>> To: "PETSc users list" < petsc-users at mcs.anl.gov >
>>>>> Sent: Tuesday, May 1, 2012 3:22:56 PM
>>>>> Subject: Re: [petsc-users] Multigrid
>>>>>
>>>>>
>>>>> On Tue, May 1, 2012 at 6:18 PM, Karthik Duraisamy < dkarthik at
>>>>> stanford.edu > wrote:
>>>>>
>>>>>
>>>>>
>>>>> Hello,
>>>>>
>>>>> Sorry (and thanks for the reply). I've attached the no multigrid case. I
>>>>> didn't include it because (at least to the untrained eye, everything
>>>>> looks the same).
>>>>>
>>>>>
>>>>>
>>>>> Did you send all the output from the MG case? There must be a PC around
>>>>> it. By default its GMRES, so there would be
>>>>> an extra GMRES loop compared to the case without MG.
>>>>>
>>>>>
>>>>> Matt
>>>>>
>>>>>
>>>>> Regards,
>>>>> Karthik
>>>>>
>>>>> KSP Object: 8 MPI processes
>>>>> type: gmres
>>>>> GMRES: restart=100, using Classical (unmodified) Gram-Schmidt
>>>>> Orthogonalization with no iterative refinement
>>>>> GMRES: happy breakdown tolerance 1e-30
>>>>> maximum iterations=1
>>>>> using preconditioner applied to right hand side for initial guess
>>>>> tolerances: relative=1e-05, absolute=1e-50, divergence=1e+10
>>>>> left preconditioning
>>>>> using nonzero initial guess
>>>>> using PRECONDITIONED norm type for convergence test
>>>>> PC Object: 8 MPI processes
>>>>> type: bjacobi
>>>>> block Jacobi: number of blocks = 8
>>>>> Local solve is same for all blocks, in the following KSP and PC objects:
>>>>> KSP Object: (sub_) 1 MPI processes
>>>>> type: preonly
>>>>> maximum iterations=10000, initial guess is zero
>>>>> tolerances: relative=1e-05, absolute=1e-50, divergence=10000
>>>>> left preconditioning
>>>>> using NONE norm type for convergence test
>>>>> PC Object: (sub_) 1 MPI processes
>>>>> type: ilu
>>>>> ILU: out-of-place factorization
>>>>> 0 levels of fill
>>>>> tolerance for zero pivot 1e-12
>>>>> using diagonal shift to prevent zero pivot
>>>>> matrix ordering: natural
>>>>> factor fill ratio given 1, needed 1
>>>>> Factored matrix follows:
>>>>> Matrix Object: 1 MPI processes
>>>>> type: seqaij
>>>>> rows=9015, cols=9015
>>>>> package used to perform factorization: petsc
>>>>> total: nonzeros=517777, allocated nonzeros=517777
>>>>> total number of mallocs used during MatSetValues calls =0
>>>>> using I-node routines: found 3476 nodes, limit used is 5
>>>>> linear system matrix = precond matrix:
>>>>> Matrix Object: 1 MPI processes
>>>>> type: seqaij
>>>>> rows=9015, cols=9015
>>>>> total: nonzeros=517777, allocated nonzeros=517777
>>>>> total number of mallocs used during MatSetValues calls =0
>>>>> using I-node routines: found 3476 nodes, limit used is 5
>>>>> linear system matrix = precond matrix:
>>>>> Matrix Object: 8 MPI processes
>>>>> type: mpiaij
>>>>> rows=75000, cols=75000
>>>>> total: nonzeros=4427800, allocated nonzeros=4427800
>>>>> total number of mallocs used during MatSetValues calls =0
>>>>> using I-node (on process 0) routines: found 3476 nodes, limit used is 5
>>>>>
>>>>>
>>>>> ----- Original Message -----
>>>>> From: "Matthew Knepley" < knepley at gmail.com >
>>>>> To: "PETSc users list" < petsc-users at mcs.anl.gov >
>>>>> Sent: Tuesday, May 1, 2012 3:15:14 PM
>>>>> Subject: Re: [petsc-users] Multigrid
>>>>>
>>>>>
>>>>> On Tue, May 1, 2012 at 6:12 PM, Karthik Duraisamy < dkarthik at
>>>>> stanford.edu > wrote:
>>>>>
>>>>>
>>>>>
>>>>> Hello Barry,
>>>>>
>>>>> Thank you for your super quick response. I have attached the output of
>>>>> ksp_view and it is practically the same as that when I don't use PCMG.
>>>>> The part I don't understand is how PCMG able to function at the zero grid
>>>>> level and still produce a much better convergence than when using the
>>>>> default PC. Is there any additional smoothing or interpolation going on?
>>>>>
>>>>>
>>>>>
>>>>> You only included one output, so I have no way of knowing what you used
>>>>> before. However, this is running GMRES/ILU.
>>>>>
>>>>>
>>>>> Also, for Algebraic Multigrid, would you recommend BoomerAMG or ML ?
>>>>>
>>>>>
>>>>>
>>>>> They are different algorithms. Its not possible to say generally that one
>>>>> is better. Try them both.
>>>>>
>>>>>
>>>>> Matt
>>>>>
>>>>>
>>>>> Best regards,
>>>>> Karthik.
>>>>>
>>>>> type: mg
>>>>> MG: type is MULTIPLICATIVE, levels=1 cycles=v
>>>>> Cycles per PCApply=1
>>>>> Not using Galerkin computed coarse grid matrices
>>>>> Coarse grid solver -- level -------------------------------
>>>>> KSP Object: (mg_levels_0_) 8 MPI processes
>>>>> type: gmres
>>>>> GMRES: restart=30, using Classical (unmodified) Gram-Schmidt
>>>>> Orthogonalization with no iterative refinement
>>>>> GMRES: happy breakdown tolerance 1e-30
>>>>> maximum iterations=1, initial guess is zero
>>>>> tolerances: relative=1e-05, absolute=1e-50, divergence=10000
>>>>> left preconditioning
>>>>> using PRECONDITIONED norm type for convergence test
>>>>> PC Object: (mg_levels_0_) 8 MPI processes
>>>>> type: bjacobi
>>>>> block Jacobi: number of blocks = 8
>>>>> Local solve is same for all blocks, in the following KSP and PC objects:
>>>>> KSP Object: (mg_levels_0_sub_) 1 MPI processes
>>>>> type: preonly
>>>>> maximum iterations=10000, initial guess is zero
>>>>> tolerances: relative=1e-05, absolute=1e-50, divergence=10000
>>>>> left preconditioning
>>>>> using NONE norm type for convergence test
>>>>> PC Object: (mg_levels_0_sub_) 1 MPI processes
>>>>> type: ilu
>>>>> ILU: out-of-place factorization
>>>>> 0 levels of fill
>>>>> tolerance for zero pivot 1e-12
>>>>> using diagonal shift to prevent zero pivot
>>>>> matrix ordering: natural
>>>>> factor fill ratio given 1, needed 1
>>>>> Factored matrix follows:
>>>>> Matrix Object: 1 MPI processes
>>>>> type: seqaij
>>>>> rows=9015, cols=9015
>>>>> package used to perform factorization: petsc
>>>>> total: nonzeros=517777, allocated nonzeros=517777
>>>>> total number of mallocs used during MatSetValues calls =0
>>>>> using I-node routines: found 3476 nodes, limit used is 5
>>>>> linear system matrix = precond matrix:
>>>>> Matrix Object: 1 MPI processes
>>>>> type: seqaij
>>>>> rows=9015, cols=9015
>>>>> total: nonzeros=517777, allocated nonzeros=517777
>>>>> total number of mallocs used during MatSetValues calls =0
>>>>> using I-node routines: found 3476 nodes, limit used is 5
>>>>> linear system matrix = precond matrix:
>>>>> Matrix Object: 8 MPI processes
>>>>> type: mpiaij
>>>>> rows=75000, cols=75000
>>>>> total: nonzeros=4427800, allocated nonzeros=4427800
>>>>> total number of mallocs used during MatSetValues calls =0
>>>>> using I-node (on process 0) routines: found 3476 nodes, limit used is 5
>>>>> linear system matrix = precond matrix:
>>>>> Matrix Object: 8 MPI processes
>>>>> type: mpiaij
>>>>> rows=75000, cols=75000
>>>>> total: nonzeros=4427800, allocated nonzeros=4427800
>>>>> total number of mallocs used during MatSetValues calls =0
>>>>> using I-node (on process 0) routines: found 3476 nodes, limit used is 5
>>>>>
>>>>>
>>>>>
>>>>> ----- Original Message -----
>>>>> From: "Barry Smith" < bsmith at mcs.anl.gov >
>>>>> To: "PETSc users list" < petsc-users at mcs.anl.gov >
>>>>> Sent: Tuesday, May 1, 2012 1:39:26 PM
>>>>> Subject: Re: [petsc-users] Multigrid
>>>>>
>>>>>
>>>>> On May 1, 2012, at 3:37 PM, Karthik Duraisamy wrote:
>>>>>
>>>>>> Hello,
>>>>>>
>>>>>> I have been using PETSc for a couple of years with good success, but
>>>>>> lately as my linear problems have become stiffer (condition numbers of
>>>>>> the order of 1.e20), I am looking to use better preconditioners. I tried
>>>>>> using PCMG with all the default options (i.e., I just specified my
>>>>>> preconditioner as PCMG and did not add any options to it) and I am
>>>>>> immediately seeing better convergence.
>>>>>>
>>>>>> What I am not sure of is why? I would like to know more about the
>>>>>> default parameters (the manual is not very explicit) and more
>>>>>> importantly, want to know why it is working even when I haven't
>>>>>> specified any grid levels and coarse grid operators. Any
>>>>>> help in this regard will be appreciated.
>>>>>
>>>>> First run with -ksp_view to see what solver it is actually using.
>>>>>
>>>>> Barry
>>>>>
>>>>>>
>>>>>> Also, ultimately I want to use algebraic multigrid so is PCML a better
>>>>>> option than BoomerAMG? I tried BoomerAMG with mixed results.
>>>>>>
>>>>>> Thanks,
>>>>>> Karthik
>>>>>>
>>>>>>
>>>>>>
>>>>>> --
>>>>>>
>>>>>> =======================================
>>>>>> Karthik Duraisamy
>>>>>> Assistant Professor (Consulting)
>>>>>> Durand Building Rm 357
>>>>>> Dept of Aeronautics and Astronautics
>>>>>> Stanford University
>>>>>> Stanford CA 94305
>>>>>>
>>>>>> Phone: 650-721-2835
>>>>>> Web: www.stanford.edu/~dkarthik
>>>>>> =======================================
>>>>>
>>>>>
>>>>>
>>>>>
>>>>>
>>>>> --
>>>>> What most experimenters take for granted before they begin their
>>>>> experiments is infinitely more interesting than any results to which
>>>>> their experiments lead.
>>>>> -- Norbert Wiener
>>>>>
>>>>>
>>>>>
>>>>>
>>>>> --
>>>>> What most experimenters take for granted before they begin their
>>>>> experiments is infinitely more interesting than any results to which
>>>>> their experiments lead.
>>>>> -- Norbert Wiener
>>>>>
>>>>>
>>>>>
>>>>>
>>>>> --
>>>>> What most experimenters take for granted before they begin their
>>>>> experiments is infinitely more interesting than any results to which
>>>>> their experiments lead.
>>>>> -- Norbert Wiener
>>>>>
>>>>
>>>
>>
>
