arXiv subjects

William Bennett

Publications and source records attributed to William Bennett.

8 recordsLinked to original sources

arXiv2026

Weak-DMD: A Galerkin approach to the problem of noise in the Dynamic Mode Decomposition algorithm

Dynamic Mode Decomposition (DMD) is a data-driven method for approximating the spatiotemporal modes of a system. The eigenvectors and eigenvalues of the system are approximated from a series of time-snapshots of the state variables. The standard formulation of DMD is subject to strict assumptions concerning the time-spacing of the snapshots and is biased by measurement noise. Variations on the method have been developed to address these shortcomings, but the problem is still open. Motivated by the effectiveness of Galerkin methods in the field of model discovery, a weak formulation of DMD is presented, weak-DMD. Weak-DMD precludes timestep considerations and also filters noise. Results for two nuclear engineering applications and the flow of fluid past a cylinder are given and compared with a state of the art DMD algorithm.

cs.CE↗

arXiv2024

MedTsLLM: Leveraging LLMs for Multimodal Medical Time Series Analysis

The complexity and heterogeneity of data in many real-world applications pose significant challenges for traditional machine learning and signal processing techniques. For instance, in medicine, effective analysis of diverse physiological signals is crucial for patient monitoring and clinical decision-making and yet highly challenging. We introduce MedTsLLM, a general multimodal large language model (LLM) framework that effectively integrates time series data and rich contextual information in the form of text to analyze physiological signals, performing three tasks with clinical relevance: semantic segmentation, boundary detection, and anomaly detection in time series. These critical tasks enable deeper analysis of physiological signals and can provide actionable insights for clinicians. We utilize a reprogramming layer to align embeddings of time series patches with a pretrained LLM's embedding space and make effective use of raw time series, in conjunction with textual context. Given the multivariate nature of medical datasets, we develop methods to handle multiple covariates. We additionally tailor the text prompt to include patient-specific information. Our model outperforms state-of-the-art baselines, including deep learning models, other LLMs, and clinical methods across multiple medical domains, specifically electrocardiograms and respiratory waveforms. MedTsLLM presents a promising step towards harnessing the power of LLMs for medical time series analysis that can elevate data-driven tools for clinicians and improve patient outcomes.

cs.LG↗

arXiv2024

High Energy Density Radiative Transfer in the Diffusion Regime with Fourier Neural Operators

Radiative heat transfer is a fundamental process in high energy density physics and inertial fusion. Accurately predicting the behavior of Marshak waves across a wide range of material properties and drive conditions is crucial for design and analysis of these systems. Conventional numerical solvers and analytical approximations often face challenges in terms of accuracy and computational efficiency. In this work, we propose a novel approach to model Marshak waves using Fourier Neural Operators (FNO). We develop two FNO-based models: (1) a base model that learns the mapping between the drive condition and material properties to a solution approximation based on the widely used analytic model by Hammer & Rosen (2003), and (2) a model that corrects the inaccuracies of the analytic approximation by learning the mapping to a more accurate numerical solution. Our results demonstrate the strong generalization capabilities of the FNOs and show significant improvements in prediction accuracy compared to the base analytic model.

physics.comp-ph↗

arXiv2023

Uncertainty benchmarks for time-dependent transport problems

Verification solutions for uncertainty quantification are presented for time dependent transport problems where $c$, the scattering ratio, is uncertain. The method of polynomial chaos expansions is employed for quick and accurate calculation of the quantities of interest and uncollided solutions are used to treat part of the uncertainty calculation analytically. We find that approximately six moments in the polynomial expansion are required to represent the solutions to these problems accurately. Additionally, the results show that if the uncertainty interval spans c=1, which means it is uncertain whether the system is multiplying or not, the confidence interval will grow in time. Finally, since the QoI is a strictly increasing function, the percentile values are known and can be used to verify the accuracy of the expansion. These results can be used to test UQ methods for time-dependent transport problems.

cs.CE↗

arXiv2023

Benchmark solutions for radiative transfer with a moving mesh and exact uncollided source treatments

The set of benchmark solutions used in the thermal radiative transfer community suffer some coverage gaps, in particular nonlinear, non-equilibrium problems. Also, there are no non-equilibrium, optically thick benchmarks. These shortcomings motivated the development of a numerical method free from the requirement of linearity and easily able to converge on smooth optically thick problems, a moving mesh Discontinuous Galerkin (DG) framework that utilizes an uncollided source treatment. Having already proven this method on time dependent scattering transport problems, we present here solutions to non-equilibrium thermal radiative transfer problems for familiar linearized systems together with more physical nonlinear systems in both optically thin and thick regimes, including both the full transport and the $S_2$/$P_1$ solution. Geometric convergence is observed for smooth sources at all times and some nonsmooth sources at late times when there is local equilibrium. Also, accurate solutions are achieved for step sources when the solution is not smooth.

cs.CE↗

arXiv2022

The Development of a Multi-Physics Approach for Modelling the Response of Aerospace Fastener Assemblies to Lightning Attachment

This work is concerned with the development of a numerical modelling approach for studying the time-accurate response of aerospace fasteners subjected to high electrical current loading from a simulated lightning strike. The electromagnetic, thermal and elastoplastic response of individual fastener components is captured by this method allowing a critical analysis of fastener design and material layering. Under high electrical current loading, ionisation of gas filled cavities in the fastener assembly can lead to viable current paths across internal voids. This ionisation can lead to localised pockets of high pressure plasma through the Joule heating effect. The multi-physics approach developed in this paper extends an existing methodology that allows a two-way dynamic non-linear coupling of the plasma arc, the titanium aerospace fastener components, the surrounding aircraft panels, the internal supporting structure and internal plasma-filled cavities. Results from this model are compared with experimental measurements of a titanium fastener holding together carbon composite panels separated by thin dielectric layers. The current distribution measurements are shown to be accurately reproduced. A parameter study is used to assess the internal cavity modelling strategy and to quantify the relation between the internal cavity plasma pressure, the electrical current distribution and changes in the internal cavity geometry.

physics.comp-ph↗

arXiv2022

Accurate solutions to time dependent transport problems with a moving mesh and exact uncollided source treatment

For the purpose of finding benchmark quality solutions to time dependent Sn transport problems, we develop a numerical method in a Discontinuous Galerkin (DG) framework that utilizes time dependent cell edges, which we call a moving mesh, and an uncollided source treatment. The DG method for discretizing space is a powerful solution technique on smooth problems and is robust on non-smooth problems. In order to realize the potential of the DG method to spectrally resolve smooth problems, our moving mesh and uncollided source treatment is devised to circumvent discontinuities in the solution or the first derivative of the solutions that are admitted in transport calculations. The resulting method achieves spectral convergence on smooth problems, like a standard DG implementation. When applied to problems with nonsmooth sources that induce discontinuities, our moving mesh, uncollided source method returns a significantly more accurate solution than the standard DG method. On problems with smooth sources, we observe spectral convergence even in problems with wave fronts. In problems where the angular flux is inherently non-smooth, as in Ganapol's (2001) well known plane pulse benchmark, we do not observe an elevated order of accuracy when compared with static meshes, but there is a reduction in error that is nearly three orders of magnitude.

cs.CE↗

arXiv2022

Benchmarks for infinite medium, time dependent transport problems with isotropic scattering

The widely used AZURV1 transport benchmarks package provides a suite of solutions to isotropic scattering transport problems with a variety of initial conditions (Ganapol 2001). Most of these solutions have an initial condition that is a Dirac delta function in space; as a result these benchmarks are challenging problems to use for verification tests in computer codes. Nevertheless, approximating a delta function in simulation often leads to low orders of convergence and the inability to test the convergence of high-order numerical methods. While there are examples in the literature of integration of these solutions as Green's functions for the transport operator to produce results for more easily simulated sources, they are limited in scope and briefly explained. For a sampling of initial conditions and sources, we present solutions for the uncollided and collided scalar flux to facilitate accurate testing of source treatment in numerical solvers. The solution for the uncollided scalar flux is found in analytic form for some sources. Since integrating the Green's functions is often nontrivial, discussion of integration difficulty and workarounds to find convergent integrals is included. Additionally, our uncollided solutions can be used as source terms in verification studies, in a similar way to the method of manufactured solutions.

cs.CE↗