Commit d87e74ee authored by Mads R. B. Kristensen's avatar Mads R. B. Kristensen
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added a sequential implementation of the heat equation

parent 4076693e
Chapel
======
Compilation instructions
------------------------
There are no specific requirements for building examples,
just standard make, working MPI environment (for MPI examples) and
OpenMP enabled C or Fortran compiler (for OpenMP examples).
Move to proper subfolder (C or Fortran) and modify the top of the **Makefile**
according to your environment (proper compiler commands and compiler flags).
All examples can be built with simple **make**, **make mpi** builds the MPI
examples and **make omp** OpenMP examples.
Chapel
======
Compilation instructions
------------------------
In order to compile and run these examples, the only requirement is a working Chapel compiler and make. You can download Chapel at http://chapel.cray.com/.
All examples can be built with simple **make**.
Heat Equation
=============
In this example, we solve the heat equation. The idea is to apply a 5-point stencil on a domain iteratively until equilibrium.
Sequential
----------
`sequential.chpl <src/sequential.chpl>` is a sequential implementation of the heat equation written in Chapel. The stencil computation is the most time consuming part of the code and look like::
for (i,j) in Interior do//Iterate over all non-border cells
{
//Assign each cell in 'T' the mean of its neighboring cells in 'A'
T[i,j] = (A[i,j] + A[i-1,j] + A[i+1,j] + A[i,j-1] + A[i,j+1]) / 5;
}
Basically, each *interior* element in ``T`` gets the mean of the corresponding element in ``A`` as well as the neighboring elements. Since ``for`` is a sequential language construct in Chapel, a single CPU-core will execute this code.
Multi-core
----------
In order to improve the performance, we can tell Chapel to use threads to execute the stencil operations in parallel (`single_machine.chpl <src/single_machine.chpl>`). We do that by replacing ``for`` with ``forall``, which tells Chapel to execute each iteration in ``Interior`` parallel.
It is our responsibility to make sure that each iteration in the ``forall`` loop is independent in order not to introduce race conditions.
Clearly in this case iteration is clearly independent since we do not read ``T``::
forall (i,j) in Interior do//Iterate over all non-border cells
{
//Assign each cell in 'T' the mean of its neighboring cells in 'A'
T[i,j] = (A[i,j] + A[i-1,j] + A[i+1,j] + A[i,j-1] + A[i,j+1]) / 5;
}
Multiple Machines
-----------------
In order to improve the performance even further, we can tell Chapel to execute the stencil operation in parallel on multiple machines (`multiple_machines.chpl <src/multiple_machines.chpl>`).
We still use the ``forall`` loop construct, be we have to tell Chapel how to distributes ``A`` and ``T`` between the multiple machines. For that, we use the ``dmapped`` language construct when defining the ``Grid`` and ``Interior`` domain::
//A n+2 by n+2 domain.
const Grid = {0..n+1, 0..n+1} dmapped Block({1..n, 1..n});
//A n by n domain that represents the interior of 'Grid'
const Interior = {1..n, 1..n} dmapped Block({1..n, 1..n});
var A, T : [Grid] real;//Zero initialized as default
We tell Chapel to use the same *block* distribution of the ``Grid`` and ``Interior`` domain such that each index in ``Grid`` has the same location as the corresponding index in ``Interior``. Because they use the same distribution, no communication is needed when accessing the same index. For example, the operations ``A[2,4] + T[2,4]`` can be done locally on the machine that *owns* index ``[2,4]``. However, it also means that a operations such as ``A[2,4] + T[3,4]`` will generally require communication.
In relation to HPC, it is very importation use ``dmapped`` such that you minimize the communication requirements of your application.
......@@ -8,7 +8,7 @@ config var iterations = 1000;//Stop condition in number of iterations
const Grid = {0..n+1, 0..n+1} dmapped Block({1..n, 1..n});
//A n by n domain that represents the interior of 'Grid'
const Interior = {1..n, 1..n};
const Interior = {1..n, 1..n} dmapped Block({1..n, 1..n});
var A, T : [Grid] real;//Zero initialized as default
......
config const n = 8;//Size of the domain squired
config const epsilon = 1.0e-10;//Stop condition in amount of change
config var iterations = 1000;//Stop condition in number of iterations
//A n+2 by n+2 domain.
const Grid = {0..n+1, 0..n+1};
//A n by n domain that represents the interior of 'Grid'
const Interior = {1..n, 1..n};
var A, T : [Grid] real;//Zero initialized as default
A[..,0] = -273.15; //Left column
A[..,n+1] = -273.15; //Right column
A[n+1,..] = -273.15; //Bottom row
A[0,..] = 40.0; //Top row
do{
//Since all iterations are independent, we can use 'forall', which allows
//the Chapel runtime system to calculate the iterations in parallel
for (i,j) in Interior do//Iterate over all non-border cells
{
//Assign each cell in 'T' the mean of its neighboring cells in 'A'
T[i,j] = (A[i,j] + A[i-1,j] + A[i+1,j] + A[i,j-1] + A[i,j+1]) / 5;
}
//Delta is the total amount of change done in this iteration
const delta = + reduce abs(A[Interior] - T[Interior]);
//Copy back the non-border cells
A[Interior] = T[Interior];
//When 'delta' is smaller than 'epsilon' the calculation has converged
iterations -= 1;
} while (delta > epsilon && iterations > 0);
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