2-3 Talk by Perla Sousi
Title: Mobile Geometric Graphs: Detection, Isolation and Percolation
Abstract: We consider the following dynamic Boolean model introduced by van den Berg, Meester and White (1997). At time 0, let the nodes of thegraphbe a Poisson point process in R^d with constant intensity and let each node move independently according to Brownian motion. At any time t, we put an edge between every pair of nodes if their distance is at most r. We study two features in this model: detection (the time until a target point–fixed or moving–is within distance r from some node of thegraph), isolation (the time until the origin is isolated) and percolation (the time until a given node belongs to the infinite connected component of thegraph). We obtain asymptotics for these features by combining ideas from stochasticgeometry, coupling and multi-scale analysis. (Based on joint works with Yuval Peres, Alistair Sinclair and Alexandre Stauffer)
Abstract: We consider the following dynamic Boolean model introduced by van den Berg, Meester and White (1997). At time 0, let the nodes of thegraphbe a Poisson point process in R^d with constant intensity and let each node move independently according to Brownian motion. At any time t, we put an edge between every pair of nodes if their distance is at most r. We study two features in this model: detection (the time until a target point–fixed or moving–is within distance r from some node of thegraph), isolation (the time until the origin is isolated) and percolation (the time until a given node belongs to the infinite connected component of thegraph). We obtain asymptotics for these features by combining ideas from stochasticgeometry, coupling and multi-scale analysis. (Based on joint works with Yuval Peres, Alistair Sinclair and Alexandre Stauffer)
3-3:30 coffee break
3.30-4.30 Talk by Kari Heine
Title:
Butterfly resampling – convergence and central limit theorems for
particle filters with constrained interactions
Abstract:
We describe a novel class of particle filters that generalizes the
classical bootstrap filter in a manner of introducing constraints
on the interaction pattern of the particles. In some instances,
the conditional independence structure of the new algorithm can
be expressed as a graph with the same structure as the butterfly
diagram of the Cooley-Tukey fast Fourier transform. The main
motivation for the interest in these algorithms with sparse
independence structure is to lay rigorous foundations for the
design of algorithms better suited to modern computing
architectures.
The law of large numbers and the central limit theorem (CLT) are
established for specific instances of the new particle filters. It
turns out, that the price to pay for the sparseness of the
conditional independence structure is increased asymptotic
variance in the CLT, and, in some cases, slower rate of
convergence that manifests itself as a non-standard scaling in the
CLT.