2. PDE-based BIP using CUQIpy and CUQIpy-FEniCS plugin

Here we build a Bayesian inverse problem to infer the conductivity in a 2D unit-square domain modelled by the Poisson equation (applications include EIT problems).

The PDE model is built using FEniCS, then we use CUQIpy-FEniCS to wrap the PDE model to interface it with CUQIpy. We use CUQIpy samplers to solve the PDE-based Bayesian problem.

Table of contents¶

2.1. Learning objectives
2.2. Building a FEniCS based Poisson problem
2.3. Building and solving the PDE-based Bayesian problem in CUQIpy
2.4. Using gradient-based sampler

2.1. Learning objectives ¶

Build a FEniCS-based Poisson problem
Build and solve the corresponding PDE-based Bayesian problem in CUQIpy
- Use Matern covariance to specify the prior
- Use pCN sampler
Use gradient-based sampler
- Identify the chain rule needed to compute the gradient of the log-likelihood
- Use NUTS sampler

⚠️ Note:

This notebook is under the GPLv3.0 license.
This notebook was run on a machine locally and not using github actions for this book. To run this notebook on your machine, you need to have CUQIpy-FEniCS installed.

Import required libraries and classes

from scipy import optimize
import matplotlib.pyplot as plt
import dolfin as dl
import numpy as np
from cuqi.model import PDEModel
from cuqi.distribution import Gaussian, Posterior
from cuqi.array import CUQIarray
from cuqipy_fenics.pde import SteadyStateLinearFEniCSPDE
from cuqipy_fenics.geometry import FEniCSContinuous, MaternKLExpansion
from cuqi.sampler import NUTS, PCN
import cuqi
import cuqipy_fenics
ufl = cuqipy_fenics.utilities._LazyUFLLoader()


# Disable progress bar dynamic update for cleaner output for the book. You can
# enable it again by setting it to True to monitor sampler progress.
cuqi.config.PROGRESS_BAR_DYNAMIC_UPDATEATE = False

2.2. Building a FEniCS based Poisson problem ¶

In this section, we use FEniCS python library to build a PDE model.

The PDE model we consider here is a 2D steady-state problem (Poisson):

\begin{align*} -\nabla \cdot \left(\kappa(\xi) \nabla u(\xi)\right) &= f(\xi) \;\;\;\;\xi=[\xi^1, \xi^2]^\mathrm{T} \in (0,1)\times(0,1),\\ u(\xi)&=0 \;\;\;\;\mathrm{on}\; \xi^1=0\\ u(\xi)&=0 \;\;\;\;\mathrm{on}\; \xi^1=1 \\ \kappa(\xi)\nabla u(\xi)\cdot n&=0 \;\;\;\;\mathrm{on}\; \xi^2=0 \\ \kappa(\xi)\nabla u(\xi)\cdot n&=0 \;\;\;\;\mathrm{on}\; \xi^2=1 \end{align*}

(6)

where $\kappa(\xi)$ is the conductivity, $u(\xi)$ is the PDE solution (potential), $f(\xi)$ is the source term.
We use the parameterization $\kappa(\xi) = e^{m(\xi)}$ , to ensure positivity of the inferred conductivity (more on this later).
We denote the discretized system that we need to solve as $\mathbf{A}(\mathbf{m})\mathbf{U} = \mathbf{F}$
- $\mathbf{A}$ is the discretized diffusion differential operator
- $\mathbf{m}$ is the discretized unknown parameter (log conductivity)
- $\mathbf{U}$ is the discretized solution (the potential)
- $\mathbf{F}$ is the discretized RHS (the source term)

2.2.1. The discretization¶

We use finite element discretization of the model above where the solution and the parameters are approximated in a second and first order Lagrange polynomial space, respectively.

Using finite element formulation requires building the weak form of the PDE, Demkowicz (2023). To formulate the weak form, we multiply the PDE by a test function and integrate by parts and substitute the Neumann boundary conditions above (the last two equations above).

For formulating the weak form, see for example this reference.

2.2.2. Set up mesh¶

We create a 2D FEniCS mesh (unit square mesh) on which the finite element solution is discretized.

n_ksi_1 = 12 # number of vertices on the ksi_1 dimension
n_ksi_2 = 12 # number of vertices on the ksi_2 dimension
mesh = dl.UnitSquareMesh(n_ksi_1, n_ksi_2) # create FEniCS mesh
dl.plot(mesh)

2.2.3. Set up function spaces¶

We define the function spaces on which the PDE solution $u$ and the quantity we want to quantify (log conductivity $m$ ) are discretized. The function spaces are, respectively, second order Lagrange polynomial space and first order Lagrange polynomial space.

solution_function_space = dl.FunctionSpace(mesh, 'Lagrange', 2) # function space for solution u
parameter_function_space = dl.FunctionSpace(mesh, 'Lagrange', 1) # function space for parameter m

2.2.4. Set up Dirichlet boundary conditions¶

We create the Dirichlet boundary conditions:

\begin{align*} u(\xi)&=0 \;\;\;\;\mathrm{on}\; \xi^1=0\\ u(\xi)&=0 \;\;\;\;\mathrm{on}\; \xi^1=1 \\ \end{align*}

(7)

# Function (where do we have Dirichlet BC)
def u_boundary(ksi, on_boundary):
    return on_boundary and ( ksi[0] < dl.DOLFIN_EPS or ksi[0] > 1.0 - dl.DOLFIN_EPS)

# Expression (what is the value on these Dirichlet BC)
dirichlet_bc_expr = dl.Expression("0", degree=1)

# FEniCS Dirichlet BC Object
dirichlet_bc = dl.DirichletBC(solution_function_space,
                              dirichlet_bc_expr,
                              u_boundary)

2.2.5. Set up source term¶

We set the source term $f(\xi)$ to a constant value 1.

f = dl.Constant(1.0)

2.2.6. Set up PDE variational form¶

After parametrizing conductivity using $\kappa(\xi) = e^{m(\xi)}$ , the variational form of the Poisson PDE above is:

\begin{align*} \int_{(0,1)\times(0,1)} \left( e^{m(\xi)} \nabla u(\xi) \cdot \nabla p(\xi) - f(\xi)p(\xi) \right){d\xi} \end{align*}

(8)

where $p(\xi)$ is a test function. We create a function that takes the unknown parameters m, and a representation of the solution function u and a test function p and returns the weak form.

# FEniCS measure for integration
dksi = dl.Measure('dx', domain=mesh)

# The weak form of the PDE
def form(m,u,p):
    return ufl.exp(m)*ufl.inner(ufl.grad(u), ufl.grad(p))*dksi - f*p*dksi

2.2.7. Create CUQIpy PDE object¶

We bundle the FEniCS PDE model that we built in a SteadyStateLinearFEniCSPDE object:

PDE = SteadyStateLinearFEniCSPDE( 
        form,
        mesh, 
        parameter_function_space=parameter_function_space,
        solution_function_space=solution_function_space,
        dirichlet_bcs=dirichlet_bc)

Let us try solving this PDE for $m(\xi)=1$ , first we create the parameter:

# Create homogeneous parameter m_1(ksi) = 1

# Create a FEniCS function for the parameter
m_1 = dl.Function(parameter_function_space)

# Assign the value 1 to the FEniCS function by interpolating a FEniCS Constant object.
m_1.interpolate(dl.Constant(1.0))

Let us check m_1 value at a given point (0.5, 0.8)

m_1(.5, .8)

1.0

Now let us use the object we created PDE to assemble (build the discretized linear system) and solve the PDE

# Assemble the PDE at m_1
PDE.assemble(m_1)

# Solve the PDE at m_1
u, _ = PDE.solve()

Plot the solution

im = dl.plot(u)
plt.colorbar(im)

2.3. Building and solving the Bayesian inverse problem in CUQIpy ¶

The goal is to infer the log conductivity profile $m(\xi)$ given observed data $y^\mathrm{obs}$ . These observation can be of the potential directly, i.e. $y^\mathrm{obs}=u(\xi)$ , or a function of the potential.

The data $y^\mathrm{obs}$ is then given by:

y^\mathrm{obs} = \mathcal{G}(m) + \eta

(9)

where

$\eta$ is the measurement noise
$\mathcal{G}$ is the forward model operator which maps $m$ to the observations.

2.3.1. Create domain geometry¶

We model $m$ as a Matern-class random field which is a Gaussian random field with a Matern covariance function that induces spatial correlation, Roininen et al. (2014). This lead to the parametrization (Karhunen-Loève (KL) expansion):

m(\xi) = \sum_{i=0}^{\infty} \sqrt{\lambda_i}x_i e_i(\xi) \approx \sum_{i=0}^{n_\mathrm{KL}} \sqrt{\lambda_i}x_i e_i(\xi)

(10)

$\lambda_i$ and $e_i$ are the eigenvalues and eigenvectors of the Matern covariance operator.
$n_\mathrm{KL}$ is the number of KL terms used to approximate the random field $m(\xi)$ (we choose $n_\mathrm{KL}=32$ here).
$x_i\sim \mathrm{Gaussian}(0,1)$ are i.i.d. standard normal random variables.
Now, $x_i$ are the unknown parameters that parameterize the conductivity field $m(\xi)$ .

To define the Matern field (which represents the domain of our forward model), we use MaternKLExpansion geometry and define the field as below, see Alghamdi et al. (2024) for further information about the Matern field geometry implementation in the CUQIpy-FEniCS plugin.

# Define CUQI geometry on which m is defined
fenics_continuous_geo = FEniCSContinuous(parameter_function_space,
                                         labels=['$\\xi_1$', '$\\xi_2$'])

# Define the MaternExpansion geometry that maps the i.i.d random variables to
# Matern field realizations
domain_geometry = MaternKLExpansion(fenics_continuous_geo,
                                    length_scale=.1,
                                    num_terms=32)

2.3.2. Create range geometry¶

We create the range geometry which represents the forward model output (the solution $u$ in the entire domain in this case)

range_geometry = FEniCSContinuous(solution_function_space,
                                  labels=['$\\xi_1$', '$\\xi_2$'])

2.3.3. Create CUQIpy forward model¶

Now we use PDEModel which is an object that belongs to the CUQIpy library and is agnostic to the FEniCS code (FEniCS code is abstracted away in the PDE object and the geometries).

cuqi_model = PDEModel(PDE, domain_geometry=domain_geometry, range_geometry=range_geometry)

2.3.4. Create prior¶

We create the prior distribution, which is a distribution of the expansion coefficients $x_i$

x = Gaussian(np.zeros(cuqi_model.domain_dim), cov=1, geometry=domain_geometry)

We can plot prior samples (realizations of Matern class Gaussian random field)

prior_samples = x.sample(5)
prior_samples.plot()

2.3.5. Create exact solution and exact data¶

We create an exact solution (for simplification in this notebook, the exact solution is created from a prior sample):

np.random.seed(1)
exact_solution = x.sample()

# plot exact solution
im = exact_solution.plot()
plt.colorbar(im[0])

Create synthesized data that corresponds to the exact_solution

exact_data = cuqi_model(exact_solution)

# plot exact data
im = range_geometry.plot(exact_data)
plt.colorbar(im[0])

2.3.6. Create likelihood and data¶

We create the data distribution

y = Gaussian(mean=cuqi_model(x), cov=.001**2, geometry=range_geometry)
y

CUQI Gaussian. Conditioning variables ['x'].

And we create the data

y_obs = y(x=exact_solution).sample()

We plot the data

# plot data
im = range_geometry.plot(y_obs)
plt.colorbar(im[0])

We create the likelihood function:

L = y(y=y_obs)

2.3.7. Create the posterior¶

We create the posterior distribution

cuqi_posterior = Posterior(L, x)

2.3.8. Sample the posterior¶

Create a preconditioned Crank-Nicolson (pCN) sampler

Ns = 100
sampler = PCN(cuqi_posterior)

Sample the posterior

sampler.warmup(10)
sampler.sample(Ns)
samples_pCN = sampler.get_samples() # ToDo: Allow burn-in removal in get_samples method

Warmup: 100%|██████████| 10/10 [00:00<00:00, 60.83it/s, acc rate: 40.00%]
Sample: 100%|██████████| 100/100 [00:01<00:00, 61.14it/s, acc rate: 2.00%]

We plot the samples mean (and the exact solution for reference)

# plot samples mean
im = samples_pCN.plot_mean(vmin=-0.2, vmax=0.5, mode='color')
cb = plt.colorbar(im[0])

# plot the exact solution
plt.figure()
im = exact_solution.plot(vmin=-0.2, vmax=0.5, mode='color')
cb = plt.colorbar(im[0])
plt.title('Exact solution')

We look at the trace plot

samples_pCN.plot_trace()

Selecting 5 randomly chosen variables

array([[<Axes: title={'center': 'v15'}>, <Axes: title={'center': 'v15'}>],
       [<Axes: title={'center': 'v23'}>, <Axes: title={'center': 'v23'}>],
       [<Axes: title={'center': 'v31'}>, <Axes: title={'center': 'v31'}>],
       [<Axes: title={'center': 'v4'}>, <Axes: title={'center': 'v4'}>],
       [<Axes: title={'center': 'v7'}>, <Axes: title={'center': 'v7'}>]],
      dtype=object)

We plot the credible interval

samples_pCN.plot_ci(exact=exact_solution, plot_par=True)
plt.xticks(np.arange(x.dim)[::5],['v'+str(i) for i in range(x.dim)][::5]);

The sampling did not go so well. We need to use a better sampling technique.

2.4. Using gradient-based sampler ¶

Having additional information about the (log of the) posterior PDF, such as its gradient, can be utilized to improve the sampling efficiency. In section 1. Sampling with CUQIpy: five little stories, we present a number of sampling techniques that utilize the gradient information including ULA, MALA and NUTS samplers. Here we explore the NUTS sampler which is a gradient-based sampler to sample the posterior distribution.

2.4.1. The chain rule¶

We compute the gradient of the log-posterior with respect to the unknown parameter $x$ using the chain rule:

\nabla_x \mathrm{log}p_\mathrm{post}(x) \propto \nabla_x \mathrm{log}p_\mathrm{likelihood}(\mathcal{G}(m(x))) + \nabla_x \mathrm{log}p_\mathrm{prior}(x),

(11)

where $p_\mathrm{likelihood}$ is the likelihood density function, $p_\mathrm{prior}$ is the prior probability density function, $p_\mathrm{post}$ is the posterior probability density function and $m$ is the Matern field.

We have the maps:

$z := m(x)$ , implemented by the domain geometry MaternKLExpansion.
$y := \mathcal{G}(z)$ , implemented by the forward model PDEModel.

By the chain rule we have (for the likelihood part):

\nabla_x \mathrm{log}p_\mathrm{likelihood}(y) = J_{z,x}^T(x) J_{y, z}^T(z) \nabla_y \mathrm{log}p_\mathrm{likelihood}(y),

(12)

where $J_{z,x}$ is the Jacobian of the map $z=m(x)$ with respect to $x$ , $J_{y, z}$ is the Jacobian of the map $y=\mathcal{G}(z)$ with respect to $z$ and $\nabla_y \mathrm{log}p_\mathrm{likelihood}(y)$ is the gradient of the log-likelihood with respect to $y$ .

We use adjoint-based method to compute the matrix vector product $J_{y, z}^T(z)v$ for some given vector $v$ .
- Costs one forward solve and one adjoint solve (cheaper than finite difference approximation)

This is done automatically by CUQIpy-FEniCS

2.4.2. Set the adjoint problem boundary conditions¶

To compute the gradient using adjoint based method, we need to define the adjoint problem (which the PDE object infers) and derive the adjoint problem boundary conditions, see Gunzburger (2002) for adjoint-based gradient derivation.

We create the adjoint problem boundary conditions.

adjoint_dirichlet_bc_expr = dl.Constant(0.0)
adjoint_dirichlet_bc = dl.DirichletBC(solution_function_space,
                                      adjoint_dirichlet_bc_expr,
                                      u_boundary) #adjoint problem bcs

We recreate the PDE object to use the adjoint boundary conditions. We then again create the PDEModel, the data distribution, and the posterior distribution to use the new PDE object.

PDE = SteadyStateLinearFEniCSPDE( 
        form,
        mesh, 
        parameter_function_space=parameter_function_space,
        solution_function_space=solution_function_space,
        dirichlet_bcs=dirichlet_bc,
        adjoint_dirichlet_bcs=adjoint_dirichlet_bc)

cuqi_model = PDEModel(PDE,
                      domain_geometry=domain_geometry,
                      range_geometry=range_geometry)

y = Gaussian(mean=cuqi_model(x), cov=.001**2, geometry=range_geometry)
L = y(y=y_obs)
cuqi_posterior = Posterior(L, x)

2.4.3. Check the gradient correctness at an input $x_\mathrm{test}$ ¶

We check the log posterior gradient correctness at an input $x_\mathrm{test}$ by comparing the gradient computed by CUQIpy-FEniCS using adjoint based method and the gradient computed using scipy optimize.approx_fprime method.

We first create the input vector $x_\mathrm{test}$

# Create x_i
x_test = CUQIarray(np.random.randn(domain_geometry.par_dim), is_par=True, geometry=domain_geometry)

Compute the posterior gradient using CUQIpy-FEniCS

print("Posterior gradient (cuqi.model)")
cuqi_grad = cuqi_posterior.gradient(x_test)

Posterior gradient (cuqi.model)

Compute the approximate gradient using optimize.approx_fprime

print("Scipy approx")
step = 1e-11   # finite diff step
scipy_grad = optimize.approx_fprime(x_test, cuqi_posterior.logpdf, step)

Scipy approx

Plot both gradients

plt.plot(cuqi_grad, label='CUQI')
plt.plot(scipy_grad , '--', label='Approximate')
plt.legend()
plt.title("Comparing gradient with numerical approximation");

2.4.4. Use gradient based sampler (NUTS)¶

Specify a gradient-based sampler (we use NUTS here)

Ns = 200 # number of samples
Nb = 10 # burn-in
sampler = NUTS(cuqi_posterior, max_depth=10)

Sample using NUTS (this may take a little while)

sampler.warmup(Nb)
sampler.sample(Ns)
samples_NUTS = sampler.get_samples().burnthin(Nb)

Warmup: 100%|██████████| 10/10 [00:04<00:00,  2.47it/s, acc rate: 60.00%]
Sample: 100%|██████████| 200/200 [13:03<00:00,  3.92s/it, acc rate: 100.00%]

Plot the mean and the exact solution

# plot samples mean
im = samples_NUTS.plot_mean(vmin=-0.2, vmax=0.5, mode='color')
cb = plt.colorbar(im[0])

# plot the exact solution
plt.figure()
im = exact_solution.plot(vmin=-0.2, vmax=0.5, mode='color')
cb = plt.colorbar(im[0])
plt.title('Exact solution')

Plot trace

samples_NUTS.plot_trace()

Selecting 5 randomly chosen variables

array([[<Axes: title={'center': 'v1'}>, <Axes: title={'center': 'v1'}>],
       [<Axes: title={'center': 'v14'}>, <Axes: title={'center': 'v14'}>],
       [<Axes: title={'center': 'v31'}>, <Axes: title={'center': 'v31'}>],
       [<Axes: title={'center': 'v6'}>, <Axes: title={'center': 'v6'}>],
       [<Axes: title={'center': 'v7'}>, <Axes: title={'center': 'v7'}>]],
      dtype=object)

We also plot the credible interval

samples_NUTS.plot_ci(exact=exact_solution, plot_par=True)
plt.xticks(np.arange(x.dim)[::5],['v'+str(i) for i in range(x.dim)][::5]);

We note that the results have improved compared to using the pCN sampler in the sense that the inferred mean is closer to the exact solution and the credible intervals for the KL expansion coefficients are narrower and mostly contain the coefficients of the exact solution. This improvement is due to utilizing gradient information of the problem.

# Your code here

References¶

Demkowicz, L. F. (2023). Mathematical theory of finite elements. SIAM. 10.1137/1.9781611977738
Roininen, L., Huttunen, J. M., & Lasanen, S. (2014). WHITTLE-MATÉRN PRIORS FOR BAYESIAN STATISTICAL INVERSION WITH APPLICATIONS IN ELECTRICAL IMPEDANCE TOMOGRAPHY. Inverse Problems & Imaging, 8(2), 561.
Alghamdi, A. M. A., Riis, N. A. B., Afkham, B. M., Uribe, F., Christensen, S. L., Hansen, P. C., & Jørgensen, J. S. (2024). CUQIpy: II. Computational uncertainty quantification for PDE-based inverse problems in Python. Inverse Problems, 40(4), 045010. 10.1088/1361-6420/ad22e8
Gunzburger, M. D. (2002). Perspectives in flow control and optimization. SIAM.