Gotowa bibliografia na temat „Asymptotic High-Dimensional Regime”

We study the interaction of in-plane elastic waves with imperfect interfaces composed of a periodic array of voids or cracks. An effective model is derived from high-order asymptotic analysis based on two-scale homogenization and matched asymptotic technique. In two-dimensional elasticity, we obtain jump conditions set on the in-plane displacements and normal stresses; the jumps involve in addition effective parameters provided by static, elementary problems being the equivalents of the cell problems in classical two-scale homogenization. The derivation of the model is conducted in the transient regime and its stability is guarantied by the positiveness of the effective interfacial energy. Spring models are envisioned as particular cases. It is shown that massless-spring models are recovered in the limit of small void thicknesses and collinear cracks. By contrast, the use of mass-spring model is justified at normal incidence, otherwise unjustified. We provide quantitative validations of our model and comparison with spring models by means of comparison with direct numerical calculations in the harmonic regime.

Abstract As modern machine learning models continue to advance the computational frontier, it has become increasingly important to develop precise estimates for expected performance improvements under different model and data scaling regimes. Currently, theoretical understanding of the learning curves (LCs) that characterize how the prediction error depends on the number of samples is restricted to either large-sample asymptotics ( m → ∞ ) or, for certain simple data distributions, to the high-dimensional asymptotics in which the number of samples scales linearly with the dimension ( m ∝ d ). There is a wide gulf between these two regimes, including all higher-order scaling relations m ∝ d r , which are the subject of the present paper. We focus on the problem of kernel ridge regression for dot-product kernels and present precise formulas for the mean of the test error, bias and variance, for data drawn uniformly from the sphere with isotropic random labels in the rth-order asymptotic scaling regime m → ∞ with m / d r held constant. We observe a peak in the LC whenever m ≈ d r / r ! for any integer r, leading to multiple sample-wise descent and non-trivial behavior at multiple scales. We include a colab (available at: https://tinyurl.com/2nzym7ym) notebook that reproduces the essential results of the paper.

Abstract Approximating significance scans of searches for new particles in high-energy physics experiments as Gaussian fields is a well-established way to estimate the trials factors required to quantify global significances. We propose a novel, highly efficient method to estimate the covariance matrix of such a Gaussian field. The method is based on the linear approximation of statistical fluctuations of the signal amplitude. For one-dimensional searches the upper bound on the trials factor can then be calculated directly from the covariance matrix. For higher dimensions, the Gaussian process described by this covariance matrix may be sampled to calculate the trials factor directly. This method also serves as the theoretical basis for a recent study of the trials factor with an empirically constructed set of Asmiov-like background datasets. We illustrate the method with studies of a H → γγ inspired model that was used in the empirical paper.

Deep networks are typically trained with many more parameters than the size of the training dataset. Recent empirical evidence indicates that the practice of overparameterization not only benefits training large models, but also assists – perhaps counterintuitively – building lightweight models. Specifically, it suggests that overparameterization benefits model pruning / sparsification. This paper sheds light on these empirical findings by theoretically characterizing the high-dimensional asymptotics of model pruning in the overparameterized regime. The theory presented addresses the following core question: ``should one train a small model from the beginning, or first train a large model and then prune?''. We analytically identify regimes in which, even if the location of the most informative features is known, we are better off fitting a large model and then pruning rather than simply training with the known informative features. This leads to a new double descent in the training of sparse models: growing the original model, while preserving the target sparsity, improves the test accuracy as one moves beyond the overparameterization threshold. Our analysis further reveals the benefit of retraining by relating it to feature correlations. We find that the above phenomena are already present in linear and random-features models. Our technical approach advances the toolset of high-dimensional analysis and precisely characterizes the asymptotic distribution of over-parameterized least-squares. The intuition gained by analytically studying simpler models is numerically verified on neural networks.

Abstract We study generalised linear regression and classification for a synthetically generated dataset encompassing different problems of interest, such as learning with random features, neural networks in the lazy training regime, and the hidden manifold model. We consider the high-dimensional regime and using the replica method from statistical physics, we provide a closed-form expression for the asymptotic generalisation performance in these problems, valid in both the under- and over-parametrised regimes and for a broad choice of generalised linear model loss functions. In particular, we show how to obtain analytically the so-called double descent behaviour for logistic regression with a peak at the interpolation threshold, we illustrate the superiority of orthogonal against random Gaussian projections in learning with random features, and discuss the role played by correlations in the data generated by the hidden manifold model. Beyond the interest in these particular problems, the theoretical formalism introduced in this manuscript provides a path to further extensions to more complex tasks.

Bulk-to-shear viscosity ratios of three orders of magnitude are often reported in carbon dioxide but are always neglected when predicting aerothermal loads in external (Mars exploration) or internal (turbomachinery, heat exchanger) turbulent flows. The recent (and first) numerical investigations of that matter suggest that the solenoidal turbulence kinetic energy is in fact well predicted despite this seemingly arbitrary simplification. The present work argues that such a conclusion may reflect limitations from the choice of configuration rather than provide a definite statement on the robustness of kinetic-energy transfers to the use of Stokes’ hypothesis. Two distinct asymptotic regimes (Euler–Landau and Stokes–Newton) in the eigenmodes of the Navier–Stokes equations are identified. In the Euler–Landau regime, the one captured by earlier studies, acoustic and entropy waves are damped by transport coefficients and the dilatational kinetic energy is dissipated, even more rapidly for high bulk-viscosity fluids and/or forcing frequencies. If the kinetic energy is initially or constantly injected through solenoidal motions, effects on the turbulence kinetic energy remain minor. However, in the Stokes–Newton regime, diffused bulk compressions and advected isothermal compressions are found to prevail and promote small-scale enstrophy via vorticity–dilatation correlations. In the absence of bulk viscosity, the transition to the Stokes–Newton regime occurs within the dissipative scales and is not observed in practice. In contrast, at high bulk viscosities, the Stokes–Newton regime can be made to overlap with the inertial range and disrupt the enstrophy at small scales, which is then dissipated by friction. Thus, flows with substantial inertial ranges and large bulk-to-shear viscosity ratios should experience enhanced transfers to small-scale solenoidal kinetic energy, and therefore faster dissipation rates leading to modifications of the heat-transfer properties. Observing numerically such transfers is still prohibitively expensive, and the present simulations are restricted to two-dimensional turbulence. However, the theory laid here offers useful guidelines to design experimental studies to track the Stokes–Newton regime and associated modifications of the turbulence kinetic energy, which are expected to persist in three-dimensional turbulence.

AbstractRecent observations of rapidly rotating stars have revealed the presence of regular patterns in their pulsation spectra. This has raised the question as to their physical origin, and, in particular, whether they can be explained by an asymptotic frequency formula for low-degree acoustic modes, as recently discovered through numerical calculations and theoretical considerations. In this context, a key question is whether compositional/density gradients can adversely affect such patterns to the point of hindering their identification. To answer this question, we calculate frequency spectra using two-dimensional ESTER stellar models. These models use a multi-domain spectral approach, allowing us to easily insert a compositional discontinuity while retaining a high numerical accuracy. We analyse the effects of such discontinuities on both the frequencies and eigenfunctions of pulsation modes in the asymptotic regime. We find that although there is more scatter around the asymptotic frequency formula, the semi-large frequency separation can still be clearly identified in a spectrum of low-degree acoustic modes.

We study convex empirical risk minimization for high-dimensional inference in binary linear classification under both discriminative binary linear models, as well as generative Gaussian-mixture models. Our first result sharply predicts the statistical performance of such estimators in the proportional asymptotic regime under isotropic Gaussian features. Importantly, the predictions hold for a wide class of convex loss functions, which we exploit to prove bounds on the best achievable performance. Notably, we show that the proposed bounds are tight for popular binary models (such as signed and logistic) and for the Gaussian-mixture model by constructing appropriate loss functions that achieve it. Our numerical simulations suggest that the theory is accurate even for relatively small problem dimensions and that it enjoys a certain universality property.

Abstract From the sampling of data to the initialisation of parameters, randomness is ubiquitous in modern Machine Learning practice. Understanding the statistical fluctuations engendered by the different sources of randomness in prediction is therefore key to understanding robust generalisation. In this manuscript we develop a quantitative and rigorous theory for the study of fluctuations in an ensemble of generalised linear models trained on different, but correlated, features in high-dimensions. In particular, we provide a complete description of the asymptotic joint distribution of the empirical risk minimiser for generic convex loss and regularisation in the high-dimensional limit. Our result encompasses a rich set of classification and regression tasks, such as the lazy regime of overparametrised neural networks, or equivalently the random features approximation of kernels. While allowing to study directly the mitigating effect of ensembling (or bagging) on the bias-variance decomposition of the test error, our analysis also helps disentangle the contribution of statistical fluctuations, and the singular role played by the interpolation threshold that are at the roots of the ‘double-descent’ phenomenon.

Cette thèse est consacrée à tester la présence de changements dans les paramètres d'un modèle de régression non-linéaire ainsi que dans un modèle de régression linéaire en très grande dimension. Tout d'abord, nous proposons une méthode basée sur la vraisemblance empirique pour tester la présence de changements dans les paramètres d'un modèle de régression non-linéaire. Sous l'hypothèse nulle, nous prouvons la consistance et la vitesse de convergence des estimateurs des paramètres de régression. La loi asymptotique de la statistique de test sous l'hypothèse nulle nous permet de trouver la valeur critique asymptotique. D'autre part, nous prouvons que la puissance asymptotique de la statistique de test proposée est égale à 1. Le modèle épidémique avec deux points de rupture est également étudié. Ensuite, on s'intéresse à construire les régions de confiance asymptotiques pour la différence entre les paramètres de deux phases d'un modèle non-linéaire avec des regresseurs aléatoires en utilisant la méthode de vraisemblance empirique. On montre que le rapport de la vraisemblance empirique a une distribution asymptotique χ2. La méthode de vraisemblance empirique est également utilisée pour construire les régions de confiance pour la différence entre les paramètres des deux phases d'un modèle non-linéaire avec des variables de réponse manquantes au hasard (Missing At Random (MAR)). Afin de construire les régions de confiance du paramètre en question, on propose trois statistiques de vraisemblance empirique : la vraisemblance empirique basée sur les données cas-complète, la vraisemblance empirique pondérée et la vraisemblance empirique par des valeurs imputées. On prouve que les trois rapports de vraisemblance empirique ont une distribution asymptotique χ2. Un autre but de cette thèse est de tester la présence d'un changement dans les coefficients d'un modèle linéaire en grande dimension, où le nombre des variables du modèle peut augmenter avec la taille de l'échantillon. Ce qui conduit à tester l'hypothèse nulle de non-changement contre l'hypothèse alternative d'un seul changement dans les coefficients de régression. Basée sur les comportements asymptotiques de la statistique de rapport de vraisemblance empirique, on propose une simple statistique de test qui sera utilisée facilement dans la pratique. La normalité asymptotique de la statistique de test proposée sous l'hypothèse nulle est prouvée. Sous l'hypothèse alternative, la statistique de test diverge
In this PHD thesis, we propose a nonparametric method based on the empirical likelihood for detecting the change in the parameters of nonlinear regression models and the change in the coefficient of linear regression models, when the number of model variables may increase as the sample size increases. Firstly, we test the null hypothesis of no-change against the alternative of one change in the regression parameters. Under null hypothesis, the consistency and the convergence rate of the regression parameter estimators are proved. The asymptotic distribution of the test statistic under the null hypothesis is obtained, which allows to find the asymptotic critical value. On the other hand, we prove that the proposed test statistic has the asymptotic power equal to 1. The epidemic model, a particular case of model with two change-points, under the alternative hypothesis, is also studied. Afterwards, we use the empirical likelihood method for constructing the confidence regions for the difference between the parameters of a two-phases nonlinear model with random design. We show that the empirical likelihood ratio has an asymptotic χ2 distribu- tion. Empirical likelihood method is also used to construct the confidence regions for the difference between the parameters of a two-phases nonlinear model with response variables missing at randoms (MAR). In order to construct the confidence regions of the parameter in question, we propose three empirical likelihood statistics : empirical likelihood based on complete-case data, weighted empirical likelihood and empirical likelihood with imputed va- lues. We prove that all three empirical likelihood ratios have asymptotically χ2 distributions. An another aim for this thesis is to test the change in the coefficient of linear regres- sion models for high-dimensional model. This amounts to testing the null hypothesis of no change against the alternative of one change in the regression coefficients. Based on the theoretical asymptotic behaviour of the empirical likelihood ratio statistic, we propose, for a deterministic design, a simpler test statistic, easier to use in practice. The asymptotic normality of the proposed test statistic under the null hypothesis is proved, a result which is different from the χ2 law for a model with a fixed variable number. Under alternative hypothesis, the test statistic diverges

Information-theoretical notions of causality provide a model-free approach to identification of the magnitude and direction of influence among sub-components of a stochastic dynamical system. In addition to detecting causal influences, any effective test should also report the level of statistical significance of the finding. Here, we focus on transfer entropy, which has recently been considered for causality detection in a variety of fields based on statistical significance tests that are valid only in the asymptotic regime, that is, with enormous amounts of data. In the interest of applications with limited available data, we develop a non-asymptotic theory for the probability distribution of the difference between the empirically-estimated transfer entropy and the true transfer entropy. Based on this result, we additionally demonstrate an approach for statistical hypothesis testing for directed information flow in dynamical systems with a given number of observed time steps.

Traditional optoelectronic manufacturing of butterfly packages involves laser welding of a fiber mount followed by a tedious realignment procedure to reverse thermally induced distortions commonly referred to as Post Weld Shift (PWS). An alternative PWS compensation technique, Laser Hammering, entails manipulation of the fiber to light alignment through deformation of the fiber housing with high precision laser beams. The goal of this study is to predict and understand fiber displacements for butterfly packages subjected to the laser hammering process using finite element analysis. A standardized, two-dimensional fiber mounting/ferrule geometry is employed in a simulation case study. Various laser waveforms are applied to focus spot diameters of 50 and 200 μm over a range of applied heat fluxes (10 to 1000 W/mm2). The primary investigation focused on the degree to which a steady state (SS) model can predict the final state of a transient response (asymptotic steady state) subjected to a periodic laser excitation. Effects of laser energy deposition location and resolution, as well as the use of multiple lasers were also studied. The results obtained to date show that the steady state solution is in good agreement with the asymptotic transient response (ATR) for the center horizontal fiber displacement and the center fiber temperature. The focus spot region surface temperature predictions of steady state and asymptotic transient simulations were also found to be in reasonable agreement. However, the vertical fiber displacement tends to be over predicted by the steady state solution, sometimes by as much as an order of magnitude. The causes, both physical and computational, of this disagreement are discussed in the paper.

The acoustic signature produced by non-pneumatic wheels with collapsible spokes is a critical design criterion for automotive and other mobility applications. During high speed rolling, acoustic noise may be produced by the interaction of vibrating spokes with a shear deformable ring as they enter the contact region, buckle and then snap back into a state of tension. In order to identify and help understand the causes of acoustic noise for a rolling non-pneumatic wheel, a two-dimensional finite element model with geometric nonlinearity has been utilized. The model consists of a shear ring modeled as two relatively inextensible membranes with high circumferential modulus separated by a hyper-elastic material. The temporal variation in spoke length as the spoke passes through the contact zone is extracted and used as input to a three-dimensional (3-D) model of a single spoke. The 3-D spoke model is able to capture out-of-plane vibration modes of the spoke which may contribute as a source of acoustic excitation and allows for modeling of edge scalloping. Natural frequencies and mode shapes of the various spoke design strategies are computed and correlated with the frequency response of the out-of-plane spoke vibrations. Results indicate that scalloping the edges of the spoke can dramatically reduce the amplitude of vibration, but does not have a strong effect on location of frequency peaks in a FFT of the time-signal. An optimal amount of scalloping was determined which reduces maximum vibration amplitude to an asymptotic value.

Abstract The impact of the mechanical formation damage in the production curve caused by the unsteady-state permeability hysteresis in pressure-sensitive reservoirs has an important role in oilfield developments. Hence, the petroleum industry has continuously sought new analytical approaches to provide adequate well-reservoir performance management and surveillance. This work proposes a new unsteady-state two-dimensional (2-D) analytical solution for modeling the oil flow in a permeability hysteretic pressure-dependent reservoir during alternating loading/unloading cycles. The nonlinear hydraulic diffusivity equation (NHDE) is perturbed through a first-order asymptotic series expansion. The reservoir engineering literature shows that this first-order expansion represents satisfactory the magnitude of the nonlinear phenomena regarding pressure-sensitive rock and fluid properties. A new hysteretic deviation factor is presented for flow (drawdown) and buildup periods. The model calibration is performed by a porous media numerical oil flow simulator named CMG IMEX, which is broadly used in formation evaluation and reservoir engineering literature. The results presented high accuracy for the drawdown and buildup of hysteretic and non-hysteretic cases. The practical uses of the model developed in this work are related to identifying flow regimes and hysteresis responses in pressure-sensitive reservoirs, estimating buildup pressure, and oil flow rates specification to prevent severe hysteretic behavior and history matching during reservoir surveillance.

Enhanced heat transfer characteristics of low Reynolds number airflows in three-dimensional sinusoidal wavy plate-fin channels are investigated. For the computational simulation, steady state, constant property, periodically developed, laminar forced convection is considered with the channel surface at the uniform heat flux condition; the wavy-fin is modeled by its two asymptotic limits of 100% and zero fin efficiency. The governing equations are solved numerically using finite-volume techniques for a non-orthogonal, non-staggered grid. Computational results for velocity and temperature distribution, isothermal Fanning friction factor f and Colburn factor j are presented for airflow rates in the range of 10 ≤ Re ≤ 1500. The numerical results are further compared with experimental data, with excellent agreement, for two different wavy-fin geometries. The influence of fin density on the flow behavior and the enhanced convection heat transfer are highlighted. Depending on the flow rate, a complex flow structure is observed, which is characterized by the generation, spatial growth and dissipation of vortices in the trough region of the wavy channel. The thermal boundary layers on the fin surface are periodically disrupted, resulting in high local heat fluxes. The overall heat transfer performance is improved considerably, compared to the straight channel with the same cross-section, with a relatively smaller increase in the associated pressure drop penalty.