Eric Hallman
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View article: Extremal bounds for Gaussian trace estimation
Extremal bounds for Gaussian trace estimation Open
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…
View article: Randomized Algorithms for Rounding in the Tensor-Train Format
Randomized Algorithms for Rounding in the Tensor-Train Format Open
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 equati…
View article: Krylov-aware stochastic trace estimation
Krylov-aware stochastic trace estimation Open
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-…
View article: Precision-aware Deterministic and Probabilistic Error Bounds for Floating Point Summation
Precision-aware Deterministic and Probabilistic Error Bounds for Floating Point Summation Open
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 …
View article: Monte Carlo Methods for Estimating the Diagonal of a Real Symmetric Matrix
Monte Carlo Methods for Estimating the Diagonal of a Real Symmetric Matrix Open
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 C…
View article: Randomized algorithms for rounding in the Tensor-Train format
Randomized algorithms for rounding in the Tensor-Train format Open
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 equati…
View article: Deterministic and Probabilistic Error Bounds for Floating Point Summation Algorithms
Deterministic and Probabilistic Error Bounds for Floating Point Summation Algorithms Open
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 probab…
View article: A Refined Probabilistic Error Bound for Sums
A Refined Probabilistic Error Bound for Sums Open
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 …
View article: A Block Bidiagonalization Method for Fixed-Accuracy Low-Rank Matrix Approximation
A Block Bidiagonalization Method for Fixed-Accuracy Low-Rank Matrix Approximation Open
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…
View article: A Block Bidiagonalization Method for Fixed-Precision Low-Rank Matrix Approximation.
A Block Bidiagonalization Method for Fixed-Precision Low-Rank Matrix Approximation. Open
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…
View article: Diffuse radio emission in the galaxy cluster SPT-CL J2031−4037: a steep-spectrum intermediate radio halo?
Diffuse radio emission in the galaxy cluster SPT-CL J2031−4037: a steep-spectrum intermediate radio halo? Open
The advent of sensitive low-frequency radio observations has revealed a number of diffuse radio objects with peculiar properties that are challenging our understanding of the physics of the intracluster medium. Here, we report the discover…
View article: Error Estimates for Least-Squares Problems
Error Estimates for Least-Squares Problems Open
In this thesis we consider error estimates for a family of iterative algorithms for solving the least-squares problem \\min_x \\|Ax-b\\|_2 based on the Golub-Kahan-Lanczos bidiagonalization process. Given a lower bound on the smallest sing…
View article: Length Scales and Turbulent Properties of Magnetic Fields in Simulated Galaxy Clusters
Length Scales and Turbulent Properties of Magnetic Fields in Simulated Galaxy Clusters Open
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 approx…