Reproducing the results of Example 3

Objective

The objective of this example is to demonstrate how the inverse temperature $\beta$ detects model form error. Namely, we see that:

if the model is correct, $\beta$ tends to be large
if the model is wrong, $\beta$ tends to be small.

Mathematical details

The ground truth problem is as in Example 2. We reproduce here the details for your convenience.

The spatial domain is $[0, 1]$. The ground truth field satisfies the equation:

$$ D\frac{d^2\phi}{dx^2} - \kappa \phi^3 = f, $$

with source term:

$$ f(x) = \cos(4x), $$

conductivity:

$$ D = 0.1, $$

non-linear coefficient:

$$ \kappa = 1, $$

and boundary conditions:

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

and

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

We sample the field at $40$ equidistant points between $0$ and $1$ and we add Gaussian noise with zero mean and standard deviation $0.01$. The observed field inputs are here and the observed field otuputs are here. If you wish to review how the observations were generated, consult the script example02_generate_observations.py.

The correct Hamiltonian for this problem is:

$$ H = \int dx \left[\frac{1}{2}D \left(\frac{d\phi}{dx}\right)^2 + \frac{1}{4}\kappa\phi^4

\phi f\right]. $$

We solve two different inverse problems of increasing complexity and ill-posedness:

Example 3.a: Model error in the source term

In this inverse problem we assume that the source term is known and we are looking for $D$ and $\kappa$. We use Jeffrey's priors on these parameters, i.e.,

$$ p(D) \propto \frac{1}{D}, $$

and

$$ p(\kappa) \propto \frac{1}{\kappa}. $$

To enforce positive, we sample in the logarithmic space:

$$ \theta_1 = \log(D), $$

and

$$ \theta_2 = \log(\kappa). $$

Example 3.b: Model error in the energy

As in Example 3.a both parameters $D$ and $\kappa$ are assumed to be uknown. We use exactly the same treatement: Jeffrey's prior and work in logarithmic space. However, we now assume that we do not know the source term $f(x)$. To parameterize the source term, we use the Karhunen-Lo`eve expansion (KLE) of a Gaussian random field which we construct using the Nystr"om, see Bilionis, et al. 2016. We assume that the covariance of this random field is the squared exponent with variance one and lengthscale $\ell = 0.3$. The number of KLE components is 10 accounting for more than 99% of the energy of the field.

Running the examples

Make sure you have compiled the code following the instructions here. The script example03_run.sh reproduces the paper figures. To run it, change in the directory ./examples and type in your terminal:

./example03_run.sh

If you wish to change any of default settings, feel free to edit the corresponding configuration files:

example03a.yml for Example 3.a, and
example03b.yml for Example 3.b.

The results

The above script creates the following figures and puts them in directories called example03a_results and example03b_results for Example 3.a and 3.b, respectively.