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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.

Now, let's run it:

```
  ./bin/heat_equation -nl 1 --size=5000*10
  Heat Equation (sequential) - n: 5000, iterations: 10, elapsed-time: 381.5 seconds
```

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;
  }
```

Now, let's run it (note that `CHPL_RT_NUM_THREADS_PER_LOCALE` tells Chapel the number of threads to use)::

```
  export CHPL_RT_NUM_THREADS_PER_LOCALE=16
  ./bin/heat_equation -nl 1 --size=5000*10
  Heat Equation (single machine) - n: 5000, iterations: 10, elapsed-time: 25.7052 seconds
```

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.

Now, let's run it (note that `-nl 8` tells Chapel to use eight locations):

```
  export CHPL_RT_NUM_THREADS_PER_LOCALE=16
  ./bin/heat_equation -nl 8 --size=5000*10
  Heat Equation (multiple machines) - n: 5000, iterations: 10, elapsed-time: 5.13 seconds
```

It is very importation that all arrays in the calculation uses similar `dmapped` layouts. For example, if we do not use `dmapped` when defines `Interior` we get horrible performance:

```
  export CHPL_RT_NUM_THREADS_PER_LOCALE=16
  ./bin/heat_equation -nl 8 --size=5000*10
  Heat Equation (multiple machines) - n: 5000, iterations: 10, elapsed-time: 1823.23 seconds
```