arXiv subjects

Paul G. Beckman

Publications and source records attributed to Paul G. Beckman.

7 recordsLinked to original sources

arXiv2026

A recursive butterfly factorization with optimality guarantees

We formalize a recursive format for representing a butterfly matrix. This new format naturally leads to a simple recursive algorithm for computing a quasi-optimal butterfly approximation to an arbitrary $N \times N$ matrix $A$. When the entries of $A$ are explicitly available, we show that the algorithm computes a butterfly matrix $B$ in $O(N^2)$ operations with approximation error $\|A - B\|_F$ at most a $O(\sqrt{\log(N)})$ factor away from that of the best possible approximation by a butterfly matrix. We also develop a matrix-free variant of the method, which uses $\widetilde{O}(\sqrt{N})$ matrix-vector products and $\widetilde{O}(N)$ working memory and, with high probability, returns a butterfly approximation with Frobenius norm error within a $O(N^{1/4})$-factor of the optimal error. We show that the algorithm is a reformulation of the hybrid butterfly factorization approach presented in [Liu et. al.; SISC, 43 (2021)]. Our paper therefore provides the first theoretical quasi-optimality guarantee for that algorithm.

math.NA↗

arXiv2026

A Butterfly-Accelerated Manifold Harmonic Transform

The eigenfunctions of the Laplacian are a natural basis of functions for many tasks in computational mathematics. On the circle and sphere, the eigenfunctions are given by complex periodic exponentials and spherical harmonics, respectively, and much work has been done to develop fast algorithms for analyzing and synthesizing data in these bases. In this work, we generalize these special-case transforms to Laplace-Beltrami eigenfunctions of arbitrary surfaces, referred to as manifold harmonics. The resulting fast algorithm for computing linear combinations of the manifold harmonics is based on a butterfly factorization, which hierarchically compresses the transform matrix by constructing nested low-rank approximations of carefully selected submatrices. Several numerical examples are provided which demonstrate the speedups and reduction in memory requirements achieved by our algorithm for a variety of geometries, discretizations, and applications. In addition, a detailed analysis of the algorithm is given in the case that the underlying manifold is the flat periodic square.

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arXiv2025

Fast nonparametric spectral density estimation from irregularly sampled data

We introduce a nonparametric spectral density estimator for continuous-time and continuous-space processes measured at fully irregular locations. Our estimator is constructed using a weighted nonuniform Fourier sum whose weights yield a high-accuracy quadrature rule with respect to a user-specified window function. The resulting estimator significantly reduces the aliasing seen in periodogram approaches and least squares spectral analysis, sidesteps the dangers of ill-conditioning of the nonuniform Fourier inverse problem, and can be adapted to a wide variety of irregular sampling settings. We describe methods for rapidly computing the necessary weights in various settings, making the estimator scalable to large datasets. We then provide a theoretical analysis of sources of bias, and close with demonstrations of the method's efficacy, including for processes that exhibit very slow spectral decay and are observed at up to a million locations in multiple dimensions.

stat.ME↗

arXiv2024

A Nonuniform Fast Hankel Transform

We describe a fast algorithm for computing discrete Hankel transforms of moderate orders from $n$ nonuniform points to $m$ nonuniform frequencies in $O((m+n)\log\min(n,m))$ operations. Our approach combines local and asymptotic Bessel function expansions with nonuniform fast Fourier transforms. The order of each expansion is adjusted automatically according to error analysis to obtain any desired precision $\varepsilon$. Several numerical examples are provided which demonstrate the speed and accuracy of the algorithm in multiple regimes and applications.

math.NA↗

arXiv2024

Fast Adaptive Fourier Integration for Spectral Densities of Gaussian Processes

The specification of a covariance function is of paramount importance when employing Gaussian process models, but the requirement of positive definiteness severely limits those used in practice. Designing flexible stationary covariance functions is, however, straightforward in the spectral domain, where one needs only to supply a positive and symmetric spectral density. In this work, we introduce an adaptive integration framework for efficiently and accurately evaluating covariance functions and their derivatives at irregular locations directly from \textit{any} continuous, integrable spectral density. In order to make this approach computationally tractable, we employ high-order panel quadrature, the nonuniform fast Fourier transform, and a Nyquist-informed panel selection heuristic, and derive novel algebraic truncation error bounds which are used to monitor convergence. As a result, we demonstrate several orders of magnitude speedup compared to naive uniform quadrature approaches, allowing us to evaluate covariance functions from slowly decaying, singular spectral densities at millions of locations to a user-specified tolerance in seconds on a laptop. We then apply our methodology to perform gradient-based maximum likelihood estimation using a previously numerically infeasible long-memory spectral model for wind velocities below the atmospheric boundary layer.

stat.CO↗

arXiv2022

Scalable Computations for Nonstationary Gaussian Processes

Nonstationary Gaussian process models can capture complex spatially varying dependence structures in spatial datasets. However, the large number of observations in modern datasets makes fitting such models computationally intractable with conventional dense linear algebra. In addition, derivative-free or even first-order optimization methods can be slow to converge when estimating many spatially varying parameters. We present here a computational framework that couples an algebraic block-diagonal plus low-rank covariance matrix approximation with stochastic trace estimation to facilitate the efficient use of second-order solvers for maximum likelihood estimation of Gaussian process models with many parameters. We demonstrate the effectiveness of these methods by simultaneously fitting 192 parameters in the popular nonstationary model of Paciorek and Schervish using 107,600 sea surface temperature anomaly measurements.

stat.CO↗

arXiv2020

A Shift Selection Strategy for Parallel Shift-Invert Spectrum Slicing in Symmetric Self-Consistent Eigenvalue Computation

The central importance of large scale eigenvalue problems in scientific computation necessitates the development of massively parallel algorithms for their solution. Recent advances in dense numerical linear algebra have enabled the routine treatment of eigenvalue problems with dimensions on the order of hundreds of thousands on the world's largest supercomputers. In cases where dense treatments are not feasible, Krylov subspace methods offer an attractive alternative due to the fact that they do not require storage of the problem matrices. However, demonstration of scalability of either of these classes of eigenvalue algorithms on computing architectures capable of expressing massive parallelism is non-trivial due to communication requirements and serial bottlenecks, respectively. In this work, we introduce the SISLICE method: a parallel shift-invert algorithm for the solution of the symmetric self-consistent field (SCF) eigenvalue problem. The SISLICE method drastically reduces the communication requirement of current parallel shift-invert eigenvalue algorithms through various shift selection and migration techniques based on density of states estimation and k-means clustering, respectively. This work demonstrates the robustness and parallel performance of the SISLICE method on a representative set of SCF eigenvalue problems and outlines research directions which will be explored in future work.

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