Dynamical Sampling

Gudrun met the USA-based mathematician Roza Aceska from Macedonia
in Turin at the Conference MicroLocal and Time-Frequency Analysis
2018.


The topic of the recorded conversation is dynamical sampling. The
situation which Roza and other mathematician study is: There is a
process which develops over time which in principle is well
understood. In mathematical terms this means we know the equation
which governs our model of the process or in other words we know
the family of evolution operators. Often this is a partial
differential equation which accounts for changes in time and in
1, 2 or 3 spatial variables. This means, if we know the initial
situation (i.e. the initial conditions in mathematical terms), we
can numerically calculate good approximations for the instances
the process will have at all places and at all times in the
future.


But in general when observing a process life is not that well
sorted. Instead we might know the principal equation but only
through (maybe only a few) measurements we can find information
about the initial condition or material constants for the
process. This leads to two questions: How many measurements are
necessary in order to obtain the full information (i.e. to have
exact knowledge)? Are there possibilities to choose the time and
the spatial situation of a measurement so clever as to gain as
much as possible new information from any measurement? These are
mathematical questions which are answered through studying the
equations.


The science of sampling started in the 1940s with Claude Shannon
who found fundamental limits of signal processing. He developed a
precise framework - the so-called information theory. Sampling
and reconstruction theory is important because it serves as a
bridge between the modern digital world and the analog world of
continuous functions. It is surprising to see how many
applications rely on taking samples in order to understand
processes. A few examples in our everyday life are: Audio signal
processing (electrical signals representing sound of speech or
music), image processing, and wireless communication. But also
seismology or genomics can only develop models by taking very
intelligent sample measurements, or, in other words, by making
the most scientific sense out of available measurements.


The new development in dynamical sampling is, that in following a
process over time it might by possible to find good options to
gain valuable information about the process at different time
instances, as well as different spatial locations. In practice,
increasing the number of spatially used sensors is more expensive
(or even impossible) than increasing the temporal sampling
density. These issues are overcome by a spatio-temporal sampling
framework in evolution processes. The idea is to use a reduced
number of sensors with each being activated more frequently. Roza
refers to a paper by Enrique Zuazua in which he and his co-author
study the heat equation and construct a series of later-time
measurements at a single location throughout the underlying
process. The heat equation is prototypical and one can use
similar ideas in a more general setting. This is one topic on
which Roza and her co-workers succeeded and want to proceed
further.


After Roza graduated with a Ph.D. in Mathematics at the
University of Vienna she worked as Assistant Professor at the
University Ss Cyril and Methodius in Skopje (Macedonia), and
after that at the Vanderbilt University in Nashville (Tennessee).
Nowadays she is a faculty member of Ball State University in
Muncie (Indiana).
References


Overview on sampling theory and applications: M. Unser:
Sampling-50 years after Shannon Proceedings of the IEEE 88 (4)
569 - 587, 2000.


Dynamical sampling in shift-invariant spaces: R. Aceska
e.a.: Dynamical Sampling in Shift-Invariant Spaces 2014
(Version at Archive}


Dynamical sampling: R. Aceska, A. Petrosyan, S. Tang:
Dynamical sampling of two-dimensional temporally-varying
signals International Conference on Sampling Theory and
Applications (SampTA), DOI:10.1109/SAMPTA.2015.7148929, 2015.

DeVore, Ronald, and Enrique Zuazua: Recovery of an initial
temperature from discrete sampling, Mathematical Models and
Methods in Applied Sciences 24.12 (2014): 2487-2501, 2014.


Evolution operators involved in dynamical sampling: S.
Tang: System identification in dynamical sampling, Advances in
Computational Mathematics 43 (3) 555–580, 2017.

On Bessel systems, bases and frames in the dynamical
sampling setup: A.Aldroubi e.a.:lterative actions of normal
operators Journal of Functional Analysis 272 (3), 1121-1146,
2017.

Related Podcasts

G. Thäter, E. Zuazua: Waves, Conversation in Modellansatz
Podcast Episode 054, Fakultät für Mathematik, Karlsruher Institut
für Technologie (KIT), 2015.