Group Meeting

Synopsis:

See presentation slides

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Synopsis:

See presentation slides .

When analyzing a Markov process, a frequent goal is identifying the functions of the process that decorrelate most slowly in time. Slowly decorrelating functions are important for dimensionality reduction and prediction, since the values of these functions can be forecast far into the future. Moreover, because of their persistent nature, slowly decorrelating functions often have scientific significance. In biomolecular systems, for example, fluctuations of bond lengths and angles decorrelate quickly, while large-scale rearrangements that determine biological activity generally decorrelate slowly.

Dynamical spectral estimation is a rigorous approach for identifying slowly decorrelating functions. The approach uses sample trajectories to estimate the eigenfunctions of the Markov transition operator. Under appropriate assumptions, a small number of eigenfunctions span all the most slowly decorrelating functions of the process, and the corresponding eigenvalues determine the slowest decorrelation rates.
Despite the prevalence of dynamical spectral estimation in biomolecular simulation studies, estimated eigenfunctions and eigenvalues can have substantial error. The goal of this talk is to identify and bound the major error sources, thereby identifying opportunities where dynamical spectral estimation can produce accurate results.

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In stochastic systems, extreme events are known to be described by "instantons", saddle point configurations of the action of the associated stochastic field theory. In this talk, I will present experimental evidence of a hydrodynamic instanton in a real world fluid system: A 270m wave channel experiment in Norway. The experiment attempts to model conditions on the ocean in order to observe so-called rogue waves, realisations of extreme ocean surface elevation out of relatively calm surroundings. These rogue waves are also observed in the ocean, where they are rare and hard to predict but pose significant danger to naval vessels. We show that the instanton approach, which is rigorously grounded in large deviation theory, offers a unified description of rogue waves in the water tank, covering the entire range of parameters for deep water waves in the ocean. In particular, this approach allows for a unified description of both the predominantly linear and the highly nonlinear regimes, and is able to predict the experimental data in the tank regardless of the strength of the nonlinearity.

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Here are notes for the talk: WormLikeChains.pdf

Info for zoom meeting and recording sent by email.

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The entropic uncertainty principle for the discrete Fourier transform
states that for any nonzero u in C^n, H(u) + H(F_n(u)) >= log(n),
where H(u) is the entropy of the discrete probability distribution 
P_j = |u_j|^2/||u||^2, and where F_n is the discrete Fourier transform
of order n.  This is a special case of a known result [1], but the
proof of the general case requires functional analysis.  Here, we give
an elementary proof of the special case (and moreover only for n
as a power of 2). The proof is based on the Fast Fourier Transform
algorithm.

Reference

[1] Dembo A, Cover TM, and Thomas JA 1991: Information Theoretic
   Inequalities. IEEE Transactions on Information Theory
   37(6):1501-1517, see Theorem 23 on page 1513.

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A collection of 10-minute talks, each one presenting a Monte Carlo algorithm that was a game changer for a previously intractable sampling problem.

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