example01.md

Reproducing the results of Example 1

Objective

The objective of this example is to demonstrate how the inverse temperature parameter $\beta$ controls prior uncertainty. Namely, we see that:

• for small $\beta$ the prior is flat over the function space
• for large $\beta$ the prior collapses to the solution of the boundary value problem.

Mathematical details

In this example the spatial domain is $[0, 1]$. The boundary value problem is:

$$ \D \frac{d^2\phi}{dx^2} = q(x), $$

with the source term being:

$$ q(x) = e^{-x}, $$

and the conductivity being:

$$ \D = 0.25. $$

The boundary conditions are:

$$ \phi(0) = 1, $$

and

$$ \phi(1) = 0.1. $$

The Hamiltonian is:

$$ H = \int dx \left[\frac{1}{2}\D \left(\frac{d\phi}{dx}\right)^2 - \phi q\right]. $$

Running the example

Make sure you have compiled the code following the instructions here. The script example01_run.sh reproduces the paper figures. To run it, change in the directory ./examples and type in your terminal:

./example01_run.sh

If you wish to change any of default settings, feel free to edit the corresponding configuration file: example01.yml.

The results

The above script creates the following figures and puts them in a directory called example01_results.