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Lévy-Driven Diffusion for time series
Abstract
Begun in 2022 due to the cancellation of the ICM in Russia, the Department mini-ICM returns to celebrate our invited speakers at the International Congress to be held in Philadelphia in July.
This year’s event will be on Monday May 11th (week 3) in L2, in the Mathematical Institute. The talks should be widely accessible, so do come along to hear about the work of our colleagues.
2.35 pm Patrick Farrell: Computing multiple solutions of systems of nonlinear equations with deflation. Chair: Mike Giles
One-Day Meeting in Combinatorics
The speakers are Penny Haxell (Waterloo), Guus Regts (University of Amsterdam), Annika Heckel (Uppsala), Standa Živný (Oxford), and Romain Tessera (Institut de Mathématiques de Jussieu-Paris Rive Gauche). Please see the event website for further details including titles, abstracts, and timings. Anyone interested is welcome to attend, and no registration is required.
12:00
A Runtime-Data-Driven Enhancement Preconditioner for PCG for a Sequence of SPD Linear Systems
Abstract
Jing-Yuan Wang is going to talk about: 'A Runtime-Data-Driven Enhancement Preconditioner for PCG for a Sequence of SPD Linear Systems'
In this work, we propose a runtime-data-driven enhancement preconditioner for improving the convergence of a preconditioned conjugate gradient method for solving a sequence of symmetric positive definite linear systems of equations. The methodology is designed for the situation where a subset of the systems has been solved and the convergence is considered too slow. In such a situation, data generated from the solved problems (residual vectors, intermediate solution vectors, approximate error vectors) are first analyzed by an unsupervised learning algorithm as a 3-step process: (1) dimension reduction; (2) classification of the slow features; (3) construction of projections to each of the feature subspaces. Based on the results of the analysis, one or more enhancement preconditioners are constructed using projection matrices corresponding to the features extracted from the slow convergence subspaces. The enhancement preconditioners are additively incorporated into the existing preconditioners and are employed to solve other systems in the sequence. The enhancement preconditioner can be further enhanced when necessary by repeating this process. Numerical experiments for time-dependent problems, including parabolic and hyperbolic equations, and stochastic elliptic equations demonstrate that the proposed approach improves the convergence considerably for other systems in the sequence when classical preconditioners are insufficient.
14:00
Random Geometric Graphs: Ramsey Bounds and Testing Thresholds
Abstract
The random geometric graph G(n,S^d,p) is obtained by placing n random points independently and uniformly on the unit sphere S^d, and connecting two points whenever they are sufficiently close, with the threshold chosen so that each edge appears with probability p. The underlying geometry of the model creates correlations between edges, making its behavior richer than that of the corresponding binomial random graph G(n,p).
A striking recent application of these correlations is due to Ma, Shen, and Xie, who used high-dimensional random geometric graphs to obtain an exponential improvement over Erdős’s celebrated lower bound for R(k,Ck), where C>1 is fixed. I will discuss a simplification of their approach using Gaussian random geometric graphs, leading to a much shorter analysis and sharper quantitative bounds.
I will then turn to a complementary question: when does the geometry disappear? More precisely, for which dimensions d is G(n,S^d,p) statistically indistinguishable from G(n,p)? This problem, introduced by Bubeck, Ding, Eldan, and Rácz, has attracted considerable interest across probability, theoretical computer science, and high-dimensional statistics. They conjectured that the threshold is governed by the signed triangle count, namely d≍n^3p^3 up to logarithmic factors. I will outline a proof of this conjecture for a wide range of p.
This talk is based on joint work with Zach Hunter and Aleksa Milojevic.