An Introduction to Finite Element Methods for Inverse Coefficient Problems in Elliptic PDEs

We consider the stationary diffusion equation with homogeneous Dirichlet boundary condition \(u|_{\partial \Omega }=0\) in a Lipschitz bounded domain \(\Omega \subset \mathbb{R}^{d}\), \(d\in \mathbb{N}\). For \(u\in H^{1}(\Omega )\), \(\sigma \in L^{\infty }_{+}(\Omega )\) and \(g\in L^{2}(\Omega )\) the equation is equivalent to finding \(u\in H_{0}^{1}(\Omega )\) with and unique solvability follows from the Lax-Milgram theorem.

We are interested in the inverse coefficient problem of determining the diffusivity coefficient \(\sigma \) in (1) from measurements of the solution for one or several source terms \(g\), cf. [9] for an application in groundwater filtration. In practical applications with finitely many measurements, it is natural to only aim for a certain pixel-based resolution and therefore assume that \(\sigma \) is piecewise constant with respect to a partition \(\Omega =\bigcup _{i=1}^{n} \mathcal{P}_{i}\), i.e. where the pixels \(\mathcal{P}_{i}\subseteq \Omega \) are assumed to be measurable subsets. The left image in Fig. 1 shows a simple example where the unit square \(\Omega =(0,1)^{2}\) is divided into \(3\times 3\) pixels. In the following, with a slight abuse of notation, we write \(\sigma =(\sigma _{1},\ldots ,\sigma _{n})^{T}\in \mathbb{R}^{n}\) for the unknown diffusivity.

The source term \(g\) in the diffusion model (1) can be identified with the linear functional on the right hand side of the variational formulation (2) which corresponds to identifying \(L^{2}(\Omega )\) with a subset of \(H^{-1}(\Omega )\). Accordingly, we consider excitations in the form of linear functionals. Also, to emphasize that the solution depends on the diffusion coefficient and the excitation, we write \(u_{\sigma }^{l}\) in the following. The left image in Fig. 2 illustrates the concentration resulting from a constant source term \(g=\chi _{D}\), i.e. \(l(v)=\int _{D} v\,\mathrm{d} x\), where \(D=D_{2}\cup D_{4}\cup D_{5}\cup D_{7}\) is a union of four circular subdomains as sketched in the right image of Fig. 1. The right image in Fig. 2 shows the corresponding plot for \(D=D_{1}\cup D_{3}\cup D_{6}\cup D_{8}\). Both images show the solution of (1) with constant diffusion coefficient \(\sigma =1\).

Natural models for measuring the solution of (1) also yield to linear functionals. Measuring the total concentration in one of the circular subdomains \(D_{j}\) corresponds to measuring \(r(u):=\int _{D_{j}} u \,\mathrm{d} x\). Hence, the inverse problem of determining finitely many information about the diffusivity coefficient from finitely many measurements of the concentration (possibly but not necessarily resulting from different excitations) leads to the finite-dimensional inverse problem to where and \(u_{\sigma }^{l_{j}}\in H_{0}^{1}(\Omega )\) solves with given \(l_{j},r_{j}\in H^{-1}(\Omega )\), \(j=1,\ldots ,m\), and

We give another example for an application that leads to an inverse elliptic coefficient problem in a similar form as the diffusion example.

EIT aims to image the inner conductivity structure of a subject by current and voltage measurements through electrodes attached to the imaging subject. Let \(\Omega \subseteq \mathbb{R}^{d}\), \(d\in \{2,3\}\), be a smoothly bounded domain denoting the imaging subject. The electrodes \(\mathcal{E}_{k}\), \(k=1,\ldots ,K\), are assumed to be open connected subsets of \(\partial \Omega \) with disjoint closures.

When currents with strength \(J=(J_{1},\ldots ,J_{K})\in \mathbb{R}^{K}\) are driven through the \(K\) electrodes (with \(\sum _{k=1}^{K} J_{k}=0\)), the resulting electric potential \(u\in H^{1}(\Omega )\) inside \(\Omega \), and the potential \(U\in \mathbb{R}^{K}\) on the electrodes, solve where \(\sigma \in L^{\infty }_{+}(\Omega )\) is the conductivity inside \(\Omega \), and \(z>0\) is the contact impedance of the electrodes.

Under the gauge condition \(U\in \mathbb{R}_{\diamond }^{K}:=\{ V\in \mathbb{R}^{K}:\ \sum _{k=1}^{K} V_{k}=0\}\), one can show (see [19]) that this so-called complete electrode model (CEM) for EIT is equivalent to the variational formulation that \((u,U)\in H^{1}(\Omega )\times \mathbb{R}^{K}_{\diamond }\) solves for all \((w,W)\in H^{1}(\Omega )\times \mathbb{R}^{K}_{\diamond }\), and unique solvability follows from the Lax-Milgram theorem.

We assume that \(z>0\) is known, and that \(\sigma (x)=\sum _{i=1}^{n} \sigma _{i} \chi _{\mathcal{P}_{i}}(x)\) is piecewise constant with respect to a pixel partition \(\Omega =\bigcup _{i=1}^{n} \mathcal{P}_{i}\), and write \(\sigma =(\sigma _{1},\ldots ,\sigma _{n})\in \mathbb{R}^{n}\) for the unknown conductivity values inside \(\Omega \).

The applied current patterns \(J=(J_{1},\ldots ,J_{K})\in \mathbb{R}^{K}\) can be identified with the functional Likewise, measuring the voltage between the \(k_{1}\)-th and the \(k_{2}\)-th electrode corresponds to measuring the linear functional of the solution \((u,U)\in H\) generated by some current pattern.

Hence, the problem of determining the interior conductivity with a fixed finite resolution from finitely many voltage-current measurements in EIT (with CEM) leads to the finite-dimensional inverse problem to where \(\mathcal{F}:\ \mathbb{R}^{n}_{+}\to \mathbb{R}^{m}\), \(\mathcal{F}(\sigma ):=(r_{j}(u_{\sigma }^{l_{j}},U_{\sigma }^{l_{j}}))_{j=1}^{m}\), and \((u_{\sigma }^{l_{j}},U_{\sigma }^{l_{j}})\in H\) solves for all \((w,W)\in H\), with given \(l_{j},r_{j}\in H'\), \(j=1,\ldots ,m\), and Clearly, one could also extend this formulation to cover the case of unknown contact impedances.

We will study problems that appear in the variational formulation of elliptic PDEs with piecewise constant coefficients on a fixed pixel partition, as in the examples in Sect. 2.

A special mathematical structure appears for measurements \(\mathcal{F}_{l,r}\), when \(l\) and \(r\) are taken from the same subset of \(H'\), and all combinations are used. In the stationary diffusion example this corresponds to using the same subsets of \(\Omega \) both for excitations and concentration measurements, in EIT this corresponds to using the same electrodes for voltage and current measurements.

Given a set of \(m\in \mathbb{N}\) excitations/measurements \(\{l_{1},\ldots ,l_{m}\}\subset H'\), we combine the measurements into a matrix-valued map \(\mathcal{F}:\ \mathbb{R}^{n}_{+}\to \mathbb{R}^{m\times m}\)

As before, we write “≥” for the elementwise order on \(\mathbb{R}^{n}\). We also write \(\mathbb{S}_{m}\subseteq \mathbb{R}^{m\times m}\) for the subset of symmetric \(m\times m\)-matrices, and “⪰” for the Loewner order on \(\mathbb{S}_{m}\), i.e. \(B\succeq A\) denotes that \(B-A\) is positive semi-definite.

From Lemma 3, we obtain a simple FEM-based implementation of the forward operator and its derivative.

We summarize the consequences of Corollary 2 in Algorithm 1. Using a FEM package that is capable of solving the considered PDE, and that allows access to the stiffness matrix and the load vector, one can simply implement the FEM-approximated forward operator and all its first derivatives by a few lines of extra code. This calculation merely requires solving two linear systems with the stiffness matrix (which is equivalent to two PDE solutions).

As in Sect. 3.2 we now consider the symmetric measurement case, where \(l\) and \(r\) are taken from the same subset of \(H'\) (and all combinations are used). Given a set of \(m\in \mathbb{N}\) excitations/measurements \(\{l_{1},\ldots ,l_{m}\}\subset H'\), we combine the measurements into a matrix-valued map \(F:\ \mathbb{R}^{n}_{+}\to \mathbb{R}^{m\times m}\)

The entries of \(F(\sigma )\) and its first derivatives \(\frac{\partial }{\partial \sigma _{i}} F(\sigma )\), \(i=1,\ldots ,n\) can be calculated as in Algorithm 1. Let us stress that this approach is particularly efficient in this symmetric case as it requires only \(m\) linear system solutions with the stiffness matrix (i.e., the equivalent of \(m\) PDE solutions) for calculating all \(m^{2}\) entries of \(F(\sigma )\in \mathbb{R}^{m\times m}\) and all \(n m^{2}\) entries of the \(n\) matrices \(\frac{\partial }{\partial \sigma _{i}} F(\sigma )\in \mathbb{R}^{m \times m}\).

As in Sect. 3.2, the FEM-approximated forward operator is monotonically non-increasing and convex in the sense of the elementwise order “≥” on \(\mathbb{R}^{n}\), and the Loewner order “⪰” on the set of symmetric \(m\times m\)-matrices.

Even for \(m\geq n\), and a noise-free measurement \(\hat{Y}=F(\hat{\sigma })\in \mathbb{R}^{m}\), it is not clear whether the measurements uniquely determine the unknown \(\hat{\sigma }\in \mathbb{R}^{n}\). To demonstrate this on a simple one-dimensional example, let us consider the stationary diffusion example with \(3\times 3\) pixels and circular excitation/measurement subdomains in each boundary pixel as in Fig. 1. We apply a source term in \(D_{1}\) in the lower left pixel, and measure the resulting total concentration in \(D_{8}\) in the top right pixel, so that \(l=\chi _{1}\in H^{-1}(\Omega )\) and \(r=\chi _{8}\in H^{-1}(\Omega )\), where we write \(\chi _{j}:=\chi _{D_{j}}\) for the ease of notation. We choose \(\sigma =1\) in all pixels except \(\mathcal{P}_{i}\), and on \(\mathcal{P}_{i}\) we vary the diffusivity in steps of 0.01 up to 3. Figure 4 shows \(F_{l,r}(\sigma )\) for all \(i=1,\ldots ,9\), in the same order as the pixels, e.g., the lower left image shows \(F_{l,r}(\sigma )\) for \(\sigma =(\sigma _{1},1,\ldots ,1)\) for varying \(\sigma _{1}\).

Intuitively speaking, one can see that rising the diffusivity in the middle pixel increases the measurement since particles can easier diffuse through the middle pixel on their way from the lower left to the top right. Rising the diffusivity in the corner pixels decreases the measurement since particles can easier diffuse to the boundary that is set to zero by the homogeneous Dirichlet condition. In the middle top, left, bottom, and right pixel, rising the diffusivity first increases the measurement since particles can easier find their way from the lower left to the top right. But at some point, this effect is reverted since it also drives particles to the boundary.

This demonstrates that changing a coefficient can have an increasing or decreasing effect on the measurements, and the effect does not have to be monotonic for all parameter values. It also indicates that an exact one-dimensional measurement \(\hat{Y}=F_{l,r}(\hat{\sigma })\) might uniquely determine one parameter in \(\hat{\sigma }\) in some cases (here: the diffusivity in the middle and corner pixels), but non-uniqueness might occur in other cases (here: in the top, left, bottom, and right pixel).

Let us also stress the following point. A single non-symmetric measurement \(F_{l,r}(\sigma )\) with \(l\neq r\) might depend non-monotonically on \(\sigma \) as demonstrated in Fig. 4. But, by Lemma 5, for all \(l,r\in H'\), the symmetric measurements \(F_{l,l}(\sigma )\), \(F_{r,r}(\sigma )\), and also the matrix-valued measurement are monotonically non-increasing and convex functions of \(\sigma \). Note that the monotonicity and convexity properties of the matrix-valued measurement hold with respect to the Loewner order even though the individual non-diagonal matrix elements might not have these properties.

The inverse problem of recovering \(\hat{\sigma }\in \mathbb{R}^{n}\) from \(F(\hat{\sigma })\in \mathbb{R}^{m}\) could be considered as a non-linear root finding problem and (for \(n=m\)) approached with Newton’s method which is only known to converge locally. However, in practice one usually takes redundant measurements (i.e., \(m>n\)), and, due to measurement or modelling errors, one cannot expect exact data fit. Hence, a common approach to reconstruct \(\hat{\sigma }\in \mathbb{R}^{n}\) from \(Y^{\delta }\approx F(\hat{\sigma })\in \mathbb{R}^{m}\) is to minimize a residuum functional, e.g., or a sum of a residuum functional together with a regularization term.

For our simple \(3\times 3\)-pixel example, the left image in Fig. 5 shows a contour plot of \(R(\sigma )\) (in a normalized logarithmic scale) for a two-dimensional measurement function \(F(\sigma ):= \begin{pmatrix} F_{\chi _{1},\chi _{7}}(\sigma ) & F_{\chi _{1},\chi _{8}}(\sigma )\end{pmatrix}^{T} \in \mathbb{R}^{2}\), exact data \(Y^{\delta }=F(\hat{\sigma })\), \(\sigma _{4}\) and \(\sigma _{6}\) are varied in steps of 0.002 up to 0.6. Again, for this choice of unknowns and measurements, we numerically observe non-uniqueness. The results indicate that there is a second point \(\tilde{\sigma }\neq \hat{\sigma }\) with \(F(\tilde{\sigma })=F(\hat{\sigma })\).

The right image in Fig. 5 shows a plot of \(R(\sigma )\) (in a normalized non-logarithmic scale) for varying \(\sigma _{4}\) while keeping \(\sigma _{6}:=\sigma _{4}\), i.e. the diagonal of the left image in Fig. 5. It indicates that in this case, one parameter in \(\hat{\sigma }\) can be uniquely reconstructed from the two-dimensional measurement \(F(\hat{\sigma })\). But it also shows that the residuum functional \(R(\sigma )\) possesses a local minimum in a wrong point. Since the curse of dimensionality makes it practically infeasible to numerically find the global minimizer of a non-linear functional in more than a few unknowns, this demonstrates that optimization-based approaches also suffer from only local convergence.

Even if \(F(\hat{\sigma })\) uniquely determines \(\hat{\sigma }\), and if this non-linear problem can be solved without running into a local minimum, the problem might be ill-posed in the sense that \(\sigma \) does not depend stably on \(F(\sigma )\). In that case, small errors in the measurements might lead to large errors in the reconstruction. The error amplification is often quantified by searching for a Lipschitz stability constant \(L>0\) with It should be stressed though, that such a stability estimate does not immediately yield an error estimate for noisy measurements \(Y^{\delta }\approx F(\hat{\sigma})\), since \(Y^{\delta }\) might not lie in the range of \(F\).

To estimate the (relative) error amplification, we calculate the condition number of for our \(3\times 3\) example, where \(F:\ \mathbb{R}^{9}\to \mathbb{R}^{64}\) now depends on all 9 pixel values, and the components of \(F\) are given by \(F_{l,r}\) with \(l\) and \(r\) running through all \(8\times 8\)-combinations of \(\chi _{1},\ldots ,\chi _{8}\), i.e., we use all combinations of circular subdomain for applying source terms and for measuring the solution. Note that, Lemma 5 implies that roughly half of the measurements are redundant by symmetry, and that unlike in Sects. 3.2 and 4.2, we simply write the measurements as a long vector in order to have as a matrix.

We repeat the calculation of the condition number of for analogous settings with \(n_{x}\times n_{x}\)-pixels, and \((4n_{x}-4)\times (4n_{x}-4)\) measurements on circular subdomains in the boundary pixels, cf. the left image in Fig. 6 for the geometry of the \(15\times 15\)-pixel case. The right image in Fig. 6 shows the condition number as a function of the total number of pixels \(n_{x}^{2}\) for \(n_{x}\in \{2,3,\ldots ,15\}\).

Our results indicate that the instability of the considered inverse problem grows roughly exponentially with the number of unknowns which is in par with theoretical results on the similar elliptic inverse coefficient problem of EIT [17].

There is vast literature on theoretical and numerical inverse coefficient problems, their practical applications, and the regularization of ill-posed problems. Let us single out just a very few results as starting points for further reading that are closely connected to the challenges addressed in this section, and the author’s own research. Arguably the most prominent inverse elliptic coefficient problem is the so-called Calderón problem [7, 8] with applications in EIT, cf. [1, 5] for surveys on EIT and the related field of diffuse optical tomography. For theoretical uniqueness proofs in the infinite-dimensional setting of (intuitively speaking) infinitely many pixels and measurements we refer to [10, 15, 20]. A survey on solving parameter identification problems for PDEs with a focus on sparsity regularization can be found in [13]. The interplay between instability, regularization and FEM discretization is studied in [14]. A result on convexification approaches to obtain globally convergent reconstruction algorithms can be found in [16]. Learning-based approaches for inverse coefficient problems and parametrized PDEs are studied in [4, 6, 18].

Results on uniqueness and Lipschitz stability for finitely many unknowns from infinitely or finitely many measurements have been obtained in, e.g., [2, 3, 11]. Let us stress that, for most problems, it is still an open question, how many (and which) measurements are required to uniquely determine an unknown PDE coefficient with a given resolution, how to explicitly quantify the error amplification, and how to obtain globally convergent reconstruction algorithms. For a relatively simple, but fully non-linear inverse problem of determining a Robin coefficient in an elliptic PDE, these question were recently answered in [12] by exploiting the convexity and monotonicity structure of symmetric measurements from Sects. 3.2 and 4.2.