*pystencils* is a package that can speed up computations on *numpy* arrays. All computations are carried out fully parallel on CPUs (single node with OpenMP, multiple nodes with MPI) or on GPUs.
*pystencils* is a package that can speed up computations on *numpy* arrays. All computations are carried out fully parallel on CPUs (single node with OpenMP, multiple nodes with MPI) or on GPUs.
It is suited for applications that run the same operation on *numpy* arrays multiple times. It can be used to accelerate computations on images or voxel fields. Its main application, however, are numerical simulations using finite differences, finite volumes, or lattice Boltzmann methods.
It is suited for applications that run the same operation on *numpy* arrays multiple times. It can be used to accelerate computations on images or voxel fields. Its main application, however, are numerical simulations using finite differences, finite volumes, or lattice Boltzmann methods.
There already exist a variety of packages to speed up numeric Python code. One could use pure numpy or solutions that compile your code, like *Cython* and *numba*. See [this page](demo_benchmark.ipynb) for a comparison of these tools.
There already exist a variety of packages to speed up numeric Python code. One could use pure numpy or solutions that compile your code, like *Cython* and *numba*. See [this page](demo_benchmark.ipynb) for a comparison of these tools.
As the name suggests, *pystencils* was developed for **stencil codes**, i.e. operations that update array elements using only a local neighborhood.
As the name suggests, *pystencils* was developed for **stencil codes**, i.e. operations that update array elements using only a local neighborhood.
It generates C code, compiles it behind the scenes, and lets you call the compiled C function as if it was a native Python function.
It generates C code, compiles it behind the scenes, and lets you call the compiled C function as if it was a native Python function.
But lets not dive too deep into the concepts of *pystencils* here, they are covered in detail in the following tutorials. Let's instead look at a simple example, that computes the average neighbor values of a *numpy* array. Therefor we first create two rather large arrays for input and output:
But lets not dive too deep into the concepts of *pystencils* here, they are covered in detail in the following tutorials. Let's instead look at a simple example, that computes the average neighbor values of a *numpy* array. Therefor we first create two rather large arrays for input and output:
%% Cell type:code id: tags:
%% Cell type:code id: tags:
``` python
``` python
input_arr=np.random.rand(1024,1024)
input_arr=np.random.rand(1024,1024)
output_arr=np.zeros_like(input_arr)
output_arr=np.zeros_like(input_arr)
```
```
%% Cell type:markdown id: tags:
%% Cell type:markdown id: tags:
We first implement a version of this algorithm using pure numpy and benchmark it.
We first implement a version of this algorithm using pure numpy and benchmark it.
Here we first have created a symbolic notation of the stencil itself. This representation is built on top of *sympy* and is explained in detail in the next section.
Here we first have created a symbolic notation of the stencil itself. This representation is built on top of *sympy* and is explained in detail in the next section.
This description is then compiled and loaded as a Python function.
This description is then compiled and loaded as a Python function.
This whole process might seem overly complicated. We have already spent more lines of code than we needed for the *numpy* implementation and don't have anything running yet! However, this multi-stage process of formulating the algorithm symbolically, and just in the end actually running it, is what makes *pystencils* faster and more flexible than other approaches.
This whole process might seem overly complicated. We have already spent more lines of code than we needed for the *numpy* implementation and don't have anything running yet! However, this multi-stage process of formulating the algorithm symbolically, and just in the end actually running it, is what makes *pystencils* faster and more flexible than other approaches.
Now finally lets benchmark the *pystencils* kernel.
Now finally lets benchmark the *pystencils* kernel.
%% Cell type:code id: tags:
%% Cell type:code id: tags:
``` python
``` python
defpystencils_kernel():
defpystencils_kernel():
kernel(src=input_arr,dst=output_arr)
kernel(src=input_arr,dst=output_arr)
```
```
%% Cell type:code id: tags:
%% Cell type:code id: tags:
``` python
``` python
%%timeit
%%timeit
pystencils_kernel()
pystencils_kernel()
```
```
%% Output
%% Output
643 µs ± 8.66 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)
643 µs ± 8.66 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)
%% Cell type:markdown id: tags:
%% Cell type:markdown id: tags:
This benchmark shows that *pystencils* is a lot faster than pure *numpy*, especially for large arrays.
This benchmark shows that *pystencils* is a lot faster than pure *numpy*, especially for large arrays.
If you are interested in performance details and comparison to other packages like Cython, have a look at [this page](demo_benchmark.ipynb).
If you are interested in performance details and comparison to other packages like Cython, have a look at [this page](demo_benchmark.ipynb).
%% Cell type:markdown id: tags:
%% Cell type:markdown id: tags:
## Short *sympy* introduction
## Short *sympy* introduction
In this tutorial we continue with a short *sympy* introduction, since the symbolic kernel definition is built on top of this package. If you already know *sympy* you can skip this section.
In this tutorial we continue with a short *sympy* introduction, since the symbolic kernel definition is built on top of this package. If you already know *sympy* you can skip this section.
You can also read the full [sympy documentation here](http://docs.sympy.org/latest/index.html).
You can also read the full [sympy documentation here](http://docs.sympy.org/latest/index.html).
%% Cell type:code id: tags:
%% Cell type:code id: tags:
``` python
``` python
importsympyassp
importsympyassp
sp.init_printing()# enable nice LaTeX output
sp.init_printing()# enable nice LaTeX output
```
```
%% Cell type:markdown id: tags:
%% Cell type:markdown id: tags:
*sympy* is a package for symbolic calculation. So first we need some symbols:
*sympy* is a package for symbolic calculation. So first we need some symbols:
%% Cell type:code id: tags:
%% Cell type:code id: tags:
``` python
``` python
x=sp.Symbol("x")
x=sp.Symbol("x")
y=sp.Symbol("y")
y=sp.Symbol("y")
type(x)
type(x)
```
```
%% Output
%% Output
sympy.core.symbol.Symbol
sympy.core.symbol.Symbol
%% Cell type:markdown id: tags:
%% Cell type:markdown id: tags:
The usual mathematical operations are defined for symbols:
The usual mathematical operations are defined for symbols:
%% Cell type:code id: tags:
%% Cell type:code id: tags:
``` python
``` python
expr=x**2*(y+x+5)+x**2
expr=x**2*(y+x+5)+x**2
expr
expr
```
```
%% Output
%% Output
$\displaystyle x^{2} \left(x + y + 5\right) + x^{2}$
$\displaystyle x^{2} \left(x + y + 5\right) + x^{2}$
2 2
2 2
x ⋅(x + y + 5) + x
x ⋅(x + y + 5) + x
%% Cell type:markdown id: tags:
%% Cell type:markdown id: tags:
Now we can do all sorts of operations on these expressions: expand them, factor them, substitute variables:
Now we can do all sorts of operations on these expressions: expand them, factor them, substitute variables:
$\displaystyle \left[ - x - 6 + \frac{1}{x^{2}}\right]$
$\displaystyle \left[ - x - 6 + \frac{1}{x^{2}}\right]$
⎡ 1 ⎤
⎡ 1 ⎤
⎢-x - 6 + ──⎥
⎢-x - 6 + ──⎥
⎢ 2⎥
⎢ 2⎥
⎣ x ⎦
⎣ x ⎦
%% Cell type:markdown id: tags:
%% Cell type:markdown id: tags:
A *sympy* expression is represented by an abstract syntax tree (AST), which can be inspected and modified.
A *sympy* expression is represented by an abstract syntax tree (AST), which can be inspected and modified.
%% Cell type:code id: tags:
%% Cell type:code id: tags:
``` python
``` python
expr
expr
```
```
%% Output
%% Output
$\displaystyle x^{2} \left(x + y + 5\right) + x^{2}$
$\displaystyle x^{2} \left(x + y + 5\right) + x^{2}$
2 2
2 2
x ⋅(x + y + 5) + x
x ⋅(x + y + 5) + x
%% Cell type:code id: tags:
%% Cell type:code id: tags:
``` python
``` python
ps.to_dot(expr,graph_style={'size':"9.5,12.5"})
ps.to_dot(expr,graph_style={'size':"9.5,12.5"})
```
```
%% Output
%% Output
<graphviz.files.Source at 0x7ff8a018e7f0>
<graphviz.files.Source at 0x7ff8a018e7f0>
%% Cell type:markdown id: tags:
%% Cell type:markdown id: tags:
Programatically the children node type is acessible as ``expr.func`` and its children as ``expr.args``.
Programatically the children node type is acessible as ``expr.func`` and its children as ``expr.args``.
With these members a tree can be traversed and modified.
With these members a tree can be traversed and modified.
%% Cell type:code id: tags:
%% Cell type:code id: tags:
``` python
``` python
expr.func
expr.func
```
```
%% Output
%% Output
sympy.core.add.Add
sympy.core.add.Add
%% Cell type:code id: tags:
%% Cell type:code id: tags:
``` python
``` python
expr.args
expr.args
```
```
%% Output
%% Output
$\displaystyle \left( x^{2}, \ x^{2} \left(x + y + 5\right)\right)$
$\displaystyle \left( x^{2}, \ x^{2} \left(x + y + 5\right)\right)$
⎛ 2 2 ⎞
⎛ 2 2 ⎞
⎝x , x ⋅(x + y + 5)⎠
⎝x , x ⋅(x + y + 5)⎠
%% Cell type:markdown id: tags:
%% Cell type:markdown id: tags:
## Using *pystencils*
## Using *pystencils*
### Fields
### Fields
*pystencils* is a module to generate code for stencil operations.
*pystencils* is a module to generate code for stencil operations.
One has to specify an update rule for each element of an array, with optional dependencies to neighbors.
One has to specify an update rule for each element of an array, with optional dependencies to neighbors.
This is done use pure *sympy* with one addition: **Fields**.
This is done use pure *sympy* with one addition: **Fields**.
Fields represent a multidimensional array, where some dimensions are considered *spatial*, and some as *index* dimensions. Spatial coordinates are given relative (i.e. one can specify "the current cell" and "the left neighbor") whereas index dimensions are used to index multiple values per cell.
Fields represent a multidimensional array, where some dimensions are considered *spatial*, and some as *index* dimensions. Spatial coordinates are given relative (i.e. one can specify "the current cell" and "the left neighbor") whereas index dimensions are used to index multiple values per cell.
%% Cell type:code id: tags:
%% Cell type:code id: tags:
``` python
``` python
my_field=ps.fields("f(3) : double[2D]")
my_field=ps.fields("f(3) : double[2D]")
```
```
%% Cell type:markdown id: tags:
%% Cell type:markdown id: tags:
Neighbors are labeled according to points on a compass where the first coordinate is west/east, second coordinate north/south and third coordinate top/bottom.
Neighbors are labeled according to points on a compass where the first coordinate is west/east, second coordinate north/south and third coordinate top/bottom.
%% Cell type:code id: tags:
%% Cell type:code id: tags:
``` python
``` python
field_access=my_field[1,0](1)
field_access=my_field[1,0](1)
field_access
field_access
```
```
%% Output
%% Output
$\displaystyle {{f}_{(1,0)}^{1}}$
$\displaystyle {{f}_{(1,0)}^{1}}$
f_E__1
f_E__1
%% Cell type:markdown id: tags:
%% Cell type:markdown id: tags:
The result of indexing a field is an instance of ``Field.Access``. This class is a subclass of a *sympy* Symbol and thus can be used whereever normal symbols can be used. It is just like a normal symbol with some additional information attached to it.
The result of indexing a field is an instance of ``Field.Access``. This class is a subclass of a *sympy* Symbol and thus can be used whereever normal symbols can be used. It is just like a normal symbol with some additional information attached to it.
%% Cell type:code id: tags:
%% Cell type:code id: tags:
``` python
``` python
isinstance(field_access,sp.Symbol)
isinstance(field_access,sp.Symbol)
```
```
%% Output
%% Output
True
True
%% Cell type:markdown id: tags:
%% Cell type:markdown id: tags:
### Building our first stencil kernel
### Building our first stencil kernel
Lets start by building a simple filter kernel. We create a field representing an image, then define a edge detection filter on the third pixel component which is blue for an RGB image.
Lets start by building a simple filter kernel. We create a field representing an image, then define a edge detection filter on the third pixel component which is blue for an RGB image.
We have mixed some standard *sympy* symbols into this expression to possibly give the different directions different weights. The complete expression is still a valid *sympy* expression, so all features of *sympy* work on it. Lets for example now fix one weight by substituting it with a constant.
We have mixed some standard *sympy* symbols into this expression to possibly give the different directions different weights. The complete expression is still a valid *sympy* expression, so all features of *sympy* work on it. Lets for example now fix one weight by substituting it with a constant.
Now lets built an executable kernel out of it, which writes the result to a second field. Assignments are created using *pystencils*`Assignment` class, that gets the left- and right hand side of the assignment.
Now lets built an executable kernel out of it, which writes the result to a second field. Assignments are created using *pystencils*`Assignment` class, that gets the left- and right hand side of the assignment.
On our way we have created an ``ast``-object. We can inspect this, to see what *pystencils* actually does.
On our way we have created an ``ast``-object. We can inspect this, to see what *pystencils* actually does.
%% Cell type:code id: tags:
%% Cell type:code id: tags:
``` python
``` python
ps.to_dot(ast,graph_style={'size':"9.5,12.5"})
ps.to_dot(ast,graph_style={'size':"9.5,12.5"})
```
```
%% Output
%% Output
<graphviz.files.Source at 0x7ff84a432e10>
<graphviz.files.Source at 0x7ff84a432e10>
%% Cell type:markdown id: tags:
%% Cell type:markdown id: tags:
*pystencils* also builds a tree structure of the program, where each `Assignment` node internally again has a *sympy* AST which is not printed here. Out of this representation *C* code can be generated:
*pystencils* also builds a tree structure of the program, where each `Assignment` node internally again has a *sympy* AST which is not printed here. Out of this representation *C* code can be generated:
Behind the scenes this code is compiled into a shared library and made available as a Python function. Before compiling this function we can modify the AST object, for example to parallelize it with OpenMP.
Behind the scenes this code is compiled into a shared library and made available as a Python function. Before compiling this function we can modify the AST object, for example to parallelize it with OpenMP.
Compare this code to the version above. In this code the loop bounds and array offsets are constants, which usually leads to faster kernels.
Compare this code to the version above. In this code the loop bounds and array offsets are constants, which usually leads to faster kernels.
%% Cell type:markdown id: tags:
%% Cell type:markdown id: tags:
### Running on GPU
### Running on GPU
If you have a CUDA enabled graphics card and [pycuda](https://mathema.tician.de/software/pycuda/) installed, *pystencils* can run your kernel on the GPU as well. You can find more details about this in the GPU tutorial.
If you have a CUDA enabled graphics card and [pycuda](https://mathema.tician.de/software/pycuda/) installed, *pystencils* can run your kernel on the GPU as well. You can find more details about this in the GPU tutorial.
ps.stencil.plot(stencil_7pt, data=[i for i in range(7)])
ps.stencil.plot(stencil_7pt, data=[i for i in range(7)])
```
```
%% Output
%% Output
%% Cell type:markdown id: tags:
%% Cell type:markdown id: tags:
This kind of plot works well for small stencils, for larger stencils this gets messy:
This kind of plot works well for small stencils, for larger stencils this gets messy:
%% Cell type:code id: tags:
%% Cell type:code id: tags:
``` python
``` python
stencil_27pt = [(i, j, k) for i in (-1, 0, 1) for j in (-1, 0, 1) for k in (-1, 0, 1)]
stencil_27pt = [(i, j, k) for i in (-1, 0, 1) for j in (-1, 0, 1) for k in (-1, 0, 1)]
ps.stencil.plot(stencil_27pt, data=[i for i in range(27)])
ps.stencil.plot(stencil_27pt, data=[i for i in range(27)])
```
```
%% Output
%% Output
%% Cell type:markdown id: tags:
%% Cell type:markdown id: tags:
Adding `slice=True` shows the three projections of the stencils instead
Adding `slice=True` shows the three projections of the stencils instead
%% Cell type:code id: tags:
%% Cell type:code id: tags:
``` python
``` python
ps.stencil.plot(stencil_27pt, data=[i for i in range(27)], slice=True)
ps.stencil.plot(stencil_27pt, data=[i for i in range(27)], slice=True)
```
```
%% Output
%% Output
%% Cell type:markdown id: tags:
%% Cell type:markdown id: tags:
## 2) Scalar fields
## 2) Scalar fields
*pystencils* also comes with helper functions to plot simulation data.
*pystencils* also comes with helper functions to plot simulation data.
The plotting functions are simple but useful wrappers around basic Matplotlib functionality.
The plotting functions are simple but useful wrappers around basic Matplotlib functionality.
*pystencils* imports all matplotlib functions as well, such that one can use
*pystencils* imports all matplotlib functions as well, such that one can use
```import pystencils.plot as plt```
```import pystencils.plot as plt```
instead of
instead of
```import matplotlib.pyplot as plt```
```import matplotlib.pyplot as plt```
The session import at the top of the notebook does this already, and also sets up inline plotting for notebooks.
The session import at the top of the notebook does this already, and also sets up inline plotting for notebooks.
This section demonstrates how to plot 2D scalar fields. Internally `imshow` from matplotlib is used, however the coordinate system is changed such that (0,0) is in the lower left corner, the $x$-axis points to the right, and the $y$-axis upward.
This section demonstrates how to plot 2D scalar fields. Internally `imshow` from matplotlib is used, however the coordinate system is changed such that (0,0) is in the lower left corner, the $x$-axis points to the right, and the $y$-axis upward.
The next function returns a scalar field for demonstration
The next function returns a scalar field for demonstration
z = np.cos(x + 0.1 * t) * np.sin(y + 0.1 * t) + 0.1 * x * y
z = np.cos(x + 0.1 * t) * np.sin(y + 0.1 * t) + 0.1 * x * y
return z
return z
```
```
%% Cell type:markdown id: tags:
%% Cell type:markdown id: tags:
To show a single scalar field use `plt.scalar_field`:
To show a single scalar field use `plt.scalar_field`:
%% Cell type:code id: tags:
%% Cell type:code id: tags:
``` python
``` python
arr = example_scalar_field()
arr = example_scalar_field()
plt.scalar_field(arr)
plt.scalar_field(arr)
plt.colorbar();
plt.colorbar();
```
```
%% Output
%% Output
%% Cell type:markdown id: tags:
%% Cell type:markdown id: tags:
A similar wrapper around `counter` from matplotlib
A similar wrapper around `counter` from matplotlib
%% Cell type:code id: tags:
%% Cell type:code id: tags:
``` python
``` python
plt.scalar_field_contour(arr);
plt.scalar_field_contour(arr);
```
```
%% Output
%% Output
%% Cell type:markdown id: tags:
%% Cell type:markdown id: tags:
A 3D surface plot is also possible:
A 3D surface plot is also possible:
%% Cell type:code id: tags:
%% Cell type:code id: tags:
``` python
``` python
plt.scalar_field_surface(arr)
plt.scalar_field_surface(arr)
```
```
%% Output
%% Output
<mpl_toolkits.mplot3d.art3d.Poly3DCollection at 0x7f38908f9828>
<mpl_toolkits.mplot3d.art3d.Poly3DCollection at 0x7f38908f9828>
%% Cell type:markdown id: tags:
%% Cell type:markdown id: tags:
For simulation results one often needs visualization of time dependent results. To show an evolving scalar field an animation can be created as shown in the next cell
For simulation results one often needs visualization of time dependent results. To show an evolving scalar field an animation can be created as shown in the next cell
Another way to display animations is as an image sequence. While the `run_func` i.e. the simulation is running the current state is then displayed as HTML image. Use this method when your `run_func` takes a long time and you want to see the status while the simulation runs. The `frames` parameter specifies how often the run function will be called.
Another way to display animations is as an image sequence. While the `run_func` i.e. the simulation is running the current state is then displayed as HTML image. Use this method when your `run_func` takes a long time and you want to see the status while the simulation runs. The `frames` parameter specifies how often the run function will be called.
A third group of plotting functions helps to display arrays as they occur in phase-field simulations. However these function may also be useful for other kinds of simulations.
A third group of plotting functions helps to display arrays as they occur in phase-field simulations. However these function may also be useful for other kinds of simulations.
They expect arrays where the last coordinate indicates the fraction of a certain component, i.e. `arr[x, y, 2]` should be a value between $0$ and $1$ and specifies the fraction of the third phase at $(x, y)$. The plotting functions expect that sum over the last coordinate gives $1$ for all cells.
They expect arrays where the last coordinate indicates the fraction of a certain component, i.e. `arr[x, y, 2]` should be a value between $0$ and $1$ and specifies the fraction of the third phase at $(x, y)$. The plotting functions expect that sum over the last coordinate gives $1$ for all cells.
The `scalar_field_alpha_value` function uses the last entry, i.e. the value between 0 and 1 as alpha value of the specified color to show where the phase is located. This visualization makes it easy to distinguish between smooth and sharp transitions.
The `scalar_field_alpha_value` function uses the last entry, i.e. the value between 0 and 1 as alpha value of the specified color to show where the phase is located. This visualization makes it easy to distinguish between smooth and sharp transitions.
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