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Waveguides – Modellansatz

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H. Wind

Everybody has experimented with resonating frequencies in a bathtub filled with water. These resonant eigenfrequencies are eigenvalues of some operator which models the flow behavior of the water. Eigenvalue problems are better known for matrices. For wave problems, we have to study eigenvalue problems in infinite dimension. Like the eigenvalues for a finite dimensional matrix the spectral theory gives access to intrinsic properties of the operator and the corresponding wave phenomena.

Anne-Sophie Bonnet-BenDhia from ENSTA in Paris is in conversation with Gudrun Thäter about transmission properties in perturbed waveguides.  This is the third of three conversations recorded during the Conference on Mathematics of Wave Phenomena July 23-27, 2018 in Karlsruhe for the Modellansatz Podcast. Anne-Sophie is interested in wave guides: Optical fibers that can guide optical waves while wind instruments are guides for acoustic waves. Electromagnetic waveguides also have important applications.

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Pattern Formation

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If one puts a pan with a layer of oil on the hot oven in order to heat it up one observes different flow patterns over time: In the beginning it is easy to see that the oil is at rest and not moving at all. But if one waits long enough the still layer breaks up into small cells which makes it more difficult to see the bottom clearly. This is due to the fact that the oil starts to move in circular patterns in these cells. In our example the temperature difference between bottom and top of the oil gets bigger as the pan is heating up. For a while the viscosity and the weight of the oil keep it still. But if the temperature difference is too big it is easier to redistribute the different temperature levels with the help of convection of the oil.

This means that the system has more than one solution and depending on physical parameters one solution is stable while the others are unstable. Mariana Haragus, Professor in Besançon at the University of Franche-Comté, is doing research on this important question for engineers as well as mathematicians.

Gudrun Thäter was in conversation with her in the context of the Modellansatz Podcast about Bernard-Rayleigh problems: Where do these convection cells evolve in theory in order to keep processes on either side of the switch? This had been one of the interesting research topics at our 2018 Conference on Mathematics of Wave Phenomena.

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Linear Sampling

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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.

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CRC-Workshop Time Integration of PDEs 2019

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During the week of the 7th to 11th October the annual CRC-workshop ‘Time Integration of PDEs’ has been held at the Marburgerhaus located at a southern slope in Hirschegg, Kleinwalsertal. In total fourteen researchers from our CRC joined the seminar. All talks were concerned with numerical and analytical aspects of time integration of partial differential equations.

In contrast to previous workshops of this series, where the current state of the own research was presented, this year the focus was set differently. Each speaker presented a paper related to his/their own work but written by others. Thanks to the different backgrounds of the speakers there was a wide range of topics from local time-stepping to well-posedness results obtained from Strichartz estimates. Due to the generous time-slots and the format of the workshop, there was also plenty of time for fruitful discussions and we thus learned a lot. We took the opportunity to think about new topics and ideas and potentially acquired a different perspective for our own research.

The seminar also included social aspects by hiking through the spectacular landscape of the Kleinwalsertal. Together we reached the 2058 meters high summit of the Kanzelwand. In addition some of us explored an ambitious via ferrata, others climbed the Hohen Ifen, and a third group took a stroll through the picturesque Breitachklamm.