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Python library implementing linear operators to allow solving large-scale optimization problems without requiring to explicitly create a dense (or sparse) matrix.

Project description


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This Python library is inspired by the MATLAB Spot – A Linear-Operator Toolbox project.

Linear operators and inverse problems are at the core of many of the most used algorithms in signal processing, image processing, and remote sensing. When dealing with small-scale problems, the Python numerical scientific libraries numpy and scipy allow to perform many of the underlying matrix operations (e.g., computation of matrix-vector products and manipulation of matrices) in a simple and compact way.

Many useful operators, however, do not lend themselves to an explicit matrix representation when used to solve large-scale problems. PyLops operators, on the other hand, still represent a matrix and can be treated in a similar way, but do not rely on the explicit creation of a dense (or sparse) matrix itself. Conversely, the forward and adjoint operators are represented by small pieces of codes that mimic the effect of the matrix on a vector or another matrix.

Luckily, many iterative methods (e.g. cg, lsqr) do not need to know the individual entries of a matrix to solve a linear system. Such solvers only require the computation of forward and adjoint matrix-vector products as done for any of the PyLops operators.

Here is simple example showing how a dense first-order first derivative operator can be created, applied and inverted using numpy/scipy commands:

import numpy as np
from scipy.linalg import lstsq

nx = 7
x = np.arange(nx) - (nx-1)/2

D = np.diag(0.5*np.ones(nx-1),k=1) - np.diag(0.5*np.ones(nx-1),-1)
D[0] = D[-1] = 0 # take away edge effects

# y = Dx
y =,x)
# x = D'y
xadj =,y)
# xinv = D^-1 y
xinv = lstsq(D, y)[0]

and similarly using PyLops commands:

from pylops import FirstDerivative

Dlop = FirstDerivative(nx, dtype='float64')

# y = Dx
y = Dlop*x
# x = D'y
xadj = Dlop.H*y
# xinv = D^-1 y
xinv = Dlop / y

Note how this second approach does not require creating a dense matrix, reducing both the memory load and the computational cost of applying a derivative to an input vector x. Moreover, the code becomes even more compact and espressive than in the previous case letting the user focus on the formulation of equations of the forward problem to be solved by inversion.

Project structure

This repository is organized as follows:

  • pylops: python library containing various linear operators and auxiliary routines
  • pytests: set of pytests
  • testdata: sample datasets used in pytests and documentation
  • docs: sphinx documentation
  • examples: set of python script examples for each linear operator to be embedded in documentation using sphinx-gallery
  • tutorials: set of python script tutorials to be embedded in documentation using sphinx-gallery

Getting started

You need Python 3.6.4 or greater.

If you want to simply use PyLops within your Python codes, type the following command in your terminal:

pip install pylops

Open a python terminal and type:

import pylops

If you do not see any error, you should be good to go, enjoy!


Feel like contributing to the project? Adding new operators or tutorial?

We advise using the Anaconda Python distribution to ensure that all the dependencies are installed via the Conda package manager. Follow the following instructions and read carefully the CONTRIBUTING file before getting started.

1. Fork and clone the repository

Execute the following command in your terminal:

git clone

2. Install PyLops in a new Conda environment

To ensure that further development of PyLops is performed within the same environment (i.e., same dependencies) as that defined by requirements-dev.txt or environment-dev.yml files, we suggest to work off a new Conda enviroment.

The first time you clone the repository run the following command:

make dev-install_conda

To ensure that everything has been setup correctly, run tests:

make tests

Make sure no tests fail, this guarantees that the installation has been successfull.

Remember to always activate the conda environment every time you open a new terminal by typing:

source activate pylops


The official documentation of PyLops is available here.

Visit this page to get started learning about different operators and their applications as well as how to create new operators yourself and make it to the Contributors list.

Moreover, if you have installed PyLops using the developer environment you can also build the documentation locally by typing the following command:

make doc

Once the documentation is created, you can make any change to the source code and rebuild the documentation by simply typing

make docupdate

Note that if a new example or tutorial is created (and if any change is made to a previously available example or tutorial) you are required to rebuild the entire documentation before your changes will be visible.


PyLops was initially written and it is currently maintained by Equinor. It is a flexible and scalable python library for large-scale optimization with linear operators that can be tailored to our needs, and as contribution to the free software community.


  • Matteo Ravasi, mrava87
  • Carlos da Costa, cako

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