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arXiv 2023年

作者： Krawciw, Benjamin Carr, Lincoln D. Behn, Cecilia Diniz Department of Applied Mathematics and Statistics Colorado School of Mines GoldenCO United States Department of Physics Colorado School of Mines GoldenCO United States Quantum Engineering Program Colorado School of Mines GoldenCO United States

Complex network theory has focused on properties of networks with real-valued edge weights. However, in signal transfer networks, such as those representing the transfer of light across an interferometer, complex-valued edge weights are needed to represent the manipulation of the signal in both magnitude and phase. These complex-valued edge weights introduce interference into the signal transfer, but it is unknown how such interference affects network properties such as small-worldness. To address this gap, we have introduced a small-world interferometer network model with complex-valued edge weights and generalized existing network measures to define the interferometric clustering coefficient, the apparent path length, and the interferometric small-world coefficient. Using high-performance computing resources, we generated a large set of small-world interferometers over a wide range of parameters in system size, nearest-neighbor count, and edge-weight phase and computed their interferometric network measures. We found that the interferometric small-world coefficient depends significantly on the amount of phase on complex-valued edge weights: for small edge-weight phases, constructive interference led to a higher interferometric small-world coefficient;while larger edge-weight phases induced destructive interference which led to a lower interferometric small-world coefficient. Thus, for the small-world interferometer model, interferometric measures are necessary to capture the effect of interference on signal transfer. This model is an example of the type of problem that necessitates interferometric measures, and applies to any wave-based network including quantum networks. Copyright © 2023, The Authors. All rights reserved.

关键词： Interferometers

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Following marginal stability manifolds in quasilinear dynamical reductions of multiscale flows in two space dimensions

arXiv

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arXiv 2024年

作者： Ferraro, Alessia Chini, Gregory P. Schneider, Gregory M. École Polytechnique Fédérale de Lausanne LausanneCH-1015 Switzerland Nordita Royal Institute of Technology Stockholm University Stockholm106 91 Sweden Department of Mechanical Engineering and Program in Integrated Applied Mathematics University of New Hampshire DurhamNH03824 United States

A two-dimensional (2D) extension of a recently developed formalism for slow-fast quasilinear (QL) systems subject to fast instabilities is derived. Prior work has demonstrated that the emergent dynamics of these systems is characterized by a slow evolution of (suitably defined) mean fields coupled to marginally stable, fast fluctuation fields. By exploiting this emergent behavior, an efficient hybrid fast-eigenvalue/slow-initial-value solution algorithm can be developed in which the amplitude of the fast fluctuations is slaved to the slowly evolving mean fields to ensure marginal stability-and temporal scale separation-is maintained. For 2D systems that are spatially-extended in one direction, the fluctuation eigenfunctions are labeled by their Fourier wavenumbers characterizing spatial variability in that direction, and the marginal mode(s) also must coincide with the fastest-growing mode(s) over all admissible Fourier wavenumbers. Here, we introduce two alternative but equivalent procedures for deriving an ordinary differential equation governing the slow evolution of the wavenumber of the fastest-growing fluctuation mode that simultaneously must be slaved to the mean dynamics to ensure the mode has zero growth rate. We illustrate the procedure in the context of a 2D model partial differential equation that shares certain attributes with the equations governing strongly stratified shear flows and other strongly constrained forms of geophysical turbulence in extreme parameter regimes. The slaved evolution follows one or more marginal stability manifolds, which constitute select state-space structures that are not invariant under the full flow dynamics yet capture quasi-coherent structures in physical space in a manner analogous to invariant solutions identified in, e.g., transitionally-turbulent shear flows. Accordingly, we propose that marginal stability manifolds are central organizing structures in a dynamical systems description of certain classes of multiscale

关键词： Ordinary differential equations

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A qualitative study of the dynamic behavior for adaptive gradient algorithms

arXiv

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arXiv 2020年

作者： Ma, Chao Wu, Lei Weinan, E. Department of Mathematics Stanford University Program in Applied and Computational Mathematics Princeton University Program in Applied and Computational Mathematics Department of Mathematics Princeton University

The dynamic behavior of RMSprop and Adam algorithms is studied through a combination of careful numerical experiments and theoretical explanations. Three types of qualitative features are observed in the training loss curve: fast initial convergence, oscillations, and large spikes in the late phase. The sign gradient descent (signGD) flow, which is the limit of Adam when taking the learning rate to 0 while keeping the momentum parameters fixed, is used to explain the fast initial convergence. For the late phase of Adam, three different types of qualitative patterns are observed depending on the choice of the hyper-parameters: oscillations, spikes, and divergence. In particular, Adam converges much smoother and even faster when the values of the two momentum factors are close to each other. This observation is particularly important for scientific computing tasks, for which the training process usually proceeds into the high precision regime. Copyright © 2020, The Authors. All rights reserved.

关键词： Gradient methods

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Regimes of stratified turbulence at low Prandtl number

arXiv

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arXiv 2023年

作者： Shah, Kasturi Chini, Gregory P. Caulfield, Colm-Cille P. Garaud, Pascale Department of Applied Mathematics and Theoretical Physics University of Cambridge CambridgeCB3 0WA United Kingdom Department of Earth Atmospheric and Planetary Sciences Massachusetts Institute of Technology CambridgeMA02139 United States Program in Integrated Applied Mathematics University of New Hampshire DurhamNH03824 United States Department of Mechanical Engineering University of New Hampshire DurhamNH03824 United States Institute for Energy and Environmental Flows University of Cambridge CambridgeCB3 0EZ United Kingdom Department of Applied Mathematics Baskin School of Engineering University of California Santa Cruz Santa CruzCA95064 United States

Quantifying transport by strongly stratified turbulence in low Prandtl number (Pr) fluids is critically important for the development of better models for the structure and evolution of stellar and planetary interiors. Motivated by recent numerical simulations showing strongly anisotropic flows suggestive of scale-separated dynamics, we perform a multiscale asymptotic analysis of the governing equations. We find that, in all cases, the resulting slow-fast systems take a quasilinear form. Our analysis also reveals the existence of several distinct dynamical regimes depending on the emergent buoyancy Reynolds and Péclet numbers, Reb = α2Re and Peb = PrReb, respectively, where α is the aspect ratio of the large-scale turbulent flow structures, and Re is the outer scale Reynolds number. Scaling relationships relating the aspect ratio, the characteristic vertical velocity, and the strength of the stratification (measured by the Froude number Fr) naturally emerge from the analysis. When Peb α, the dynamics at all scales is dominated by buoyancy diffusion, and our results recover the scaling laws empirically obtained from direct numerical simulations by Cope et al. (2020). For Peb ≥ O(1), diffusion is negligible (or at least subdominant) at all scales and our results are consistent with those of Chini et al. (2022) for strongly stratified geophysical turbulence at Pr = O(1). Finally, we have identified a new regime for α Peb 1, in which slow, large scales are diffusive while fast, small scales are not. We conclude by presenting a map of parameter space that clearly indicates the transitions between isotropic turbulence, non-diffusive stratified turbulence, diffusive stratified turbulence and viscously-dominated flows, and by proposing parameterisations of the buoyancy flux, mixing efficiency and turbulent diffusion coefficient for each regime. © 2023, CC BY.

关键词： Buoyancy

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Aspect-ratio dependent heat transport by baroclinic acoustic streaming

arXiv

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arXiv 2024年

作者： Massih, Jacques Abdul Mushthaq, Remil Michel, Guillaume Chini, Gregory P. Program in Integrated Applied Mathematics University of New Hampshire DurhamNH03824 United States Department of Mechanical Engineering University of New Hampshire DurhamNH03824 United States Sorbonne Université CNRS Institut Jean Le Rond d’Alembert ParisF-75005 France

Standing acoustic waves have been known to generate Eulerian time-mean ‘streaming’ flows at least since the seminal investigation of Lord Rayleigh in the 1880s. Nevertheless, a recent body of numerical and experimental evidence has shown that inhomogeneities in the ambient density distribution lead to much faster flows than arise in classical Rayleigh streaming. The emergence of these unusually strong flows creates new opportunities to enhance heat transfer in systems in which convective cooling cannot otherwise be easily achieved. To assess this possibility, a theoretical study of acoustic streaming in an ideal gas confined in a rectangular channel with top and bottom walls maintained at fixed but differing temperatures is performed. A two time-scale system of equations is utilized to efficiently capture the coupling between the fast acoustic waves and the slowly evolving streaming flow, enabling strongly nonlinear regimes to be accessed. A large suite of numerical simulations is carried out to probe the streaming dynamics, to highlight the critical role played by baroclinically-generated wave vorticity and to quantify the additional heat flux induced by the standing acoustic wave. Proper treatment of the two-way coupling between the waves and mean flow is found to be essential for convergence to a self-consistent steady-state, and the variation of the resulting acoustically-enhanced steady-state heat flux with both the amplitude of the acoustic wave and the O(1) aspect ratio of the channel is documented. For certain parameters, heat fluxes almost two orders of magnitude larger than those realizable by conduction alone can be attained. © 2024, CC BY-SA.

关键词： Heat flux

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Modeling of Measurement Error in Financial Returns Data

arXiv

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arXiv 2024年

作者： Jasra, Ajay Maama, Mohamed Mijatović, Aleksandar School of Data Science The Chinese University of Hong Kong Shenzhen China Applied Mathematics and Computational Science Program Computer Electrical and Mathematical Sciences and Engineering Division King Abdullah University of Science and Technology Thuwal23955-6900 Saudi Arabia Department of Statistics University of Warwick United Kingdom

In this paper we consider the modeling of measurement error for fund returns data. In particular, given access to a time-series of discretely observed log-returns and the associated maximum over the observation period, we develop a stochastic model which models the true log-returns and maximum via a Lévy process and the data as a measurement error there-of. The main technical difficulty of trying to infer this model, for instance Bayesian parameter estimation, is that the joint transition density of the return and maximum is seldom known, nor can it be simulated exactly. Based upon the novel stick breaking representation of [12] we provide an approximation of the model. We develop a Markov chain Monte Carlo (MCMC) algorithm to sample from the Bayesian posterior of the approximated posterior and then extend this to a multilevel MCMC method which can reduce the computational cost to approximate posterior expectations, relative to ordinary MCMC. We implement our methodology on several applications including for real data. Copyright © 2024, The Authors. All rights reserved.

关键词： Markov chains

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Generalization error of GAN from the discriminator's perspective

arXiv

引用

arXiv 2021年

作者： Yang, Hongkang Weinan, E. Program in Applied and Computational Mathematics Princeton University United States Department of Mathematics Princeton University United States Beijing Institute of Big Data Research China

The generative adversarial network (GAN) is a well-known model for learning high-dimensional distributions, but the mechanism for its generalization ability is not understood. In particular, GAN is vulnerable to the memorization phenomenon, the eventual convergence to the empirical distribution. We consider a simplified GAN model with the generator replaced by a density, and analyze how the discriminator contributes to generalization. We show that with early stopping, the generalization error measured by Wasserstein metric escapes from the curse of dimensionality, despite that in the long term, memorization is inevitable. In addition, we present a hardness of learning result for *** Codes 68T07, 62G07, 60-08 Copyright © 2021, The Authors. All rights reserved.

关键词： Probability distributions

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Hole Statistics of Equilibrium 2D and 3D Hard-Sphere Crystals

arXiv

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arXiv 2024年

作者： Wang, Haina Huse, David A. Torquato, Salvatore Department of Chemistry Princeton University PrincetonNJ08544 United States Department of Physics Princeton University PrincetonNJ08544 United States Department of Chemistry Department of Physics Princeton Institute of Materials and Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

The probability of finding a spherical "hole" of a given radius r contains crucial structural information about many-body systems. Such hole statistics, including the void conditional nearest-neighbor probability functions GV (r), have been well studied for hard-sphere fluids in d-dimensional Euclidean space Rd. However, little is known about these functions for hard-sphere crystals for values of r beyond the hard-sphere diameter, as large holes are extremely rare in crystal phases. To overcome these computational challenges, we introduce a biased-sampling scheme that accurately determines hole statistics for equilibrium hard spheres on ranges of r that far extend those that could be previously explored. We discover that GV (r) in crystal and hexatic states exhibits oscillations whose amplitudes increase rapidly with the packing fraction, which stands in contrast to GV (r) for fluid states that is monotonic in r. The oscillations in GV (r) for 2D crystals are strongly correlated with the local orientational order metric in the vicinity of the holes, and variations in GV (r) for 3D states indicate a transition between tetrahedral and octahedral holes, demonstrating the power of GV (r) as a probe of local coordination geometry. To further study the statistics of interparticle spacing in hard-sphere systems, we compute the local packing fraction distribution f(l) of Delaunay cells, and find that for d ≤ 3, the excess kurtosis of f(l) switches sign at a certain transitional global packing fraction. Our investigation facilitates the study of structural and bulk properties of materials that involve the creation of rare large holes, such as the solubility of alloys. © 2024, CC BY.

关键词： Spheres

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Deep Learning based Fault Diagnosis in a Grid-Connected Photovoltaic Systems 19

Deep Learning based Fault Diagnosis in a Grid-Connected Phot...

引用

19th IEEE International Multi-Conference on Systems, Signals and Devices, SSD 2022

作者： Hichri, Amal Hajji, Mansour Mansouri, Majdi Bouzrara, Kais Nounou, Hazem Nounou, Mohamed Higher Institute of Applied Sciences and Technology of Kasserine Kasserine Tunisia Electrical and Computer Engineering Program Texas A&m University at Qatar Doha23874 Qatar Prince Sultan University Department of Mathematics and Sciences Riyadh11586 Saudi Arabia Signal Processing and Image National Engineering School of Monastir Research Laboratory of Automation 5019 Tunisia Chemical Engineering Program Texas A&m University at Qatar Doha23874 Qatar

ISBN: (数字)9781665471084

ISBN: (纸本)9781665471084

Photovoltaic (PV) system fault diagnosis is crucial because it helps PV system operators reduce energy and income losses. It also decreases the risk of fire and electric shock from PV system failures. Thus, the implementation of fault diagnosis tool is required. The most well-known techniques are machine learning models. Although these techniques provided well in terms of diagnosis and classification accuracy, they rely on manual feature extraction, which is time consuming, expensive, and requires diagnostic skill. Manual feature extraction is a problem that has to be addressed. This paper therefore presents a PV system fault diagnosis techniques that can automatically extract features from raw data in order to classify PV system faults. The implemented methods include long short-term memory (LSTM) bidirectional long short-term memory (BiLSTM) networks, which are deep learning based algorithms. Finally, the presented fault diagnosis approaches showed a good diagnosis performances when applied to an emulated GCPV system. © 2022 IEEE.

关键词： Failure analysis

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Refined least squares for support recovery

arXiv

引用

arXiv 2021年

作者： Lindenbaum, Ofir Steinerberger, Stefan Program in Applied Mathematics Yale University New HavenCT06511 United States Department of Mathematics University of Washington SeattleWA98195 United States

We study the problem of exact support recovery based on noisy observations and present Refined Least Squares (RLS). Given a set of noisy measurement y = Xθ∗ + ω, and X ∈ RN×D which is a (known) Gaussian matrix and ω ∈ RN is an (unknown) Gaussian noise vector, our goal is to recover the support of the (unknown) sparse vector θ∗ ∈ {-1, 0, 1}D. To recover the support of the θ∗ we use an average of multiple least squares solutions, each computed based on a subset of the full set of equations. The support is estimated by identifying the most significant coefficients of the average least squares solution. We demonstrate that in a wide variety of settings our method outperforms state-of-the-art support recovery algorithms. © 2021, CC BY.

关键词： Recovery

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