# Inverse problems

## Back and forth nudging algorithm for data assimilation

Data assimilation consists in estimating the state of a system by combining via numerical methods two different sources of information: models and observations. Nudging can be seen as a relaxation method, by applying a Newtonian recall of the state value toward its direct observation. The Back and Forth Nudging algorithm consists in iteratively and alternately solving forward and backward in time the model equation, with a feedback term to the observations. We propose theoretical and numerical convergence results and comparisons with other data assimilation methods.

## Why the obstacle reconstruction by topological sensitivity may work

This work investigates the performance of topological sensitivity as a tool for dealing with the inverse scattering of scalar waves in the high-frequency regime, when the wave length of the incident field is small relative to remaining length scales in the problem. To provide a focus in the study, it is assumed that the obstacle is convex and impenetrable (of either Dirichlet or Neumann type), and that the full-waveform measurements of the scattered field are taken over a sphere whose radius is finite, yet large relative to the size of the sampling region. In this setting, the formula for topological sensitivity is expressed a pair of nested surface integrals -- one taken the measurement sphere, and the other over the surface of a hidden obstacle. By way of multipole expansion, the inner integral (over the measurement surface) is reduced to a set of antilinear forms in terms of the Green’s function and its gradient. The remaining expression is distilled by evaluating the scattered field on the surface of the obstacle via Kirchhoff approximation, and deploying the method of stationary phase to evaluate the remaining integral. In this way the topological sensitivity is expressed as a sum of the closed-form expressions, signifying the contribution of (isolated) stationary points, and a contour integral over the edge of the "illuminated" part of the obstacle’s surface (when applicable). Thus obtained result explicitly demonstrates the localizing nature of the topological sensitivity and, via numerical simulations, helps better understand some of the reconstruction patterns observed in previous works.

## Determination of defects with the Robin condition

In this talk I shall deal with some inverse problems arising in non-destructing testing. In particular, I will focus on an inverse elliptic problem consisting in the localization of an unknown corroded portion of a specimen boundary and the identification of an unknown boundary impedance coefficient. I will mostly discuss the stable determination of both the unknown boundary and the Robin term by means of finite electrostatic measurements providing a logarithmic rate of stability for the above mentioned inverse problem. Moreover, I shall present some new logarithmic stability results obtained in joint work with V. Bacchelli, M. Di Cristo and S. Vessella for the corresponding parabolic case.