arXiv subjects

Aaron Myers

Publications and source records attributed to Aaron Myers.

4 recordsLinked to original sources

arXiv2026

Hierarchical Spline-Based Bayesian Beta-Binomial Regression for Estimating Time-Varying Risk in Power Outages

We propose a hierarchical Bayesian model for estimating time-varying outage risk from county-level power outage data. The model combines cubic B-spline basis functions with a Beta-Binomial likelihood to capture smooth, nonlinear recovery trajectories while accommodating overdispersion in observed customer counts. A shared hyperprior on the Beta-Binomial concentration parameter enables hierarchical shrinkage across geographically indexed groups, allowing sparse or short-lived events to borrow statistical strength from the broader population. Posterior inference is conducted via the No-U-Turn Sampler (NUTS) in PyMC, yielding full posterior distributions over latent outage probabilities and derived resilience metrics including the area under the risk curve (AUC). We assess predictive performance using posterior predictive coverage, RMSE, and leave-one-out cross-validation, and demonstrate the model across a heterogeneous set of outage events in southern Wisconsin drawn from the EAGLE-I power outage monitoring platform. A direct comparison against naive trapezoidal AUC estimation confirms that the posterior mean recovers the same point estimates as deterministic integration while providing calibrated uncertainty quantification that deterministic approaches structurally cannot. The framework offers utilities and emergency planners a principled tool for benchmarking recovery dynamics and comparing outage events under uncertainty.

stat.AP↗

arXiv2020

Sequential Ensemble Transform for Bayesian Inverse Problems

We present the Sequential Ensemble Transform (SET) method, an approach for generating approximate samples from a Bayesian posterior distribution. The method explores the posterior distribution by solving a sequence of discrete optimal transport problems to produce a series of transport plans which map prior samples to posterior samples. We prove that the sequence of Dirac mixture distributions produced by the SET method converges weakly to the true posterior as the sample size approaches infinity. Furthermore, our numerical results indicate that, when compared to standard Sequential Monte Carlo (SMC) methods, the SET approach is more robust to the choice of Markov mutation kernels and requires less computational efforts to reach a similar accuracy when used to explore complex posterior distributions. Finally, we describe adaptive schemes that allow to completely automate the use of the SET method.

stat.ME↗

arXiv2018

Scalable matrix-free adaptive product-convolution approximation for locally translation-invariant operators

We present an adaptive grid matrix-free operator approximation scheme based on a "product-convolution" interpolation of convolution operators. This scheme is appropriate for operators that are locally translation-invariant, even if these operators are high-rank or full-rank. Such operators arise in Schur complement methods for solving partial differential equations (PDEs), as Hessians in PDE-constrained optimization and inverse problems, as integral operators, as covariance operators, and as Dirichlet-to-Neumann maps. Constructing the approximation requires computing the impulse responses of the operator to point sources centered on nodes in an adaptively refined grid of sample points. A randomized a-posteriori error estimator drives the adaptivity. Once constructed, the approximation can be efficiently applied to vectors using the fast Fourier transform. The approximation can be efficiently converted to hierarchical matrix ($H$-matrix) format, then inverted or factorized using scalable $H$-matrix arithmetic. The quality of the approximation degrades gracefully as fewer sample points are used, allowing cheap lower quality approximations to be used as preconditioners. This yields an automated method to construct preconditioners for locally translation-invariant Schur complements. We directly address issues related to boundaries and prove that our scheme eliminates boundary artifacts. We test the scheme on a spatially varying blurring kernel, on the non-local component of an interface Schur complement for the Poisson operator, and on the data misfit Hessian for an advection dominated advection-diffusion inverse problem. Numerical results show that the scheme outperforms existing methods.

math.NA↗

arXiv2016

A Data-Scalable Randomized Misfit Approach for Solving Large-Scale PDE-Constrained Inverse Problems

A randomized misfit approach is presented for the efficient solution of large-scale PDE-constrained inverse problems with high-dimensional data. The purpose of this paper is to offer a theory-based framework for random projections in this inverse problem setting. The stochastic approximation to the misfit is analyzed using random projection theory. By expanding beyond mean estimator convergence, a practical characterization of randomized misfit convergence can be achieved. The theoretical results developed hold with any valid random projection in the literature. The class of feasible distributions is broad yet simple to characterize compared to previous stochastic misfit methods. This class includes very sparse random projections which provide additional computational benefit. A different proof for a variant of the Johnson-Lindenstrauss lemma is also provided. This leads to a different intuition for the $O(\epsilon^{-2})$ factor in bounds for Johnson-Lindenstrauss results. The main contribution of this paper is a theoretical result showing the method guarantees a valid solution for small reduced misfit dimensions. The interplay between Johnson-Lindenstrauss theory and Morozov's discrepancy principle is shown to be essential to the result. The computational cost savings for large-scale PDE-constrained problems with high- dimensional data is discussed. Numerical verification of the developed theory is presented for model problems of estimating a distributed parameter in an elliptic partial differential equation. Results with different random projections are presented to demonstrate the viability and accuracy of the proposed approach.

math.NA↗