Operators and solvers for high-performance computing.
Project description
PyOperators
The PyOperators package defines operators and solvers for high-performance computing. These operators are multi-dimensional functions with optimised and controlled memory management. If linear, they behave like matrices with a sparse storage footprint.
Documentaion
https://pchanial.github.io/pyoperators
Installation
pip install pyoperators[fft,wavelets]
On some platforms, it might be more convenient to install pyfftw through Conda beforehand to use the FFTOperator:
conda install pyfftw
For MPI communication, an MPI library needs to be installed, for example on Ubuntu:
sudo apt install libopenmpi-dev
pip install pyoperators[fft,wavelets,mpi]
Getting started
To define an operator, one needs to define a direct function which will replace the usual matrix-vector operation:
>>> def f(x, out):
... out[...] = 2 * x
Then, you can instantiate an Operator:
>>> A = pyoperators.Operator(direct=f, flags='symmetric')
An alternative way to define an operator is to define a subclass:
>>> from pyoperators import flags, Operator
... @flags.symmetric
... class MyOperator(Operator):
... def direct(x, out):
... out[...] = 2 * x
...
... A = MyOperator()
This operator does not have an explicit shape, it can handle inputs of any shape:
>>> A(np.ones(5))
array([ 2., 2., 2., 2., 2.])
>>> A(np.ones((2,3)))
array([[ 2., 2., 2.],
[ 2., 2., 2.]])
By setting the symmetric flag, we ensure that A's transpose is A:
>>> A.T is A
True
For non-explicit shape operators, we get the corresponding dense matrix by specifying the input shape:
>>> A.todense(shapein=2)
array([[2, 0],
[0, 2]])
Operators do not have to be linear. Many operators are already predefined, such as the DiagonalOperator, the FFTOperator or the nonlinear ClipOperator.
The previous A matrix could be defined more easily like this:
>>> from pyoperators import I
>>> A = 2 * I
where I is the identity operator with no explicit shape.
Operators can be combined together by addition, element-wise multiplication or composition. Note that the operator * stands for matrix multiplication if the two operators are linear, or for element-wise multiplication otherwise:
>>> from pyoperators import I, DiagonalOperator
>>> B = 2 * I + DiagonalOperator(range(3))
>>> B.todense()
array([[2, 0, 0],
[0, 3, 0],
[0, 0, 4]])
Algebraic rules can easily be attached to operators. They are used to simplify expressions to speed up their execution. The B Operator has been reduced to:
>>> B
DiagonalOperator(array([2, ..., 4], dtype=int64), broadcast='disabled', dtype=int64, shapein=3, shapeout=3)
Many simplifications are available. For instance:
>>> from pyoperators import Operator
>>> C = Operator(flags='idempotent,linear')
>>> C * C is C
True
>>> D = Operator(flags='involutary')
>>> D(D)
IdentityOperator()
Requirements
- python 3.8
Optional requirements:
- PyWavelets: wavelet transforms
- pyfftw: Fast Fourier transforms
- mpi4py: For MPI communication
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