Inverse Problem in Electric Impedance Tomography¶

Problem Formulation¶

For this demo, we investigate an inverse problem in the setting of electric impedance tomography. It is based on the one used in the preprint Blauth, Nonlinear Conjugate Gradient Methods for PDE Constrained Shape Optimization Based on Steklov-Poincaré-Type Metrics. The problem reads

\[\begin{split}&\min_{\Omega} J(u, \Omega) = \sum_{i=1}^{M} \frac{\nu_i}{2} \int_{\partial D} \left( u_i - m_i \right)^2 \text{ d}s \\ &\text{subject to} \quad \left\lbrace \quad \begin{alignedat}{2} -\kappa^\text{in} \Delta u_i^\text{in} &= 0 \quad &&\text{ in } \Omega,\\ -\kappa^\text{out} \Delta u_i^\text{out} &= 0 \quad &&\text{ in } D \setminus \Omega, \\ \kappa^\text{out} \partial_n u^\text{out}_i &= f_i \quad &&\text{ on } \partial D, \\ u^\text{out}_i &= u^\text{in}_i \quad &&\text{ on } \Gamma, \\ \kappa^\text{out} \partial_{n^\text{in}} u^\text{out}_i &= \kappa^\text{in} \partial_{n^\text{in}} u^\text{in}_i \quad &&\text{ on } \Gamma, \\ \int_{\partial D} u_i^\text{out} \text{ d}s &= 0. \end{alignedat} \right.\end{split}\]

The setting is as follows. We have an object \(\Omega\) that is located inside another one, \(D \setminus \Omega\). Our goal is to identify the shape of the interior from measurements of the electric potential, denoted by \(u\), on the outer boundary of \(D\), to which we have access. In particular, we consider the case of having several measurements at our disposal, as indicated by the index \(i\). The PDE constraint now models the electric potential in experiment \(i\), namely \(u_i\), which results from an application of the electric current \(f_i\) on the outer boundary \(\partial D\). Our goal of identifying the interior body \(\Omega\) is modeled by the tracking type cost functional, which measures the \(L^2(\Gamma)\) distance between the simulated electric potential \(u_i\) and the measured one, \(m_i\). In particular, note that the outer boundaries, i.e. \(\partial D\) are fixed, and only the internal boundary \(\Gamma = \partial \Omega\) is deformable. For a detailed description of this problem as well as its physical interpretation, we refer to the preprint Blauth, Nonlinear Conjugate Gradient Methods for PDE Constrained Shape Optimization Based on Steklov-Poincaré-Type Metrics.

For our demo, we use as domain the unit square \(D = (0,1)^2\), and the initial geometry of \(\Omega\) is a square inside \(D\). We consider the case of three measurements, so that \(M = 3\), given by

\[\begin{split}f_1 = 1 \quad \text{ on } \Gamma^l \cup \Gamma^r \qquad \text{ and } \qquad f_1 = -1 \quad \text{ on } \Gamma^t \cup \Gamma^b,\\ f_2 = 1 \quad \text{ on } \Gamma^l \cup \Gamma^t \qquad \text{ and } \qquad f_2 = -1 \quad \text{ on } \Gamma^r \cup \Gamma^b,\\ f_3 = 1 \quad \text{ on } \Gamma^l \cup \Gamma^b \qquad \text{ and } \qquad f_3 = -1 \quad \text{ on } \Gamma^r \cup \Gamma^t,\end{split}\]

where \(\Gamma^l, \Gamma^r, \Gamma^t, \Gamma^b\) are the left, right, top, and bottom sides of the unit square.

Implementation¶

The complete python code can be found in the file demo_inverse_tomography.py, and the corresponding config can be found in config.ini.

Initialization and generation of synthetic measurements¶

We start our code by importing FEniCS and CASHOCS

from fenics import *
import cashocs

Next, we directly define the sample values of \(\kappa^\text{in}\) and \(\kappa^\text{out}\) since these are needed later on

kappa_out = 1e0
kappa_in = 1e1

In the next part, we generate synthetic measurements, which correspond to \(m_i\). To do this, we define a function generate_measurements() as follows

def generate_measurements():
    mesh, subdomains, boundaries, dx, ds, dS = cashocs.import_mesh('./mesh/reference.xdmf')

    cg_elem = FiniteElement('CG', mesh.ufl_cell(), 1)
    r_elem = FiniteElement('R', mesh.ufl_cell(), 0)
    V = FunctionSpace(mesh, MixedElement([cg_elem, r_elem]))

    u, c = TrialFunctions(V)
    v, d = TestFunctions(V)

    a = kappa_out*inner(grad(u), grad(v))*dx(1) + kappa_in*inner(grad(u), grad(v))*dx(2) + u*d*ds + v*c*ds
    L1  = Constant(1)*v*(ds(3) + ds(4)) + Constant(-1)*v*(ds(1) + ds(2))
    L2  = Constant(1)*v*(ds(3) + ds(2)) + Constant(-1)*v*(ds(1) + ds(4))
    L3  = Constant(1)*v*(ds(3) + ds(1)) + Constant(-1)*v*(ds(2) + ds(4))

    meas1 = Function(V)
    meas2 = Function(V)
    meas3 = Function(V)
    solve(a==L1, meas1)
    solve(a==L2, meas2)
    solve(a==L3, meas3)

    m1, _ = meas1.split(True)
    m2, _ = meas2.split(True)
    m3, _ = meas3.split(True)

    return [m1, m2, m3]

Note

The code executed in generate_measurements() is used to solve the state problem on a reference domain, given by the mesh ./mesh/reference.xdmf. This mesh has the domain \(\Omega\) as a circle in the center of the unit square. To distinguish between these two, we note that \(D \setminus \Omega\) has the index / marker 1 and that \(\Omega\) has the index / marker 2 in the corresponding GMSH file, which is then imported into subdomains.

Note, that we have to use a mixed finite element method to incorporate the integral constraint on the electric potential. The second component of the corresponding fenics.FunctionSpace V is just a scalar, one-dimensional, real element. The actual PDE constraint is then given by the part

kappa_out*inner(grad(u), grad(v))*dx(1) + kappa_in*inner(grad(u), grad(v))*dx(2)

and the integral constraint is realized with the saddle point formulation

u*d*ds + v*c*ds

The right hand sides L1, L2, and L3 are just given by the Neumann boundary conditions as specified above.

Finally, these PDEs are then solved via the fenics.solve() command, and then only the actual solution of the PDE (and not the Lagrange multiplier for the integral constraint) is returned.

As usual, we load the config into CASHOCS with the line

config = cashocs.load_config('./config.ini')

Afterwards, we import the mesh into CASHOCS

mesh, subdomains, boundaries, dx, ds, dS = cashocs.import_mesh('./mesh/mesh.xdmf')

Next, we define the fenics.FunctionSpace object, which consists of CG1 elements together with a scalar, real element, which acts as a Lagrange multiplier for the integral constraint

cg_elem = FiniteElement('CG', mesh.ufl_cell(), 1)
r_elem = FiniteElement('R', mesh.ufl_cell(), 0)
V = FunctionSpace(mesh, MixedElement([cg_elem, r_elem]))

Next, we compute the synthetic measurements via

measurements = generate_measurements()

The PDE constraint¶

Let us now investigate how the PDE constraint is defined. As we have a mixed finite element problem due to the integral constraint, we proceed similarly to Coupled Problems - Monolithic Approach and define the first state equation with the following lines

uc1 = Function(V)
u1, c1 = split(uc1)
pd1 = Function(V)
p1, d1 = split(pd1)
e1 = kappa_out*inner(grad(u1), grad(p1))*dx(1) + kappa_in*inner(grad(u1), grad(p1))*dx(2) + u1*d1*ds + p1*c1*ds \
     - Constant(1)*p1*(ds(3) + ds(4)) - Constant(-1)*p1*(ds(1) + ds(2))

The remaining two experiments are defined completely analogously:

uc2 = Function(V)
u2, c2 = split(uc2)
pd2 = Function(V)
p2, d2 = split(pd2)
e2 = kappa_out*inner(grad(u2), grad(p2))*dx(1) + kappa_in*inner(grad(u2), grad(p2))*dx(2) + u2*d2*ds + p2*c2*ds \
     - Constant(1)*p2*(ds(3) + ds(2)) - Constant(-1)*p2*(ds(1) + ds(4))

uc3 = Function(V)
u3, c3 = split(uc3)
pd3 = Function(V)
p3, d3 = split(pd3)
e3 = kappa_out*inner(grad(u3), grad(p3))*dx(1) + kappa_in*inner(grad(u3), grad(p3))*dx(2) + u3*d3*ds + p3*c3*ds \
     - Constant(1)*p3*(ds(3) + ds(1)) - Constant(-1)*p3*(ds(2) + ds(4))

Finally, we group together the state equations as well as the state and adjoint variables to (ordered) lists, as in Using Multiple Variables and PDEs

e = [e1, e2, e3]
u = [uc1, uc2, uc3]
p = [pd1, pd2, pd3]

Since the problem only has Neumann boundary conditions, we use

bcs = None

the specify this.

The shape optimization problem¶

The cost functional is then defined by first creating the individual summands, and then summing them up:

J1 = Constant(0.5)*pow(u1 - measurements[0], 2)*ds
J2 = Constant(0.5)*pow(u2 - measurements[1], 2)*ds
J3 = Constant(0.5)*pow(u3 - measurements[2], 2)*ds

J = J1 + J2 + J3

where we use a coefficient of \(\nu_i = 1\) for all cases.

Before we can define the shape optimization properly, we have to take a look at the config file to specify which boundaries are fixed, and which are deformable. There, we have the following lines

[ShapeGradient]
shape_bdry_def = []
shape_bdry_fix = [1, 2, 3, 4]

Note, that the boundaries 1, 2, 3, 4 are the sides of the unit square, as defined in the .geo file for the geometry (located in the ./mesh/ directory), and they are fixed due to the problem definition (recall that \(\partial D\) is fixed). However, at the first glance it seems that there is no deformable boundary. This is, however, wrong. In fact, there is still an internal boundary, namely \(\Gamma\), which is not specified here, and which is, thus, deformable (this is the default behavior).

Warning

As stated in Documentation of the Config Files for Shape Optimization Problems, we have to use the config file setting

use_pull_back = False

This is due to the fact that the measurements are defined / computed on a different mesh / geometry than the remaining objects, and FEniCS is not able to do some computations in this case. However, note that the cost functional is posed on \(\partial D\) only, which is fixed anyway. Hence, the deformation field vanishes there, and the corresponding diffeomorphism, which maps between the deformed and original domain, is just the identity mapping. In particular, no material derivatives are needed for the measurements, which is why it is safe to disable use_pull_back for this particular problem.

The shape optimization problem can now be created as in Shape Optimization with a Poisson Problem and can be solved as easily, with the commands

sop = cashocs.ShapeOptimizationProblem(e, bcs, J, u, p, boundaries, config)
sop.solve()

The results should look like this

../../_images/img_inverse_tomography.png

and we observe that we are indeed able to identify the shape of the circle which was used to create the measurements.