arXiv subjects

Eric Hallman

Publications and source records attributed to Eric Hallman.

16 recordsLinked to original sources

arXiv2026

A Variational Equation and Lower Bound for the Linear Least-Squares Backward Error

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 results from indefinite linear algebra. For problems with multiple right-hand sides, the variational equation also shows that the backward error can be decomposed as a sum of smaller backward error problems. Applications to stopping criteria for iterative methods are considered, and a new sketching-based lower bound is proposed which is provably of quality comparable to the sketched Karlson-Wald\'{e}n estimate.

math.NA↗

arXiv2025

Two Variations on the XTrace Algorithm

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 variance reduction step that averages the output of XTrace over right-multiplications of the test vectors by random orthogonal matrices. The second is to form a low-rank approximation to the matrix using the whole Krylov space produced by the test vectors, rather than the output of a single power iteration as is used by XTrace. Experiments on synthetic data show that the first modification offers only slight benefits in practice, while the second can lead to significant improvements depending on the spectrum of the matrix.

math.NA↗

arXiv2024

Extremal bounds for Gaussian trace estimation

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 has greater spectrum under this ordering, its estimator will have worse tail bounds. This is done for two families of matrices: positive semidefinite matrices with bounded effective rank, and indefinite matrices with bounded 2-norm and fixed Frobenius norm. In each case, the tail region is defined rigorously and is constant for a given family.

math.ST↗

arXiv2022

Krylov-aware stochastic trace estimation

We introduce an algorithm for estimating the trace of a matrix function $f(\mathbf{A})$ using implicit products with a symmetric matrix $\mathbf{A}$. Existing methods for implicit trace estimation of a matrix function tend to treat matrix-vector products with $f(\mathbf{A})$ as a black-box to be computed by a Krylov subspace method. Like other recent algorithms for implicit trace estimation, our approach is based on a combination of deflation and stochastic trace estimation. However, we take a closer look at how products with $f(\mathbf{A})$ are integrated into these approaches which enables several efficiencies not present in previously studied methods. In particular, we describe a Krylov subspace method for computing a low-rank approximation of a matrix function by a computationally efficient projection onto Krylov subspace.

math.NA↗

arXiv2022

Precision-aware Deterministic and Probabilistic Error Bounds for Floating Point Summation

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 precision. We present a systematic recurrence for a martingale on a computational tree, which leads to explicit and interpretable bounds without asymptotic big-O terms. Two probability parameters strengthen the precision-awareness of our bounds: one parameter controls the first order terms in the summation error, while the second one is designed for controlling higher order terms in low precision or extreme-scale problem dimensions. Our systematic approach yields new deterministic and probabilistic error bounds for three classes of mono-precision algorithms: general summation, shifted general summation, and compensated (sequential) summation. Extension of our systematic error analysis to mixed-precision summation algorithms that allow any number of precisions yields the first probabilistic bounds for the mixed-precision FABsum algorithm. Numerical experiments illustrate that the probabilistic bounds are accurate, and that among the three classes of mono-precision algorithms, compensated summation is generally the most accurate. As for mixed precision algorithms, our recommendation is to minimize the magnitude of intermediate partial sums relative to the precision in which they are computed.

math.NA↗

arXiv2022

Monte Carlo Methods for Estimating the Diagonal of a Real Symmetric Matrix

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 for normwise absolute and relative errors apply to Monte Carlo estimators based on random Rademacher, sparse Rademacher, normalized and unnormalized Gaussian vectors, and to vectors with bounded fourth moments. The novel use of matrix concentration inequalities in our proofs represents a systematic model for future analyses. Our bounds mostly do not depend on the matrix dimension, target different error measures than existing work, and imply that the accuracy of the estimators increases with the diagonal dominance of the matrix. An application to derivative-based global sensitivity metrics corroborates this, as do numerical experiments on synthetic test matrices. We recommend against the use in practice of sparse Rademacher vectors, which are the basis for many randomized sketching and sampling algorithms, because they tend to deliver barely a digit of accuracy even under large sampling amounts.

math.NA↗

arXiv2021

Randomized algorithms for rounding in the Tensor-Train format

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 of certain types of parametrized partial differential equations. For many of these problems, computing the solution explicitly would require an infeasible amount of memory and computational time. While the TT format makes these problems tractable, iterative techniques for solving the PDEs must be adapted to perform arithmetic while maintaining the implicit structure. The fundamental operation used to maintain feasible memory and computational time is called rounding, which truncates the internal ranks of a tensor already in TT format. We propose several randomized algorithms for this task that are generalizations of randomized low-rank matrix approximation algorithms and provide significant reduction in computation compared to deterministic TT-rounding algorithms. Randomization is particularly effective in the case of rounding a sum of TT-tensors (where we observe 20x speedup), which is the bottleneck computation in the adaptation of GMRES to vectors in TT format. We present the randomized algorithms and compare their empirical accuracy and computational time with deterministic alternatives.

math.NA↗

arXiv2021

Deterministic and Probabilistic Error Bounds for Floating Point Summation Algorithms

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 informative explicit expressions, and new deterministic and probabilistic bounds for errors in three classes of algorithms: general summation,shifted general summation, and compensated (sequential) summation. Our probabilistic bounds for general and shifted general summation hold to all orders. For compensated summation, we also present deterministic and probabilistic first and second order bounds, with a first order bound that differs from existing ones. Numerical experiments illustrate that the bounds are informative and that among the three algorithm classes, compensated summation is generally the most accurate method.

math.NA↗

arXiv2021

A Refined Probabilistic Error Bound for Sums

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 probabilistic bound is derived for the forward error of the computed sum of n real numbers. This work improves upon existing probabilistic bounds by holding to all orders, and as a result provides informative bounds for larger problem sizes.

math.NA↗

arXiv2021

A Multilevel Approach to Stochastic Trace Estimation

This article presents a randomized matrix-free method for approximating the trace of $f({\bf A})$, where ${\bf A}$ is a large symmetric matrix and $f$ is a function analytic in a closed interval containing the eigenvalues of ${\bf A}$. Our method uses a combination of stochastic trace estimation (i.e., Hutchinson's method), Chebyshev approximation, and multilevel Monte Carlo techniques. We establish general bounds on the approximation error of this method by extending an existing error bound for Hutchinson's method to multilevel trace estimators. Numerical experiments are conducted for common applications such as estimating the log-determinant, nuclear norm, and Estrada index, and triangle counting in graphs. We find that using multilevel techniques can substantially reduce the variance of existing single-level estimators.

math.NA↗

arXiv2021

A Block Bidiagonalization Method for Fixed-Accuracy Low-Rank Matrix Approximation

We present randUBV, a randomized algorithm for matrix sketching based on the block Lanzcos bidiagonalization process. Given a matrix $\bf{A}$, it produces a low-rank approximation of the form ${\bf UBV}^T$, where $\bf{U}$ and $\bf{V}$ have orthonormal columns in exact arithmetic and $\bf{B}$ is block bidiagonal. In finite precision, the columns of both ${\bf U}$ and ${\bf V}$ will be close to orthonormal. Our algorithm is closely related to the randQB algorithms of Yu, Gu, and Li (2018) in that the entries of $\bf{B}$ are incrementally generated and the Frobenius norm approximation error may be efficiently estimated. Our algorithm is therefore suitable for the fixed-accuracy problem, and so is designed to terminate as soon as a user input error tolerance is reached. Numerical experiments suggest that the block Lanczos method is generally competitive with or superior to algorithms that use power iteration, even when $\bf{A}$ has significant clusters of singular values.

math.NA↗

arXiv2021

Faster Stochastic Trace Estimation with a Chebyshev Product Identity

Methods for stochastic trace estimation often require the repeated evaluation of expressions of the form $z^T p_n(A)z$, where $A$ is a symmetric matrix and $p_n$ is a degree $n$ polynomial written in the standard or Chebyshev basis. We show how to evaluate these expressions using only $\lceil n/2\rceil$ matrix-vector products, thus substantially reducing the cost of existing trace estimation algorithms that use Chebyshev interpolation or Taylor series.

math.NA↗

arXiv2016

Length Scales and Turbulent Properties of Magnetic Fields in Simulated Galaxy Clusters

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 consider the role of magnetic fields and the way the fluid approximation breaks down. This paper represents a first step towards developing a self-consistent model of the ICM by characterizing the statistical properties of magnetic fields in cosmological simulations of galaxy clusters. We find that plasma conditions are largely homogeneous across a range of cluster masses and relaxation states. We also find that the magnetic field length scales are resolution dependent and not based on any particular physical process. Energy transfer mechanisms and scales are also identified, and imply the existence of small scale dynamo action. The scales of the small scale dynamo are resolution limited and driven by numerical resistivity and viscosity.

astro-ph.CO↗

arXiv2012

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

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). In previous work, we have demonstrated using cosmological $N$-body + Eulerian hydrodynamic simulations that NCCs are often consistent with early major mergers events that destroy embryonic CCs. In this paper we extend those results and conduct a series of simulationsusing different methods of gas cooling, and of energy and metal feedback from supernovae, where we attempt to produce a population of clusters with realistic central cooling times, entropies, and temperatures. We find that the use of metallicity-dependent gas cooling is essential to prevent early overcooling,and that adjusting the amount of energy and metal feedback can have a significant impact on observable X-ray quantities of the gas. We are able to produce clusters with more realistic central observable quantities than have previously been attained. However, there are still significant discrepancies between the simulated clusters and observations, which indicates that a different approach to simulating galaxies in clusters is needed. We conclude by looking towards a promising subgrid method of modeling galaxy feedback in clusters which may help to ameliorate the discrepancies between simulations and observations.

astro-ph.CO↗

arXiv2004

Chandra Observation of the Merging Cluster A168: A Late Stage in the Evolution of a Cold Front

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. We derive a temperature map for A168, which shows that the merger has proceeded beyond the core passage and is near subcluster turnaround. It also reveals an unusual feature -- the gas core of one of the subclusters forms a tongue-like structure extending ahead (in the direction of motion) of the subcluster center. The coolest cluster gas is found in a crescent-shaped region at the tip of this tongue, and forms a cold front in pressure equilibrium with the external gas. In contrast with this feature's forward location, previously observed merger cold fronts (e.g., A3667, 1E0657--56) lagged behind their host subclusters, as expected in the presense of ram pressure. We propose that A168 illustrates a much later stage in the evolution of a cold front, when its host subcluster approaches the apocenter of the merger orbit where the ram pressure on its gas drops sharply. As a result, a large chunk of the subcluster gas ``slingshots'' past the dark matter center, becomes unbound from the subcluster and expands adiabatically, as seen in some recent hydrodynamic simulations.

astro-ph↗

arXiv2003

Cosmological Shock Waves and Their Role in the Large Scale Structure of the Universe

We study the properties of cosmological shock waves identified in high-resolution, N-body/hydrodynamic simulations of a $Λ$CDM universe and their role on thermalization of gas and acceleration of nonthermal, cosmic ray (CR) particles. External shocks form around sheets, filaments and knots of mass distribution when the gas in void regions accretes onto them. Within those nonlinear structures, internal shocks are produced by infall of previously shocked gas to filaments and knots, and during subclump mergers, as well as by chaotic flow motions. Due to the low temperature of the accreting gas, the Mach number of external shocks is high, extending up to $M\sim 100$ or higher. In contrast, internal shocks have mostly low Mach numbers. For all shocks of $M\ge1.5$ the mean distance between shock surfaces over the entire computed volume is $\sim4 h^{-1}$ Mpc at present, or $\sim 1 h^{-1}$ Mpc for internal shocks within nonlinear structures. Identified external shocks are more extensive, with their surface area $\sim2$ times larger than that of identified internal shocks at present. However, especially because of higher preshock densities, but also due to higher shock speeds, internal shocks dissipate more energy. Hence, the internal shocks are mainly responsible for gas thermalization as well as CR acceleration. In fact, internal shocks with $2 \la M \la 4$ contribute $\sim 1/2$ of the total dissipation. Using a nonlinear diffusive shock acceleration model for CR protons, we estimate the ratio of CR energy to gas thermal energy dissipated at cosmological shock waves to be $\sim1/2$ through the history of the universe. Our result supports scenarios in which the intracluster medium contains energetically significant populations of CRs.

astro-ph↗