Linear Sampling

On closer inspection, we find science and especially mathematics throughout our everyday life, from the tap to automatic speed regulation on motorways, in medical technology or on our mobile phone. As part of the Podcast “Modellansatz” Gudrun Thäter talked to Fioralba Cakoni about the Linear Sampling Method and Scattering.

For her Ph.D. Fioralba worked with George Dassios from the University of Patras and since 2015 she has been Professor at Rutgers University in New Jersey.

Der Modellansatz: Linear Sampling. Photo: G. Thäter, Komposition: S. Ritterbusch

She introduces us to the linear sampling method. Its aim is to reconstruct the shape of an obstacle from its scattering without a priori knowledge of either the physical properties or the number of disconnected components of the scatterer. The principal problem is to detect objects inside an object without seeing it with our eyes. So we send waves of a certain frequency range into an object and then measure the response on the surface of the body. The waves can be absorbed, reflected and scattered inside the body. From this answer we would like to detect if there is something like a tumor inside the body and if yes where. Or to be more precise what is the shape of the tumor. Since the problem is non-linear and ill posed this is a difficult question and needs several mathematical steps on the analytical as well as the numerical side.

In 1996 Colton and Kirsch proposed a new method for the obstacle reconstruction problem in inverse scattering which is today known as the linear sampling method. It is a method to solve the above stated problem, which scientists call an inverse scattering problem. The method of linear sampling combines the answers to lots of frequencies but stays linear. So the problem in itself is not approximated but the interpretation of the response is.

The central idea is to invert a bounded operator which is constructed with the help of the integral over the boundary of the body.

Read more (external link).