pygeoinf 1.6.4

A package for solving geophysical inference and inverse problems

pygeoinf: A Python Library for Geophysical Inference

pygeoinf is a Python library for solving geophysical inference and inverse problems in a coordinate-free, abstract framework. It leverages the mathematics of Hilbert spaces to provide a robust and flexible foundation for Bayesian and optimisation-based inference.

Overview

The core philosophy of pygeoinf is to separate the abstract mathematical structure of an inverse problem from its concrete numerical implementation. Instead of manipulating NumPy arrays directly, you work with high-level objects like HilbertSpace, LinearOperator, and GaussianMeasure. This allows you to write code that is more readable, less error-prone, and closer to the underlying mathematics.

The library is built on a few key concepts:

• HilbertSpace: The foundational class. It represents a vector space with an inner product, but it abstracts away the specific representation of vectors (e.g., NumPy arrays, pyshtools grids).
• LinearOperator: Represents linear mappings between Hilbert spaces. These are the workhorses of the library, supporting composition, adjoints, and matrix representations.
• GaussianMeasure: Generalizes the multivariate normal distribution to abstract Hilbert spaces, providing a way to define priors and noise models.
• ForwardProblem: Encapsulates the mathematical model d = A(u) + e, linking the unknown model u to the observed data d.
• Inversion Classes: High-level classes like LinearBayesianInversion and LinearLeastSquaresInversion provide ready-to-use algorithms for solving the inverse problem.

Key Features

• Abstract Coordinate-Free Formulation: Write elegant code that mirrors the mathematics of inverse problems.
• Bayesian Inference: Solve inverse problems in a probabilistic framework to obtain posterior distributions over models.
• Optimisation Methods: Includes Tikhonov-regularized least-squares and minimum-norm solutions.
• Extensive Solver Suite: Choose from direct factorisations or matrix-free iterative solvers (e.g., CG, MINRES, LSQR, BiCGStab, FCG) designed for abstract vectors.
• Coupled Systems: Effortlessly build block matrices and direct sum spaces using HilbertSpaceDirectSum for joint inversions.
• Geometric Constraints: Constrain your inversions using affine subspaces, bounds, and convex sublevel sets.
• Probabilistic Modelling: Define priors and noise models using GaussianMeasure objects on abstract spaces.
• Randomized Algorithms: Utilizes randomized SVD and Cholesky decompositions for efficient low-rank approximations of large operators.
• Specialized Operator Variants: Efficient implementations for SparseMatrixLinearOperator and DiagonalSparseMatrixLinearOperator
• Application-Specific Spaces: Provides concrete HilbertSpace implementations for functions on a line, circle, and the two-sphere.
• High-Quality Visualisation: Built-in plotting methods for functions on symmetric spaces, now featuring multi-dimensional corner plots for joint posterior distributions.

Advanced Features

• Block Operators: Construct complex operators from smaller components using BlockLinearOperator, ColumnLinearOperator, and RowLinearOperator. This is ideal for coupled inverse problems.
• Parallelisation: Many expensive operations are parallelized with joblib, including dense matrix construction and randomized algorithms.

Installation

The package can be installed directly using pip. By default, this will perform a minimal installation.

    # Minimal installation
    pip install pygeoinf

To include the functionality for functions on the sphere, you can install the sphere extra. This provides support for pyshtools and Cartopy.

    # Installation with sphere-related features
    pip install pygeoinf[sphere]

For development, you can clone the repository and install using Poetry:

    git clone https://github.com/da380/pygeoinf.git
    cd pygeoinf
    poetry install

You can install all optional dependencies for development—including tools for running the test suite, building the documentation, and running the Jupyter tutorials—by using the --with flag and specifying the dev group.

    # Install all development dependencies (for tests, docs, and tutorials)
    poetry install --with dev

Documentation

The full documentation for the library, including the API reference and tutorials, is available at pygeoinf.readthedocs.io.

Tutorials

You can run the interactive tutorials directly in Google Colab to get started with the core concepts of the library.

Tutorial Name	Link to Colab
Tutorial 1 - A first example
Tutorial 2 - Hilbert spaces
Tutorial 3 - Dual spaces
Tutorial 4 - Linear operators
Tutorial 5 - Linear solvers
Tutorial 6 - Gaussian measures
Tutorial 7 - Minimum norm inversions
Tutorial 8 - Bayesian inversions
Tutorial 9 - Direct sums
Tutorial 10 - Symmetric spaces

Dependencies

• Python >=3.12
• NumPy >=1.26.0
• SciPy >=1.16.1
• Matplotlib >=3.0.0
• PyQt6 >=6.0.0
• joblib ^1.5.2

Optional (sphere group)

• pyshtools >=4.0.0
• Cartopy ^0.23.0
• threadpoolctl ^3.6.0
• numba ^0.63.1
• ipympl ^0.10.0

Recent Updates

pygeoinf is under active development. Recent updates have expanded its capabilities to handle massive, constrained, and coupled geophysical datasets:

• Advanced Linear Solvers & Preconditioning: The library now includes a robust suite of matrix-free iterative solvers, including MINRES, LSQR, BiCGStab, and Flexible Conjugate Gradient (FCG). These are backed by a flexible preconditioning framework supporting Jacobi and spectral (low-rank) methods.
• Joint Inversions & Block Operators: You can now construct complex, coupled inverse problems using HilbertSpaceDirectSum, BlockLinearOperator, and ColumnLinearOperator. This is specifically designed for joint inversions where a single model is constrained by disparate data types.
• Geometric Subspaces and Constraints: We have introduced a Constructive Solid Geometry (CSG) approach to abstract Hilbert spaces. Users can now seamlessly define affine subspaces, hyper-ellipsoids, and convex subsets to enforce strict prior constraints or project unconstrained posteriors.
• Advanced Posterior Visualisation: The plotting module has been expanded to include professional-grade corner plots (plot_corner_distributions) for visualising joint posterior distributions, as well as dual-axis 1D marginal distributions.

Future Plans

Future development will focus on the following areas:

• Non-linear Bayesian Inference: We plan to develop methods for non-linear Bayesian problems, including techniques for linearizing the problem around the maximum a posteriori (MAP) solution to estimate posterior uncertainty. This will also involve constructing efficient proposal distributions for Markov chain Monte Carlo (MCMC) sampling methods.
• New Geophysical Hilbert Spaces: We will be adding more HilbertSpace implementations for specific geophysical applications. A key focus will be on creating spaces for functions defined within a spherical annulus (spherical shell), which is crucial for problems in global seismology and mantle tomography.

Contributing

Contributions are welcome! If you would like to contribute, please feel free to fork the repository, make your changes, and submit a pull request. For major changes, please open an issue first to discuss what you would like to change.

License

This project is licensed under the BSD-3-Clause License - see the LICENSE file for details.