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)

• What are inverse problems, why are they important?

• Inverse methods

  • Bayesian inversion, Bayesian inference, and Bayesian calibration
  • Markov Chain Monte Carlo method
  • Sequential Monte Carlo method
  • Kalman filters and Gaussian approximation algorithm
  • Unscented Kalman inversion and its variants
  • Ensemble Kalman inversion and its variants
  • When is posterior distribution close to Gaussian
  • All models are wrong

• 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
  • High-dimensional inverse problem: Bernoulli random vector

• Posterior distribution estimation

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

• Chaotic system

  • Chaos and butterfly effects
  • Lorenz63 model
  • Lorenz96 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)

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 Z. Huang, Tapio Schneider, and Andrew M. Stuart. "Unscented Kalman Inversion."

2. Daniel Z. Huang, Jiaoyang Huang. "Improve Unscented Kalman Inversion With Low-Rank Approximation and Reduced-Order Model."

3. Daniel Z. Huang, Jiaoyang Huang. "Unscented Kalman Inversion: Efficient Gaussian Approximation to the Posterior Distribution."

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