The geometrically rigorous nonlinear analysis of elastic shells is considered in the context of finite, but small, strain theory. The research is focused on the introduction of the full shell metric and examination of its influence on the nonlinear structural response. The exact relation between the reference and equidistant strains is employed and the complete analytic elastic constitutive relation between energetically conjugated forces and strains is derived via the reciprocal shift tensor. Utilizing these strict relations, the geometric stiffness matrix is derived explicitly by the variation of the unknown metric. Moreover, a compact form of this matrix is presented. Despite the linear displacement distribution due to the Kirchhoff-Love hypothesis, a nonlinear strain distribution arises along the shell thickness. This fact is sometimes disregarded for the nonlinear analysis of thin shells based on the initial geometry, thereby ignoring the strong curviness of a shell at some subsequent configuration. We show that the curviness of a shell at each configuration determines the appropriate shell formulation. For shells that become strongly curved at some configurations during deformation, the nonlinear distribution of strain throughout the thickness must be considered in order to obtain accurate results. We investigate four computational models: one based on the full analytical constitutive relation, and three simplified ones. Robustness, efficiency and accuracy of the presented formulation are examined via selected numerical experiments. Our main finding is that the employment of the full metric is often required when the complete response of the shells is sought, even for the initially thin shells. Finally, the simplified model that provided the best balance between efficiency and accuracy is suggested for the nonlinear analysis of strongly curved shells.

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Given an optimization problem, the Hessian matrix and its eigenspectrum can be used in many ways, ranging from designing more efficient second-order algorithms to performing model analysis and regression diagnostics. When nonlinear models and non-convex problems are considered, strong simplifying assumptions are often made to make Hessian spectral analysis more tractable. This leads to the question of how relevant the conclusions of such analyses are for more realistic nonlinear models. In this paper, we exploit deterministic equivalent techniques from random matrix theory to make a \emph{precise} characterization of the Hessian eigenspectra for a broad family of nonlinear models, including models that generalize the classical generalized linear models, without relying on strong simplifying assumptions used previously. We show that, depending on the data properties, the nonlinear response model, and the loss function, the Hessian can have \emph{qualitatively} different spectral behaviors: of bounded or unbounded support, with single- or multi-bulk, and with isolated eigenvalues on the left- or right-hand side of the bulk. By focusing on such a simple but nontrivial nonlinear model, our analysis takes a step forward to unveil the theoretical origin of many visually striking features observed in more complex machine learning models.

This paper studies computational methods for quasi-stationary distributions (QSDs). We first proposed a data-driven solver that solves Fokker-Planck equations for QSDs. Similar as the case of Fokker-Planck equations for invariant probability measures, we set up an optimization problem that minimizes the distance from a low-accuracy reference solution, under the constraint of satisfying the linear relation given by the discretized Fokker-Planck operator. Then we use coupling method to study the sensitivity of a QSD against either the change of boundary condition or the diffusion coefficient. The 1-Wasserstein distance between a QSD and the corresponding invariant probability measure can be quantitatively estimated. Some numerical results about both computation of QSDs and their sensitivity analysis are provided.

Spectral measures arise in numerous applications such as quantum mechanics, signal processing, resonances, and fluid stability. Similarly, spectral decompositions (pure point, absolutely continuous and singular continuous) often characterise relevant physical properties such as long-time dynamics of quantum systems. Despite new results on computing spectra, there remains no general method able to compute spectral measures or spectral decompositions of infinite-dimensional normal operators. Previous efforts focus on specific examples where analytical formulae are available (or perturbations thereof) or on classes of operators with a lot of structure. Hence the general computational problem is predominantly open. We solve this problem by providing the first set of general algorithms that compute spectral measures and decompositions of a wide class of operators. Given a matrix representation of a self-adjoint or unitary operator, such that each column decays at infinity at a known asymptotic rate, we show how to compute spectral measures and decompositions. We discuss how these methods allow the computation of objects such as the functional calculus, and how they generalise to a large class of partial differential operators, allowing, for example, solutions to evolution PDEs such as Schr\"odinger equations on $L^2(\mathbb{R}^d)$. Computational spectral problems in infinite dimensions have led to the SCI hierarchy, which classifies the difficulty of computational problems. We classify computation of measures, measure decompositions, types of spectra, functional calculus, and Radon--Nikodym derivatives in the SCI hierarchy. The new algorithms are demonstrated to be efficient on examples taken from OPs on the real line and the unit circle (e.g. giving computational realisations of Favard's theorem and Verblunsky's theorem), and are applied to evolution equations on a 2D quasicrystal.

Finite Sample Smeariness (FSS) has been recently discovered. It means that the distribution of sample Fr\'echet means of underlying rather unsuspicious random variables can behave as if it were smeary for quite large regimes of finite sample sizes. In effect classical quantile-based statistical testing procedures do not preserve nominal size, they reject too often under the null hypothesis. Suitably designed bootstrap tests, however, amend for FSS. On the circle it has been known that arbitrarily sized FSS is possible, and that all distributions with a nonvanishing density feature FSS. These results are extended to spheres of arbitrary dimension. In particular all rotationally symmetric distributions, not necessarily supported on the entire sphere feature FSS of Type I. While on the circle there is also FSS of Type II it is conjectured that this is not possible on higher-dimensional spheres.

In this paper, distributed dynamics are deployed to solve resource allocation over time-varying multi-agent networks. The state of each agent represents the amount of resources used/produced at that agent while the total amount of resources is fixed. The idea is to optimally allocate the resources among the group of agents by reducing the total cost functions subject to fixed amount of total resources. The information of each agent is restricted to its own state and cost function and those of its immediate neighbors. This is motivated by distributed applications such as in mobile edge-computing, economic dispatch over smart grids, and multi-agent coverage control. The non-Lipschitz dynamics proposed in this work shows fast convergence as compared to the linear and some nonlinear solutions in the literature. Further, the multi-agent network connectivity is more relaxed in this paper. To be more specific, the proposed dynamics even reaches optimal solution over time-varying disconnected undirected networks as far as the union of these networks over some bounded non-overlapping time-intervals includes a spanning-tree. The proposed convergence analysis can be applied for similar 1st-order resource allocation nonlinear dynamics. We provide simulations to verify our results.

In this paper, we are concerned with a time-dependent transmission problem for a thermo-piezoelectric elastic body immersed in a compressible fluid. It is shown that the problem can be treated by the boundary-field equation method, provided an appropriate scaling factor is employed. As usual, based on estimates for solutions in the Laplace-transformed domain, we may obtain properties of corresponding solutions in the time-domain without having to perform the inversion of the Laplace-domain solutions.

This paper develops a general causal inference method for treatment effects models with noisily measured confounders. The key feature is that a large set of noisy measurements are linked with the underlying latent confounders through an unknown, possibly nonlinear factor structure. The main building block is a local principal subspace approximation procedure that combines $K$-nearest neighbors matching and principal component analysis. Estimators of many causal parameters, including average treatment effects and counterfactual distributions, are constructed based on doubly-robust score functions. Large-sample properties of these estimators are established, which only require relatively mild conditions on the principal subspace approximation. The results are illustrated with an empirical application studying the effect of political connections on stock returns of financial firms, and a Monte Carlo experiment. The main technical and methodological results regarding the general local principal subspace approximation method may be of independent interest.

Modeling the distribution of high dimensional data by a latent tree graphical model is a common approach in multiple scientific domains. A common task is to infer the underlying tree structure given only observations of the terminal nodes. Many algorithms for tree recovery are computationally intensive, which limits their applicability to trees of moderate size. For large trees, a common approach, termed divide-and-conquer, is to recover the tree structure in two steps. First, recover the structure separately for multiple randomly selected subsets of the terminal nodes. Second, merge the resulting subtrees to form a full tree. Here, we develop Spectral Top-Down Recovery (STDR), a divide-and-conquer approach for inference of large latent tree models. Unlike previous methods, STDR's partitioning step is non-random. Instead, it is based on the Fiedler vector of a suitable Laplacian matrix related to the observed nodes. We prove that under certain conditions this partitioning is consistent with the tree structure. This, in turn leads to a significantly simpler merging procedure of the small subtrees. We prove that STDR is statistically consistent, and bound the number of samples required to accurately recover the tree with high probability. Using simulated data from several common tree models in phylogenetics, we demonstrate that STDR has a significant advantage in terms of runtime, with improved or similar accuracy.

Locimetric authentication is a form of graphical authentication where users validate their identity by selecting predetermined points on a predetermined image. Its primary advantage over the ubiquitous text-based approach stems from users' superior ability to remember visual information over textual information, coupled with the authentication process being transformed to one requiring recognition (instead of recall). Ideally, these differentiations enable users to create more complex passwords, which theoretically are more secure. Yet, locimetric authentication has one significant weakness, hot-spots, that is, areas in an image that users gravitate towards and consequently have a higher probability of being selected. This paper investigates whether the hot-spot problem persists with high-resolution images, as well as whether user characteristics and password length play a role. Our findings confirm the presence of hot-spots in high-resolution images, thus influencing the locimetric authentication scheme's effectiveness. Furthermore, we find that neither user characteristics (such as age, gender, and income) nor password length radically influence their extent. We conclude by proposing strategies to mitigate the hot-spot phenomenon.

Methods that align distributions by minimizing an adversarial distance between them have recently achieved impressive results. However, these approaches are difficult to optimize with gradient descent and they often do not converge well without careful hyperparameter tuning and proper initialization. We investigate whether turning the adversarial min-max problem into an optimization problem by replacing the maximization part with its dual improves the quality of the resulting alignment and explore its connections to Maximum Mean Discrepancy. Our empirical results suggest that using the dual formulation for the restricted family of linear discriminators results in a more stable convergence to a desirable solution when compared with the performance of a primal min-max GAN-like objective and an MMD objective under the same restrictions. We test our hypothesis on the problem of aligning two synthetic point clouds on a plane and on a real-image domain adaptation problem on digits. In both cases, the dual formulation yields an iterative procedure that gives more stable and monotonic improvement over time.

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