Near-linear Time Approximation Schemes for Clustering in Doubling Metrics

We consider the classic Facility Location, k -Median, and k -Means problems in metric spaces of doubling dimension d . We give nearly linear-time approximation schemes for each problem. The complexity of our algorithms is Õ(2 (1/ε) O(d2) n) , making a significant improvement over the state-of-the-art algorithms that run in time n (d/ε) O(d) . Moreover, we show how to extend the techniques used to get the first efficient approximation schemes for the problems of prize-collecting k -Median and k -Means and efficient bicriteria approximation schemes for k -Median with outliers, k -Means with outliers and k -Center.

High performance simulations of a single X-pinch

The dynamics of plasmas produced by low current X-pinch devices are explored. This comprehensive computational study is the first step in the preparation of an experimental campaign aiming to understand the formation of plasma jets in table-top pulsed power X-pinch devices. Two state-of-the-art Magneto-Hydro-Dynamic codes, GORGON and PLUTO, are used to simulate the evolution of the plasma and describe its key dynamic features. GORGON and PLUTO are built on different approximation schemes and the simulation results obtained are discussed and analyzed in relation to the physics adopted by each code. Both codes manage to accurately handle the numerical demands of the X-pinch plasma evolution and provide precise details on the mechanisms of the plasma expansion, the jet-formation, and the pinch generation. Furthermore, the influence of electrical resistivity, radiation transport and optically thin losses on the dynamic behaviour of the simulated X-pinch produced plasma is studied in PLUTO. Our findings highlight the capabilities of the GORGON and PLUTO codes in simulating the wide range of plasma conditions found in X-pinch experiments, enabling for the direct comparison to the scheduled experiments.

Analysis of Tensor Approximation Schemes for Continuous Functions

In this article, we analyze tensor approximation schemes for continuous functions. We assume that the function to be approximated lies in an isotropic Sobolev space and discuss the cost when approximating this function in the continuous analogue of the Tucker tensor format or of the tensor train format. We especially show that the cost of both approximations are dimension-robust when the Sobolev space under consideration provides appropriate dimension weights.

Diffusion approximation of multi-class Hawkes processes: Theoretical and numerical analysis

Oscillatory systems of interacting Hawkes processes with Erlang memory kernels were introduced by Ditlevsen and Löcherbach (Stoch. Process. Appl., 2017). They are piecewise deterministic Markov processes (PDMP) and can be approximated by a stochastic diffusion. In this paper, first, a strong error bound between the PDMP and the diffusion is proved. Second, moment bounds for the resulting diffusion are derived. Third, approximation schemes for the diffusion, based on the numerical splitting approach, are proposed. These schemes are proved to converge with mean-square order 1 and to preserve the properties of the diffusion, in particular the hypoellipticity, the ergodicity, and the moment bounds. Finally, the PDMP and the diffusion are compared through numerical experiments, where the PDMP is simulated with an adapted thinning procedure.