IPBayesML: Programs for Inverse Problems of PDEs with Bayesian and Machine Learning Methods

The program depends on FEniCS Version 2019 (https://fenicsproject.org/download/archive/)

core: provide the basic components of coding statistical inverse problems

• probability.py
• noise.py
• model.py
• eigensystem.py
• optimizer.py
• sample.py
• approximate_sample.py
• linear_eq_solver.py
• misc.py

SimpleSmooth

In this folder, we provide a simple example.

• The forward problem $d = (Id - \alpha\Delta)^{-1}u + \epsilon$;
• The inverse problem is that given the data $d$ find the function parameter $u$.

The common.py in the folder of the SimpleSmooth provides the classes

• EquSolver: contains forward equation solver, adjoint equation solver, incremental forward solver, and incremental adjoint solver
• ModelSS: composed by prior measure, differential equations, and noise distributions

SteadyStateDarcyFlow

In this folder, we provide codes for inverse problems of the steady state Darcy flow equation. Details of the inverse problems of Darcy flow can be found in a some articles:

1. M. Dashti, A. M. Stuart, The Bayesian Approch to Inverse Problems, Hankbook of Uncertainty Quantification, 2017 [Section 1.3 Elliptic Inverse Problem]
2. Junxiong Jia, Peijun Li, Deyu Meng, Stein variational gradient descent on infinite-dimensional space and applications to statistical inverse problems, SIAM Journal on Numerical Analysis, 60(4): 2225-2252, 2022.

The common.py in the folder of the SimpleSmooth provides the classes

• EquSolver: contains forward equation solver, adjoint equation solver, incremental forward solver, and incremental adjoint solver
• ModelDarcyFlow: composed by prior measure, differential equations, and noise distributions

Citation:
@article{IPBayesML,
title = {IPBayesML: Programs for Inverse Problems of PDEs with Bayesian and Machine Learning Methods },
author = {Junxiong Jia},
year = {2022},
url = {https://github.com/jjx323/IPBayesML}
}