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.

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.