The linear Helmholtz equation is used to model the propagation of sound waves or electromagnetic waves of small amplitude in inhomogeneous isotropic media in the time-harmonic regime. However, if the amplitudes are large then intensity-dependent material laws are required and nonlinear Helmholtz equations are more appropriate. A prominent example are Kerr-type nonlinear media. In this talk we discuss an inverse medium scattering problem for a class of nonlinear Helmholtz equations \begin{equation*} \Delta u + k^2 u \,=\, - k^2 q(x,|u|)u \,, \qquad x\in\mathbb{R}^d \,, \;d=2,3 \,, \end{equation*} that covers generalized Kerr-type nonlinear media of the form \begin{equation*} q(x,|z|) \,=\, q_0(x) + \sum_{l=1}^L q_l(x)|z|^{\alpha_l} \,, \qquad x\in\mathbb{R}^d \,,\; z\in\mathbb{C} \,, \end{equation*} where $q_0,\ldots,q_L\in L^\infty(\mathbb{R}^d)$ with support in some bounded open set $D\subset\mathbb{R}^d$, the lowest order term satisfies $\mathrm{essinf} q_0>-1$ in $\mathbb{R}^d$, and the exponents fulfill $0<\alpha_1<\cdots<\alpha_L<\infty$.

Assuming the knowledge of a nonlinear far field operator, which maps Herglotz incident waves to the far field patterns of the corresponding unique small solutions of the nonlinear scattering problem, we show that the nonlinear index of refraction is uniquely determined.

This is joint work with Marvin Knöller and Rainer Mandel (KIT).

The linearised inverse conductivity problem is investigated in a two-dimensional bounded simply connected domain with a smooth enough boundary. After extending the linearised problem for square integrable perturbations, the space of perturbations is orthogonally decomposed and Lipschitz stability, with explicit Lipschitz constants, is proven for each of the infinite-dimensional subspaces. The stability estimates are based on using the Hilbert-Schmidt norm for the Neumann-to-Dirichlet boundary map and its Fréchet derivative with respect to the conductivity coefficient. A direct reconstruction method that inductively yields the orthogonal projections of a conductivity coefficient onto the aforementioned subspaces is devised and numerically tested with data simulated by solving the original nonlinear forward problem.

Electrical impedance tomography is an imaging modality for deducing information about the conductivity inside a physical body from boundary measurements of current and voltage by a finite number of contact electrodes. This work applies techniques of Bayesian experimental design to the linearized forward model of impedance tomography in order to select optimal positions for the available electrodes. The aim is to place the electrodes so that the conditional probability distribution of the (discretized) conductivity given the electrode measurements is as localized as possible in the sense of the A- or D-optimality criterion of Bayesian experimental design. The focus is on difference imaging of a human head under the assumption that an MRI or CT image of the patient in question is available. The algorithm is developed in the computational framework introduced in [1].

[1] V. Candiani, A. Hannukainen, N. Hyvönen. Computational framework for applying electrical impedance tomography to head imaging. SIAM J. Sci. Comput. 41 (2019), no. 5, B1034–B1060. https://doi.org/10.1137/19M1245098

EIT is a non-invasive imaging technique that aims, to reconstruct the electrical conductivity distribution inside a domain by imposing electrical currents on the boundary of this domain, and measuring the resulting voltages on the same boundary. It has several applications in the medical field, in particular in lung monitoring and stroke detection. Mathematically, the problem, known as Calderón’s problem or inverse conductivity problem, is a severely ill-posed inverse problem. In practical experiments, the currents are driven inside the body of interest through a collection of surface electrodes, no current being driven between the electrodes. For each current pattern, the potential differences between the electrodes are measured. This practical setting is accurately modeled by the Complete Electrode Model (CEM). It takes into account the shape of the electrodes as well as the shunting effect, that is the thin resistive layer that appears at the interface between the electrodes and the object during the measurements. The CEM is known to correctly predict experimental data, and therefore is widely used in the numerical resolution of both direct and inverse problems related to EIT. The CEM is as follows :

Find the potentials $u\in H^1(\Omega)$ and $U \in \mathbb{R}^M_\diamond$ such that, \begin{equation*} \left\lbrace \begin{array}{rcl} \nabla \cdot (\sigma \nabla u) & = & 0 \text{ in} \ \Omega \\ u + z_m \sigma \partial_\nu u & = & U_m \text{ on } E_m, \\ \sigma \partial_\nu u & = & 0 \text{ on } \partial \Omega \setminus \overline{E}, \\ \displaystyle \int_{E_m} \sigma \partial_\nu u \, ds & = & I_m, \\%\text{ for all } m \text{ in } \left\lbrace 1 ,\ldots , M\right\rbrace. \end{array} \right. \end{equation*} with $E_m$ the $m$th electrode , $z_m$ the associated contact impedance, $I \in \mathbb{R}^M_\diamond$ the current pattern. Where $$ \mathbb{R}^M_\diamond = \left\lbrace I \in \mathbb{R}^M, \sum_{k=1}^M I_k = 0 \right\rbrace. $$

We propose an immersed boundary method (IBM) for the numerical resolution of the CEM in Electrical Impedance Tomography, that we use as a main ingredient in the resolution of inverse problems in medical imaging. Such method allows to use a Cartesian mesh without accurate discretization of the boundary, which is useful in situations where the boundary is complicated and/or changing. We prove the convergence of our method, and illustrate its efficiency with two dimensional direct and inverse problems.

[1] J. Dardé, N. Nasr, L. Weynans. Immersed boundary method for the complete electrode model in electrical impedance tomography. 2022.

[2] M. Cisternino, L. Weynans. A parallel second order Cartesian method for elliptic interface problems. Addison Wesley, Massachusetts, 2nd ed.