Publications (17)
Randomized algorithms for rounding in the Tensor-Train format
Hussam Al Daas, Grey Ballard, Paul Cazeaux +5
The Tensor-Train (TT) format is a highly compact low-rank representation for high-dimensional tensors. TT is particularly useful when representing approximations to the solutions o…
On The Road To More Realistic Galaxy Cluster Simulations: The Effects of Radiative Cooling and Thermal Feedback Prescriptions on the Observational Properties of Simulated Galaxy Clusters
Stephen Skory, Eric Hallman, Jack O. Burns +3
Flux limited X-ray surveys of galaxy clusters show that clusters come in two roughly equally proportioned varieties: "cool core" clusters (CCs) and non-"cool core" clusters (NCCs).…
Two Variations on the XTrace Algorithm
Eric Hallman
This paper studies two potential modifications of XTrace (Epperly et al., SIMAX 45(1):1-23, 2024), a randomized algorithm for estimating the trace of a matrix. The first is a varia…
Monte Carlo Methods for Estimating the Diagonal of a Real Symmetric Matrix
Eric Hallman, Ilse C. F. Ipsen, Arvind Saibaba
For real symmetric matrices that are accessible only through matrix vector products, we present Monte Carlo estimators for computing the diagonal elements. Our probabilistic bounds…
Length Scales and Turbulent Properties of Magnetic Fields in Simulated Galaxy Clusters
Hilary Egan, Brian W. O'Shea, Eric Hallman +5
Additional physics beyond standard hydrodynamics is needed to fully model the intracluster medium (ICM); however, as we move to more sophisticated models, it is important to consid…
A Multilevel Approach to Stochastic Trace Estimation
Eric Hallman, Devon Troester
This article presents a randomized matrix-free method for approximating the trace of , where is a large symmetric matrix and is a function analytic in a c…
Precision-aware Deterministic and Probabilistic Error Bounds for Floating Point Summation
Eric Hallman, Ilse C. F. Ipsen
We analyze the forward error in the floating point summation of real numbers, for computations in low precision or extreme-scale problem dimensions that push the limits of the prec…
Cosmological Shock Waves and Their Role in the Large Scale Structure of the Universe
Dongsu Ryu, Hyesung Kang, Eric Hallman +1
We study the properties of cosmological shock waves identified in high-resolution, N-body/hydrodynamic simulations of a CDM universe and their role on thermalization of gas and…
Chandra Observation of the Merging Cluster A168: A Late Stage in the Evolution of a Cold Front
Eric Hallman, Maxim Markevitch
We present Chandra observations of the cool cluster A168, for which previous X-ray imaging and optical studies indicated a merger of two subclusters nearly in the plane of the sky.…
Extremal bounds for Gaussian trace estimation
Eric Hallman
This work derives extremal tail bounds for the Gaussian trace estimator applied to a real symmetric matrix. We define a partial ordering on the eigenvalues, so that when a matrix h…
Faster Stochastic Trace Estimation with a Chebyshev Product Identity
Eric Hallman
Methods for stochastic trace estimation often require the repeated evaluation of expressions of the form , where is a symmetric matrix and is a degree po…
A Refined Probabilistic Error Bound for Sums
Eric Hallman
This paper considers a probabilistic model for floating-point computation in which the roundoff errors are represented by bounded random variables with mean zero. Using this model,…
A Block Bidiagonalization Method for Fixed-Accuracy Low-Rank Matrix Approximation
Eric Hallman
We present randUBV, a randomized algorithm for matrix sketching based on the block Lanzcos bidiagonalization process. Given a matrix , it produces a low-rank approximation…
Humanity's Last Exam
Long Phan, Alice Gatti, Ziwen Han +1144
Benchmarks are important tools for tracking the rapid advancements in large language model (LLM) capabilities. However, benchmarks are not keeping pace in difficulty: LLMs now achi…
Krylov-aware stochastic trace estimation
Tyler Chen, Eric Hallman
We introduce an algorithm for estimating the trace of a matrix function using implicit products with a symmetric matrix . Existing methods for implicit…
Deterministic and Probabilistic Error Bounds for Floating Point Summation Algorithms
Eric Hallman, Ilse C. F. Ipsen
We analyse the forward error in the floating point summation of real numbers, from algorithms that do not require recourse to higher precision or better hardware. We derive informa…
A Variational Equation and Lower Bound for the Linear Least-Squares Backward Error
Eric Hallman
This paper derives a new variational equation for the linear least-squares backward error by expressing the backward error in terms of a generalized eigenvalue problem and using re…