This talk is devoted to some inverse boundary problems associated with a time-dependent semilinear hyperbolic equation, where both nonlinearity and sources (including initial displacement and initial velocity) are unknown. It is shown in several generic scenarios that one can uniquely determine the nonlinearity and/or the sources by using passive or active boundary observations. In order to exploit the nonlinearity and the sources simultaneously, we develop a new technique, which combines the observability for linear wave equations and an approximation property with higher order linearization for the semilinear hyperbolic equation

We study an inverse problem for fractional elasticity. In analogy to the classical problem of linear elasticity, we consider the unique recovery of the Lamé parameters associated to a linear, isotropic fractional elasticity operator from fractional Dirichlet-to-Neumann data. In our analysis we make use of a fractional matrix Schrödinger equation via a generalization of the so-called Liouville reduction, a technique classically used in the study of the scalar conductivity equation. We conclude that unique recovery is possible if the Lamé parameters agree and are constant in the exterior, and their Poisson ratios agree everywhere. Our study is motivated by the significant recent activity in the field of nonlocal elasticity.

This is a joint work with Prof. Maarte de Hoop and Prof. Mikko Salo.

Let $(M, g)$ be a Riemannian manifold embedded (up to a conformal factor) into the product $\mathbb{R}^2 \times (M_0, g_0)$, let $A$ be a skew-Hermitian matrix of $1$-forms and let $Q$ be a matrix potential. In this talk, I will explain how to simultaneously recover the pair $(A, Q)$, up to gauge-equivalence, from the associated Dirichlet-to-Neumann map of the Schroedinger operator $d_A^*d_A + Q := (d + A)^* (d + A) + Q$. Techniques involve constructing complex geometric optics (CGO) solutions and analysing a complex ray transform that arises. This improves on the previously known results.

We consider two inverse shape problems coming from diffuse optical tomography and inverse scattering. For both problems, we assume that there are small volume subregions that we wish to recover using the measured Cauchy data. We will derive an asymptotic expansion involving their respective fields. Using the asymptotic expansion, we derive a MUSIC-type algorithm for the Reciprocity Gap Functional, which we prove can recover the subregion(s) with a finite amount of Cauchy data. Numerical examples will be presented for both problems in two dimensions in the unit circle.