InverseProblems

InverseProblems is a

• educational Inverse Problem or Numerical Method library. The goal is to provide students with a light-weighted code to explore these areas and interactive lectures with amazing Jupyter Notebook.
• benchmark repository originally designed to test unscented Kalman inversion and other derivative-free inverse methods. The goal is to provide reseachers with access to various inverse problems, while enabling researchers to quickly and easily develop and test novel inverse methods.

Code Struction

• All the inverse methods are in Inversion folder
• Each other folder contains one category of inverse problems

Tutorial

Let's start! (⚠️ under construction)

• Overview
  • What are inverse problems, why are they important?
  • Bayesian inversion, Bayesian inference, and Bayesian calibration
  • Probability density function space
• Probabilistic approaches
  • Invariant and ergodic measures
    • Langevin dynamics
    • Markov Chain Monte Carlo methods
    • Interacting particle methods
  • Variational inference
    • Gaussian variational inference
    • Stein variational inference
    • Wasserstein gradient flow
    • Affine invariant gradient flows
    • Derivative Free Gaussian Mixture Variational Inference
  • Coupling ideas
    • Filtering
      • Kalman filters
    • Inversion
      • Sequential Monte Carlo method
      • Kalman inversion part I : stochastic dynamical system
      • Kalman inversion part II : implementation
    • Transport map
  • When is posterior distribution close to Gaussian
  • All models are wrong
• Examples

  • Linear inverse problems

    • Well-determined, under-determined, and over-determined inverse problems
    • Ill-conditioned matrix: inverse of Hilbert matrix
    • High-dimensional inverse problem: 1-D elliptic equation

  • Chaotic systems

    • Chaos and butterfly effects
    • Lorenz63 model
    • Lorenz96 model
    • Kuramoto-Sivashinksy equation model

  • Structure mechanics problems

    • Damage detection of a "bridge"
    • Consitutive modeling of a multiscale fiber-reinforced plate

  • Fluid mechanics problems

    • 2D Darcy flow
    • Navier-Stokes initial condition recovery

  • Fluid structure interaction problems

    • Piston problem
      • Receding piston (analytical solution)
      • Piston system calibration
    • Airfoil damage detection during transonic buffeting

  • Climate modeling

    • Barotropic climate model
    • Idealized general circulation model (Held-Suarez benchmark)

  • Other posterior distribution estimations

    • Some nonlinear maps
    • 2-parameter elliptic equation
    • 1D Darcy flow

Submit an issue

You are welcome to submit an issue for any questions related to InverseProblems.

Here are some research papers using InverseProblem

1. Daniel Zhengyu Huang, Tapio Schneider, and Andrew M. Stuart. "Iterated Kalman Methodology For Inverse Problems / Unscented Kalman Inversion."

2. Daniel Zhengyu Huang, Jiaoyang Huang, Sebastian Reich, and Andrew M. Stuart. "Efficient Derivative-free Bayesian Inference for Large-Scale Inverse Problems."

3. Shunxiang Cao, Daniel Zhengyu Huang. "Bayesian Calibration for Large-Scale Fluid Structure Interaction Problems Under Embedded/Immersed Boundary Framework."

4. Yifan Chen, Daniel Zhengyu Huang, Jiaoyang Huang, Sebastian Reich, and Andrew M. Stuart. "Gradient Flows for Sampling: Mean-Field Models, Gaussian Approximations and Affine Invariance."

5. Daniel Zhengyu Huang, Jiaoyang Huang, and Zhengjiang Lin. "Convergence Analysis of Probability Flow ODE for Score-based Generative Models."

6. Yifan Chen, Daniel Zhengyu Huang, Jiaoyang Huang, Sebastian Reich, and Andrew M. Stuart. "Efficient, Multimodal, and Derivative-Free Bayesian Inference With Fisher-Rao Gradient Flows."