Uncertainty Quantification in Inverse Problems with CUQIpy

Welcome to the book on “Uncertainty Quantification in Inverse Problems with CUQIpy”. This book contains training material on how to use the CUQIpy library for uncertainty quantification in inverse problems. It also covers some of the theoretical background behind the methods implemented in CUQIpy.

⚠️ Disclaimer: The code in this book runs without a fixed random seed, so results will vary with each re-compilation and user run. While conclusions should remain consistent, specific values may differ due to the probabilistic nature of Bayesian sampling, which is fundamental when working with uncertainty quantification.

Table of contents¶

• Uncertainty Quantification in Inverse Problems with CUQIpy
• Part I: Foundations of CUQIpy and Bayesian Inverse Problems
  • Chapter 1: Introduction to uncertainty, priors, and Bayesian inverse problems
    • 1. Errors and uncertainties and why they matter
    • 2. Inverse problems and ill-posed problems
    • 3. Prior information and Bayesian inverse problems
    • 4. Gaussians priors
    • 5. Computational UQ: the need for sampling
  • Chapter 2: Introduction to CUQIpy
    • 1. Overview of CUQIpy
    • 2. Running the book code examples
  • Chapter 3: Introduction to Bayesian Inverse Problems (BIPs) in CUQIpy
    • 1. Probably the simplest BIP in the world (the short story)
    • 2. Probably the simplest BIP in the world (the long story)
  • Chapter 4: Distributions, random variables, and priors in CUQIpy
    • 1. Introduction to distributions and basic sampling in CUQIpy
    • ⚠️ 2. Distribution gallery
    • ⚠️ 3. Random fields in CUQIpy
    • ⚠️ 4. Random variables and hierarchical priors
    • ⚠️ 5. Implicit priors
    • 6. Prior entropy
  • Chapter 5: Forward models, data generation, and forward UQ
    • 1. Forward models in CUQIpy and data generation
    • 2. Forward UQ
    • 3. More forward model examples
  • Chapter 6: Solving BIPs in CUQIpy
    • 1. Bayesian Inverse Problems in CUQIpy
    • 2. More BIPs examples
  • Chapter 7: More on CUQIpy technical details
    • 1. Geometry objects: Representation, parametrization, and mapping
• Part II: Inference and Sampling Methods
  • Chapter 8: Background on sampling methods
    • 1. Sampling with CUQIpy: five little stories
  • Chapter 9: Hybrid Gibbs sampling in CUQIpy
    • 1. Gibbs sampling for hierarchical Bayesian models
    • Sampling BIP with multiple-input forward models
  • Chapter 10: Sampling with CUQIpy-PyTorch
    • 1. Hamiltonian Monte Carlo with CUQIpy-PyTorch
  • ⚠️ Chapter 11: BIPs with implicit prior
    • 1. Regularized linear RTO
    • 2. 1D Poisson problem with TV denoising using MYULA
• Part III: Applications and Research Using CUQIpy
  • Chapter 12: X-ray Computed Tomography (CT)
    • 1. X-ray CT using CUQIpy and CUQIpy-CIL plugin
  • Chapter 13: PDE-based BIP
    • 1. Solving PDE-based BIP using core CUQIpy
    • 2. PDE-based BIP using CUQIpy and CUQIpy-FEniCS plugin
  • Chapter 14: More applications and benchmarks using CUQIpy
    • 1. User showcase collection
    • 2. CUQIpy benchmarks
  • Chapter 15: Research based on CUQIpy
    • 1. CUQIpy – I. Computational uncertainty quantification for inverse problems in Python
    • 2. CUQIpy – II. Computational uncertainty quantification for PDE-based inverse problems in Python
    • 3. A Computational Framework and Implementation of Implicit Priors in Bayesian Inverse Problems
    • 4. Integration of CUQIpy and UM-Bridge
    • Prior modeling and uncertainty quantification in X-ray computed tomography with application to defect detection in subsea pipes
    • 6. Cochlear aqueduct advection and diffusion inferred from computed tomography imaging with a Bayesian approach
    • 7. Efficient monotonic Gaussian processes via Randomize-then-Optimize
  • Chapter 16: Resources and bibliography

Authors¶

• Amal Mohammed A Alghamdi

• Nicolai André Brogaard Riis

• Jakob Sauer Jørgensen

• Chao Zhang

• Felipe Uribe

• Jasper M. Everink

• Silja L. Christensen