Conference on Applied Inverse Problems, July 20-24, 2009, Vienna, Austria
Invited Talk
Gen Nakamura: Inverse problems for reconstructing the medium discontinuities

Fri, 24 July, 2009, 09:00-10:00, C1

In about 15 years or so several reconstruction schemes have been proposed for the inverse scattering problems. This is the problem to recover the unknown discontinuities such as cracks, cavities and inclusions inside a known medium from the measurements which measure the so called far field patterns of scattered waves generated by the unknown discontinuity and giving incident waves of plane wave type from infinitely many directions. The other familiar schemes are the factorization method, singular sources method, probe method, no-response method, range and enclosure method.

As far as the unknown discontinuities are compact, it is known that this inverse scattering problem is equivalent to the so called inverse boundary value problem. That is taking a domain with Lipschitz boundary which contains all the unknown discontinuities of the medium, recover the discontinuities from the so called Dirichlet to Neumann map defined on the boundary of this domain whose graph is the set of all the Cauchy data on the boundary of this medium for all the solutions to the governing equation which describes the waves propagating inside the medium. Here it should be remarked that the waves considered here so far are all stationary waves with fixed wave number. For the inverse boundary value problem, there are also corresponding schemes for the above mentioned reconstruction schemes.

Recently, dynamical probe method for the inverse boundary value problem has been developed. This is reconstruction scheme for identifying unknown discontinuities inside a known medium. The measured data for this method are infinitely many sets of temperature and the corresponding heat flux at the boundary of the mediums. For simplicity, such an inverse boundary value problem will be referred as inverse boundary value problem for heat equation. The advantage of this method is that it is easy to repeat the measurement in a short time so that many measured data can be easily collected. Because of this, infinitely many measurements can become more realistic for this method.

In my talks I will mainly focus on the inverse scattering problem for stationary waves with fixed wave number and inverse boundary value problem for heat equation. For inverse scattering problem, several links between aforementioned reconstruction schemes and a framework which can unify these reconstruction schemes will be given. Although the idea is not new, the dynamical probe method has been newly developed. Hence, I will try to give a short overview of the method and current state of its study.

Presentation slides (pdf, 464 KB)

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