Applied Mathematics Seminar

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The active Brownian particle (ABP) model describes a swimmer,
synthetic or living, whose direction of swimming is a Brownian motion.
The swimming is due to a propulsion force, and the fluctuations are
typically thermal in origin.  We present a 2D model where the
fluctuations arise from nonthermal noise in a propelling force acting
at a single point, such as that due to a flagellum.  We take the
overdamped limit and find several modifications to the traditional ABP
model.  Since the fluctuating force causes a fluctuating torque, the
diffusion tensor describing the process has a coupling between
translational and rotational degrees of freedom.  An anisotropic
particle also exhibits a noise-induced drift.  We show that these
effects have measurable consequences for the long-time diffusivity of
active particles, in particular adding a contribution that is
independent of where the force acts.  This is joint work with Jiajia
Guo.

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Solution manifolds induced by transport-dominated problems
such as hyperbolic conservation laws typically exhibit nonlinear
structures. This means that traditional model reduction methods based
on linear approximations in subspaces are inefficient when applied to
these problems. This presentation discusses model reduction methods
for constructing nonlinear reduced models that seek approximations on
manifolds, rather than in subspaces, and so lead to efficient
dimensionality reduction even for transport-dominated problems. First,
we will discuss an online adaptive approach that exploits locality in
space and time to efficiently adapt piecewise linear approximations of
the solution manifolds. Second, we present an approach that derives
reduced approximations that are nonlinear by explicitly composing
global transport dynamics with locally linear approximations of the
solution manifolds. The compositions can be interpreted as
one-hidden-layer neural networks. Numerical results demonstrate that
the proposed approaches achieve speedups even for problems where
traditional, linear reduced models are more expensive to solve than
the high-dimensional, full model.
 

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Abstract: Contact tracing mobile applications are clear candidates enabling us to slow down an epidemics and keep the society running while holding the health risks down. Most of the currently discussed and developed mobile applications aim to notify individuals who were recently in a significant contact with an individual who tested positive. The contacted individuals would then be tested or/and put in isolation. In our work, we aim to quantify the epidemiological gain one would obtain if, additionally, individuals who were recently in contact could exchange messages of information. With such a message passing the risk of contracting the infection could be estimated with much better accuracy than simple contact tracing. Our results show that in some range of epidemic spreading (typically when the manual tracing of all contacts of infected people becomes practically impossible, but before the fraction of infected people reaches the scale where a lock-down becomes unavoidable), this inference of individuals at risk could be an efficient way to mitigate the epidemic. Our approaches translate into fully distributed algorithms that only require communication between individuals who have recently been in contact. We conclude that probabilistic risk estimation is capable of enhancing the performance of digital contact tracing and should be considered in the currently developed mobile applications.

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Diverse multi-cellular animals encode a breathtaking diversity of natural behaviors. Non local interactions in traditional nervous systems make the study of underlying origins of behavior in animals difficult (and fascinating). It is a well-known fact that simple dynamical systems can also encode perplexing complexity with purely local update rules. In this talk, using a variety of toy models and systems, we will explore how complex behavior can arise in non-neuronal ensembles; or in short "how do animals with no brains (neurons), decide, compute or think?"