On some inverse problems with partial data
Dmitry Ponomarev
Inverse Problems and Mathematical Imaging Group
In this talk, I will outline several inverse problems and relevant issues that I had occasions to deal with in the past.
First, I will show a complex-analytic approach to a Cauchy problem for the Laplace equation. Such a problem arises in crack and corrosion detection contexts. The proposed treatment allows incorporating additional measurements from interior of the domain.
Then, I will show some results on the inverse obstacle problem for the wave equation from partial space-time data. A peculiarity here is that measurements are available on the part of the boundary and only for a finite time interval, the geometry of a reconstructed obstacle is a priori unknown.
Third problem comes from a concrete lab set-up: reconstruction of the overall magnetisation of a paleomagnetic sample. Interestingly enough, this problem admits a closed-form asymptotic solution, with an asymptotic parameter related to the size of the measurement area.
Finally, if time permits, I will also show my recent results on the convolution integral equations which are naturally pertinent to some basic inverse problem settings.