Some info about the MG benchmark
(Note: this info applies to the parallel version and mostly concerns
the processor decomposition.  Info not concerning the decomposition
still applies to the serial version.)
================================
    
'mg_demo' demonstrates the capabilities of a very simple multigrid
solver in computing a three dimensional potential field.  This is
a simplified multigrid solver in two important respects:

  (1) it solves only a constant coefficient equation,
  and that only on a uniform cubical grid,
    
  (2) it solves only a single equation, representing
  a scalar field rather than a vector field.

We chose it for its portability and simplicity, and expect that a
supercomputer which can run it effectively will also be able to
run more complex multigrid programs at least as well.
     
     Eric Barszcz                         Paul Frederickson
     RIACS
     NASA Ames Research Center            NASA Ames Research Center

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Running the program:  (Note: also see parameter lm information in the
                       two sections immediately below this section)

The program may be run with or without an input deck (called "mg.input"). 
The following describes a few things about the input deck if you want to 
use one. 

The four lines below are the "mg.input" file required to run a
problem of total size 256x256x256, for 4 iterations (Class "A"),
and presumes the use of 8 processors:

   8 = top level
   256 256 256 = nx ny nz
   4 = nit
   0 0 0 0 0 0 0 0 = debug_vec

The first line of input indicates how many levels of multi-grid
cycle will be applied to a particular subpartition.  Presuming that
8 processors are solving this problem (recall that the number of 
processors is specified to MPI as a run parameter, and MPI subsequently
determines this for the code via an MPI subroutine call), a 2x2x2 
processor grid is  formed, and thus each partition on a processor is 
of size 128x128x128.  Therefore, a maximum of 8 multi-grid levels may 
be used.  These are of size 128,64,32,16,8,4,2,1, with the coarsest 
level being a single point on a given processor.


Next, consider the same size problem but running on 1 processor.  The
following "mg.input" file is appropriate:

    9 = top level
    256 256 256 = nx ny nz
    4 = nit
    0 0 0 0 0 0 0 0 = debug_vec

Since this processor must solve the full 256x256x256 problem, this
permits 9 multi-grid levels (256,128,64,32,16,8,4,2,1), resulting in 
a coarsest multi-grid level of a single point on the processor


Next, consider the same size problem but running on 2 processors.  The
following "mg.input" file is required:

    8 = top level
    256 256 256 = nx ny nz
    4 = nit
    0 0 0 0 0 0 0 0 = debug_vec

The algorithm for partitioning the full grid onto some power of 2 number 
of processors is to start by splitting the last dimension of the grid
(z dimension) in 2: the problem is now partitioned onto 2 processors.
Next the middle dimension (y dimension) is split in 2: the problem is now
partitioned onto 4 processors.  Next, first dimension (x dimension) is
split in 2: the problem is now partitioned onto 8 processors.  Next, the
last dimension (z dimension) is split again in 2: the problem is now
partitioned onto 16 processors.  This partitioning is repeated until all 
of the power of 2 processors have been allocated.

Thus to run the above problem on 2 processors, the grid partitioning 
algorithm will allocate the two processors across the last dimension, 
creating two partitions each of size 256x256x128. The coarsest level of 
multi-grid must be a single point surrounded by a cubic number of grid 
points.  Therefore, each of the two processor partitions will contain 4 
coarsest multi-grid level points, each surrounded by a cube of grid points 
of size 128x128x128, indicated by a top level of 8.


Next, consider the same size problem but running on 4 processors.  The
following "mg.input" file is required:

    8 = top level
    256 256 256 = nx ny nz
    4 = nit
    0 0 0 0 0 0 0 0 = debug_vec

The partitioning algorithm will create 4 partitions, each of size
256x128x128.  Each partition will contain 2 coarsest multi-grid level
points each surrounded by a cube of grid points of size 128x128x128, 
indicated by a top level of 8.


Next, consider the same size problem but running on 16 processors.  The
following "mg.input" file is required:

    7 = top level
    256 256 256 = nx ny nz
    4 = nit
    0 0 0 0 0 0 0 0 = debug_vec

On each node a partition of size 128x128x64 will be created.  A maximum
of 7 multi-grid levels (64,32,16,8,4,2,1) may be used, resulting in each 
partions containing 4 coarsest multi-grid level points, each surrounded 
by a cube of grid points of size 64x64x64, indicated by a top level of 7.


Note that non-cubic problem sizes may also be considered:

The four lines below are the "mg.input" file appropriate for running a
problem of total size 256x512x512, for 20 iterations and presumes the 
use of 32 processors (note: this is NOT a class C problem):

    8 = top level
    256 512 512 = nx ny nz
    20 = nit
    0 0 0 0 0 0 0 0 = debug_vec

The first line of input indicates how many levels of multi-grid
cycle will be applied to a particular subpartition.  Presuming that
32 processors are solving this problem, a 2x4x4 processor grid is
formed, and thus each partition on a processor is of size 128x128x128.
Therefore, a maximum of 8 multi-grid levels may be used.  These are of
size 128,64,32,16,8,4,2,1, with the coarsest level being a single 
point on a given processor.