Waves

Prof. Enrique Zuazua is a Distinguished Professor of Ikerbasque
(Basque Foundation for Science) and Founding Scientific Director
at the Basque Center for Applied Mathematics (BCAM), which he
pushed into life in 2008.


He is also Professor in leave of Applied Mathematics at the
Universidad Autónoma de Madrid (UAM) and a Humboldt Awardee at
the University of Erlangen-Nuremberg (FAU) as well. He was
invited by the PDE-group of our Faculty in Karlsruhe to join our
work on Wave Phenomena for some days in May 2015.


In our conversation he admits that waves have been holding his
interest since his work as a PhD student in Paris at the
Université Pierre-et-Marie-Curie in the world famous group of
Jacques-Louis Lions.


Indeed, waves are everywhere. They are visible in everything
which vibrates and are an integral part of life itself. In our
work as mathematician very often the task is to influence waves
and vibrating structures like houses or antennae such that they
remain stable. This leads to control problems like feedback
control for elastic materials.


In these problems it is unavoidable to always have a look at the
whole process. It starts with modelling the problem into
equations, analysing these equations (existence, uniqueness and
regularity of solutions and well-posedness of the problem),
finding the right numerical schemes and validating the results
against the process which has been modelled. Very often there is
a large gap between the control in the discrete process and the
numerical approximation of the model equations and some of these
differences are explainable in the framework of the theory for
hyperbolic partial differential equations and not down to
numerical or calculation errors.


In the study of Prof. Zuazua the interaction between the
numerical grid and the propagation of waves of different
frequencies leads to very intuitive results which also provide
clear guidelines what to do about the so-called spurious wave
phenomena produced by high frequencies, an example of which is
shown in this podcast episode image.


This is an inherent property of that sort of equations which are
able to model the many variants of waves which exist. They are
rich but also difficult to handle. This difficulty is visible in
the number of results on existence, uniqueness and regularity
which is tiny compared to elliptic and parabolic equations but
also in the difficulty to find the right numerical schemes for
them. On the other hand they have the big advantage that they are
best suited for finding effective methods in massively parallel
computers. Also there is a strong connection to so-called Inverse
Problems on the theoretical side and through applications where
the measurement of waves is used to find oil and water in the
ground, e.g (see, e.g. our Podcast Modell004 on Oil Exploration).


Prof. Zuazua has a lot of experience in working together with
engineers. His first joint project was shape optimization for
airfoils. The geometric form and the waves around it interact in
a lot of ways and on different levels. Also water management has
a lot of interesting and open questions on which he is working
with colleagues in Zaragoza. At the moment there is a strong
collaboration with the group of Prof. Leugering in Erlangen which
is invested in a Transregio research initiative on gasnets which
is a fascinating topic ranging from our everyday expectations to
have a reliable water and gas supply at home to the latest
mathematical research on control. Of course, in working with
engineers there is always a certain delay (in both directions)
since the culture and the results and questions have to be
translated and formulated in a relevant form between engineers
and mathematicians.


In dealing with theses questions there are two main risks:
Firstly, one finds wrong results which are obviously wrong and
secondly wrong results which look right but are wrong
nonetheless. Here it is the crucial role of mathematicians to
have the right framework to find these errors.


Prof. Zuazua is a proud Basque. Of the 2.5 Mill. members of the
basque people most are living in Spain with a minority status of
their culture and language. But since the end of the Franco era
this has been translated into special efforts to push culture and
education in the region. In less than 40 years this transformed
the society immensely and led to modern universities, relevant
science and culture which grew out of "nothing". Now Spain and
the Basque country have strong bonds to the part of Europe on the
other side of the Pyrenees and especially with industry and
research in Germany. The Basque university has several campuses
and teaches 40.000 students. This success could be a good example
how to extend our education system and provide possibilities for
young people which is so much a part of our culture in Europe
across the boundaries of our continent.