This is a software package written in C++ and Python for solving Bayesian inverse problems that occur in the gas industry, e.g., to identify friction coefficients in gas pipes. The flow of the gas in a pipe is modelled using a quasi-linear iso-thermal model, and the gas is subject to friction. The behavior of the gas is described by its pressure and momentum along the pipe.
The task is to obtain statistical properties of the friction
coefficient based on finite noisy observation of the pressure drop at
both ends of the gas pipe, and a prior knowledge on the friction
coefficient. As an example, we assume that the friction coefficient is
uniformly distributed in the interval [0.0,0.5] while the true value
is 0.075, then the UQ package using Markov-Chain Monte Carlo (MCMC)
method and a discretization of the gas model, obtain a posterior
distribution for the friction coefficient. In the following Figure we
observe that the posterior distribution is clustered around the true
friction coefficient, i.e., 0.075.
For more information see References.
requirements: make, g++, python2.7 and pip
Download the package either through git,
git clone https://github.com/hajianOne/UQ.gitor download it directly from GitHub and decompress it. Go into the package directory. You should see the following folder structure:
UQ
|-- makefile
|-- README.md
|-- LICENSE.md
|-- requirements.txt
|-- setup.py
|-- build
|-- build_cpp
|-- examples
|-- notebooks
|-- lib
|-- results
|-- src
\-- UQuant
In order to compile and install the package execute the following commands (make sure you have the right permissions):
make lib/CWrapper.dylib
pip install -r requirements.txt
python setup.py installThe first command generates a shared library and save a file called
CWrapper.dylib in lib/ directory and CWrapper.so in
UQuant/lib/ directory. The second and third commands installs
python packages in UQuant folder and the shared library.
If you didn't receive any error, you should be able to import the libraries in the python. For instance the following code should produce an output describing the pipe:
from UQuant.SemilinearSystem import SemiLinSystem
from UQuant.mcmc import MCMC
c_sound, t_final, x_l, x_r, dx, boundary_eps = [1.0, 5.0, 0.0, 1.0, 0.005, 0.05]
expan_coef = 1
true_friction = [0.075]
# build the pipe
pipe_true = SemiLinSystem(c_sound, t_final, x_l, x_r, dx, expan_coef, boundary_eps)
# get the info of the pipe
pipe_true.info()
# compute the solution of the semilinear system with the prescribed friction coefficients
pipe_true.run(true_friction)
# get the pressure drop
y_obs = pipe_true.get_presure_drop(time_instance=time_ins, inplace=False)In order to compile and install the package execute the following commands (make sure you have the right permissions):
make lib/CWrapper.so
pip install -r requirements.txt
python setup.py installThe first command generates a shared library and save a file called
CWrapper.so in lib/ directory and CWrapper.so in
UQuant/lib/ directory. The second and third commands installs python
packages in UQuant folder and the shared library.
If you didn't receive any error, you should be able to import the libraries in the python. For instance the following code should produce an output describing the pipe:
from UQuant.SemilinearSystem import SemiLinSystem
from UQuant.mcmc import MCMC
c_sound, t_final, x_l, x_r, dx, boundary_eps = [1.0, 5.0, 0.0, 1.0, 0.005, 0.05]
expan_coef = 1
true_friction = [0.075]
# build the pipe
pipe_true = SemiLinSystem(c_sound, t_final, x_l, x_r, dx, expan_coef, boundary_eps)
# get the info of the pipe
pipe_true.info()
# compute the solution of the semilinear system with the prescribed friction coefficients
pipe_true.run(true_friction)
# get the pressure drop
y_obs = pipe_true.get_presure_drop(time_instance=time_ins, inplace=False)If everything goes well you should be able to test the package. First
go to the root directory of the package, e.g., UQ and then execute
the following command
python examples/test.pyThe output in the console looks like
Address of the pipe in the memory from cpp: 0x1328660
Address of the pipe in the memory from python: 0x1328660
========================================== info ==
Omega _____________________________(0,1)
Length of the domain ______________1
Dt ________________________________0.005
Dx ________________________________0.005
Number of cells ___________________200
Current time ______________________0
Final time ________________________5
Epsilon for the boundary __________0.05
N_epsilon _________________________10
Size of friction coef. vec. _______7
==================================================
>> Computation Done
and also you should be able to see the following figure which shows continuous dependence of the pressure drop at both ends of the pipe with respect to the friction coefficient.
In the second test we will run the Markov-Chain Monte-Carlo (MCMC) algorithm for the case where the friction coefficient is a scalar number:
python examples/test0.pyThe output is saved in results/friction_scalar.png:
In this example, the true friction coefficient is 0.075 and we
provide a prior density function with uniform distribution in the
interval [0.0, 0.5]. The initial sample is located at 0.45 (far
from the true value). Observe that the posterior distribution captures
the truth and samples are clustered around 0.075.
We can also perform an experiment when the friction coefficient is a
function. The script is called uq.py and is located in examples
folder. In order to test, execute
python examples/uq.pyIn the following figure we can see the true friction coefficient as a function of spatial coordinates and the sequence of samples obtained from MCMC.
In addition to MCMC, a dimension robust algorithm called pCN is also
implemented. For a test case please see examples/pcn_uq.py or the
jupyter notebook notebooks/pcn.ipynb.
The UQ package is developed by S. Hajian, N. Strogies and M. Hintermüller. The UQ package is supported financially by the German Research Foundation DFG through the SFB-TRR 154 and funds from the Einstein Center for Mathematics Berlin through C-SE5 and D-OT1. The aforementioned package (including code modifications) may only be used for NON-COMMERCIAL RESEARCH purposes. For inquiries concerning a different use, please contact the authors. See the license.
- S. Hajian, M. Hintermüller, C. Schillings and N. Strogies, A Bayesian approach for parameter identification in gas networks, PDF.