Smoothness Assumptions Research Articles

Convergence Analysis and Numerical Studies for Linearly Elastic Peridynamics with Dirichlet-Type Boundary Conditions

The nonlocal models of peridynamics have successfully predicted fractures and deformations for a variety of materials. In contrast to local mechanics, peridynamic boundary conditions must be defined on a finite volume region outside the body. Therefore, theoretical and numerical challenges arise in order to properly formulate Dirichlet-type nonlocal boundary conditions, while connecting them to the local counterparts. While a careless imposition of local boundary conditions leads to a smaller effective material stiffness close to the boundary and an artificial softening of the material, several strategies were proposed to avoid this unphysical surface effect. In this work, we study convergence of solutions to nonlocal state-based linear elastic model to their local counterparts as the interaction horizon vanishes, under different formulations and smoothness assumptions for nonlocal Dirichlet-type boundary conditions. Our results provide explicit rates of convergence that are sensitive to the compatibility of the nonlocal boundary data and the extension of the solution for the local model. In particular, under appropriate assumptions, constant extensions yield \(\frac{1}{2}\) order convergence rates and linear extensions yield \(\frac{3}{2}\) order convergence rates. With smooth extensions, these rates are improved to quadratic convergence. We illustrate the theory for any dimension \(d\ge 2\) and numerically verify the convergence rates with a number of two-dimensional benchmarks, including linear patch tests, manufactured solutions, and domains with curvilinear surfaces. Numerical results show a first-order convergence for constant extensions and second-order convergence for linear extensions, which suggests a possible room of improvement in the future convergence analysis.

Deep Probabilistic Learning for Process Quality Evaluation With a Case Study of Gear Hobbing Process

Conventional semisupervised learning approaches are limited by the reliance on assumptions of samples (manifold assumption, cluster assumption, and smoothness assumption) and may fail to reach their full potential when it comes to industrial scenarios with complicated data characteristics. In this article, a semisupervised process quality evaluation framework is proposed based on conditional variational auto-encoder. Label inference process is utilized to handle unlabeled samples during training stage and to directly predict the label during the evaluation stage. Real-valued nonvolume preserving (Real NVP) is introduced to optimize the posterior distribution in order to increase flexibility. The proposed framework does not rely on assumptions of data distributions and it learns the hidden distributions from samples. Gear hobbing simulation and experiment investigations are conducted to verify the effectiveness of the proposed framework. The results indicate that the proposed method can achieve remarkable classification performance compared with other state-of-the-art approaches.

The Bayesian Spatial Bradley–Terry Model: Urban Deprivation Modelling in Tanzania

Abstract Identifying the most deprived regions of any country or city is key if policy makers are to design successful interventions. However, locating areas with the greatest need is often surprisingly challenging in developing countries. Due to the logistical challenges of traditional household surveying, official statistics can be slow to be updated; estimates that exist can be coarse, a consequence of prohibitive costs and poor infrastructures; and mass urbanization can render manually surveyed figures rapidly out-of-date. Comparative judgement models, such as the Bradley–Terry model, offer a promising solution. Leveraging local knowledge, elicited via comparisons of different areas’ affluence, such models can both simplify logistics and circumvent biases inherent to household surveys. Yet widespread adoption remains limited, due to the large amount of data existing approaches still require. We address this via development of a novel Bayesian Spatial Bradley–Terry model, which substantially decreases the number of comparisons required for effective inference. This model integrates a network representation of the city or country, along with assumptions of spatial smoothness that allow deprivation in one area to be informed by neighbouring areas. We demonstrate the practical effectiveness of this method, through a novel comparative judgement data set collected in Dar es Salaam, Tanzania.

Sign Stochastic Gradient Descents without bounded gradient assumption for the finite sum minimization

Sign-based Stochastic Gradient Descents (Sign-based SGDs) use the signs of the stochastic gradients for communication costs reduction. Nevertheless, current convergence results of sign-based SGDs applied to the finite sum optimization are established on the bounded assumption of the gradient, which fails to hold in various cases. This paper presents a convergence framework about sign-based SGDs with the elimination of the bounded gradient assumption. The ergodic convergence rates are provided only with the smooth assumption of the objective functions. The Sign Stochastic Gradient Descent (signSGD) and its two variants, including majority vote and zeroth-order version, are developed for different application settings. Our framework also removes the bounded gradient assumption used in the previous analysis of these three algorithms.

Asymptotic normality of the regression mode in the nonparametric random design model for censored data

In the nonparametric regression model, where the regression function is given by for a measurable function estimation of the location (mode) of a unique maximum of by the location of a maximum of the Nadaraya-Watson kernel estimator for the curve is considered. Within the setting of the censored data, we obtain the consistency with rate and the asymptotic normality results for under mild local smoothness assumptions on the regression and the design density of Simulation results are performed to illustrate the performance of the procedure.

Truncated estimation in functional generalized linear regression models

Functional generalized linear models investigate the effect of functional predictors on a scalar response. An interesting case is when the functional predictor is thought to exert an influence on the conditional mean of the response only through its values up to a certain point in the domain. In the literature, models with this type of restriction on the functional effect have been termed truncated or historical regression models. A penalized likelihood estimator is formulated by combining a structured variable selection method with a localized B-spline expansion of the regression coefficient function. In addition to a smoothing penalty that is typical for functional regression, a nested group lasso penalty is also included which guarantees the sequential entering of B-splines and thus induces the desired truncation on the estimator. An optimization scheme is developed to compute the solution path efficiently when varying the truncation tuning parameter. The convergence rate of the coefficient function estimator and consistency of the truncation point estimator are given under suitable smoothness assumptions. The proposed method is demonstrated through simulations and an application involving the effects of blood pressure values in patients who suffered a spontaneous intracerebral hemorrhage.

Semi-controlled Learning in Information Processing Problems

The article substantiates the need for further research of known methods and the development of new methods of machine learning – semi-supervized learning. It is shown that knowledge of the probability distribution density of the initial data obtained using unlabeled data should carry information useful for deriving the conditional probability distribution density of labels and input data. If this is not the case, semi-supervised learning will not provide any improvement over supervised learning. It may even happen that the use of unlabeled data reduces the accuracy of the prediction. For semi-supervised learning to work, certain assumptions must hold, namely: the semi-supervised smoothness assumption, the clustering assumption (low-density partitioning), and the manifold assumption. A new hybrid semi-supervised learning algorithm using the label propagation method has been developed. An example of using the proposed algorithm is given.

Hyperfast second-order local solvers for efficient statistically preconditioned distributed optimization

Statistical preconditioning enables fast methods for distributed large-scale empirical risk minimization problems. In this approach, multiple worker nodes compute gradients in parallel, which are then used by the central node to update the parameter by solving an auxiliary (preconditioned) smaller-scale optimization problem. The recently proposed Statistically Preconditioned Accelerated Gradient (SPAG) method [1] has complexity bounds superior to other such algorithms but requires an exact solution for computationally intensive auxiliary optimization problems at every iteration. In this paper, we propose an Inexact SPAG (InSPAG) and explicitly characterize the accuracy by which the corresponding auxiliary subproblem needs to be solved to guarantee the same convergence rate as the exact method. We build our results by first developing an inexact adaptive accelerated Bregman proximal gradient method for general optimization problems under relative smoothness and strong convexity assumptions, which may be of independent interest. Moreover, we explore the properties of the auxiliary problem in the InSPAG algorithm assuming Lipschitz third-order derivatives and strongly convexity. For such problem class, we develop a linearly convergent Hyperfast second-order method and estimate the total complexity the InSPAG method with hyperfast auxiliary problem solver. Finally, we illustrate the proposed method's practical efficiency by performing large-scale numerical experiments on logistic regression models. To the best of our knowledge, these are the first empirical results on implementing high-order methods on large-scale problems, we work with data where the dimension is of the order of 3 million, and the number of samples is 700 million. • Inexact Statistically Preconditioned Accelerated Gradient Method for large-scale distributed convex empirical risk minimization (ERM) problems. • Hyperfast second-order algorithm for minimizing strongly convex functions with Lipschitz third-order derivatives. • Inexact adaptive accelerated Bregman proximal gradient method for general optimization problems under relative smoothness and strong convexity assumptions. • Empirical evidence for the efficiency of tensor optimization methods for large-scale ERM problems.

Almost automorphy of minimal sets for C1${C}^{1}$‐smooth strongly monotone skew‐product semiflows on Banach spaces

We focus on the presence of almost automorphy in strongly monotone skew-product semiflows on Banach spaces. Under the $C^1$-smoothness assumption, it is shown that any linearly stable minimal set must be almost automorphic. This extends the celebrated result of Shen and Yi [Mem. Amer. Math. Soc. 136(1998), No. 647] for the classical $C^{1,\alpha}$-smooth systems. Based on this, one can reduce the regularity of the almost periodically forced differential equations and obtain the almost automorphic phenomena in a wider range.

Gaussian Process Convolutional Dictionary Learning

Convolutional dictionary learning (CDL), the problem of estimating shift-invariant templates from data, is typically conducted in the absence of a prior/structure on the templates. In data-scarce or low signal-to-noise ratio (SNR) regimes, learned templates overfit the data and lack smoothness, which can affect the predictive performance of downstream tasks. To address this limitation, we propose GPCDL, a convolutional dictionary learning framework that enforces priors on templates using Gaussian Processes (GPs). With the focus on smoothness, we show theoretically that imposing a GP prior is equivalent to Wiener filtering the learned templates, thereby suppressing high-frequency components and promoting smoothness. We show that the algorithm is a simple extension of the classical iteratively reweighted least squares algorithm, independent of the choice of GP kernels. This property allows one to experiment flexibly with different smoothness assumptions. Through simulation, we show that GPCDL learns smooth dictionaries with better accuracy than the unregularized alternative across a range of SNRs. Through an application to neural spiking data, we show that GPCDL learns a more accurate and visually-interpretable smooth dictionary, leading to superior predictive performance compared to non-regularized CDL, as well as parametric alternatives.

Asynchronous Parallel Nonconvex Optimization Under the Polyak-Łojasiewicz Condition

Communication delays and synchronization are major bottlenecks for parallel computing, and tolerating asynchrony is therefore crucial for accelerating parallel computation. Motivated by optimization problems that do not satisfy convexity assumptions, we present an asynchronous block coordinate descent algorithm for nonconvex optimization problems whose objective functions satisfy the Polyak-Łojasiewicz condition. This condition is a generalization of strong convexity to nonconvex problems and requires neither convexity nor uniqueness of minimizers. Under only assumptions of mild smoothness of objective functions and bounded delays, we prove that a linear convergence rate is obtained. Numerical experiments for logistic regression problems are presented to illustrate the impact of asynchrony upon convergence.

Entanglement island, miracle operators and the firewall

In this paper, we obtain some general results on information retrieval from the black hole interior, based on the recent progress on quantum extremal surface formula and entanglement island. We study an AdS black hole coupled to a bath with generic dynamics, and ask whether it is possible to retrieve information about a small perturbation in the interior from the bath system. We show that the one-norm distance between two reduced states in a bath region A is equal to the same quantity in the bulk quantum field theory for region AI where I is the entanglement island of A. This is a straightforward generalization of bulk-boundary correspondence in AdS/CFT. However, we show that a contradiction arises if we apply this result to a special situation when the bath dynamics includes a unitary operation that carries a particular measurement to a region A and send the result to another region W. Physically, the contradiction arises between transferability of classical information during the measurement, and non-transferability of quantum information which determines the entanglement island.We propose that the resolution of the contradiction is to realize that the state reconstruction formula does not apply to the special situation involving interior-information-retrieving measurements. This implies that the assumption of smooth replica AdS geometry with boundary condition set by the flat space bath has to break down when the particular measurement operator is applied to the bath. Using replica trick, we introduce an explicitly construction of such operator, which we name as “miracle operators”. From this construction we see that the smooth replica geometry assumption breaks down because we have to introduce extra replica wormholes connecting with the “simulated blackholes” introduced by the miracle operator. We study the implication of miracle operators in understanding the firewall paradox.

Designing truncated priors for direct and inverse Bayesian problems

The Bayesian approach to inverse problems with functional unknowns, has received significant attention in recent years. An important component of the developing theory is the study of the asymptotic performance of the posterior distribution in the frequentist setting. The present paper contributes to the area of Bayesian inverse problems by formulating a posterior contraction theory for linear inverse problems, with truncated Gaussian series priors, and under general smoothness assumptions. Emphasis is on the intrinsic role of the truncation point both for the direct as well as for the inverse problem, which are related through the modulus of continuity as this was recently highlighted by Knapik and Salomond (2018).

An Improved ARIMA Method Based on Functional Principal Component Analysis and Bidirectional Bootstrap and Its Application to Stock Price Forecasting

This paper proposes an improved ARIMA method based on functional principal component analysis and bi-directional bootstrap. The proposed method does not require a smoothness assumption, uses intraday prices as auxiliary information and considers their functional characteristics, and effectively performs a bias-variance trade-off in the forecasting model by using a bi-directional bootstrap method. This is achieved by forming a paired sample of ARIMA forecast residuals and functional characteristics, and then fitting the forecast residuals to the regression model using the two-way bootstrap method, thereby improving the forecast accuracy.In addition, the choice of regression model is free. The empirical results show that the proposed method has better predictive performance and is more robust than the ARIMA model. Finally, the proposed method can be extended to environmental science, social science and other fields to help deal with various prediction problems.

On some inverse problem for bi-parabolic equation with observed data in L^{p} spaces

The bi-parabolic equation has many practical significance in the field of heat transfer. The objective of the paper is to provide a regularized problem for bi-parabolic equation when the observed data are obtained in \(L^p\). We are interested in looking at three types of inverse problems. Regularization results in the \(L^2\) space appears in many related papers, but the survey results are rare in \(L^p\), \(p \neq 2\). The first problem related to the inverse source problem when the source function has split form. For this problem, we introduce the error between the Fourier regularized solution and the exact solution in \(L^p\) spaces. For the inverse initial problem for both linear and nonlinear cases, we applied the Fourier series truncation method. Under the terminal input data observed in \(L^p\), we obtain the approximated solution also in the space \(L^p\). Under some reasonable smoothness assumptions of the exact solution, the error between the the regularized solution and the exact solution are derived in the space \(L^p\). This paper seems to generalize to previous results for bi-parabolic equation on this direction.

Relaxing the Smoothness Assumption in Optimization: Applications to Finance

Relaxing the Smoothness Assumption in Optimization: Applications to Finance

A Novel Ellipsoidal Semisupervised Extreme Learning Machine Algorithm and Its Application in Wind Turbine Blade Icing Fault Detection

The conventional semi-supervised extreme learning machine (SS-ELM) algorithm can provide a solution to the lack of labeled samples in wind turbine blade icing fault detection, but its performance is limited by the irrationality of the spherical nearest neighbor graph (SNNG) calculation strategy. To solve this problem, a novel ellipsoidal semi-supervised extreme learning machine (ESS-ELM) algorithm is proposed in this paper and applied to wind turbine blade icing fault detection. In this study, we creatively propose a novel ellipsoidal nearest neighbor graph (ENNG) calculation strategy that considers the distribution information of the labeled samples to construct the ESS-ELM algorithm. Different from the conventional SNNG, the ENNG can adaptively assign corresponding calculation weights to each feature in the data space according to the Fisher criterion, which allows it to better match the smoothness assumption of the manifold regularization learning framework. In addition, the two adjustable parameters of the enhancement factor and the degradation factor are ingeniously introduced into the ENNG, which further ensure the applicability and security of the proposed ESS-ELM. The superiority of the ESS-ELM algorithm is verified by extensive benchmark industrial fault datasets and the real-world icing dataset of two wind turbines. It is demonstrated that the proposed ESS-ELM algorithm achieves better performance than the ELM, its existing variants, and some state-of-the-art wind turbine blade icing detection methods.

ADAPTIVE STRATIFIED SAMPLING FOR NONSMOOTH PROBLEMS

Science and engineering problems subject to uncertainty are frequently both computationally expensive and feature non-smooth parameter dependence, making standard Monte Carlo too slow, and excluding efficient use of accelerated uncertainty quantification methods relying on strict smoothness assumptions. To remedy these challenges, we propose an adaptive stratification method suitable for non-smooth problems and with significantly reduced variance compared to Monte Carlo sampling. The stratification is iteratively refined and samples are added sequentially to satisfy an allocation criterion combining the benefits of proportional and optimal sampling. Theoretical estimates are provided for the expected performance and probability of failure to correctly estimate essential statistics. We devise a practical adaptive stratification method with strata of the same kind of geometrical shapes, cost-effective refinement satisfying a greedy variance reduction criterion. Numerical experiments corroborate the theoretical findings and exhibit speedups of up to three orders of magnitude compared to standard Monte Carlo sampling.

Nonlinear systems of parabolic IBVP: A stable super-supraconvergent fully discrete piecewise linear FEM

The main purpose of this paper is the design of discretizations for second order nonlinear parabolic initial boundary value problems which are stable and present second order of convergence with respect to H1-discrete norms. The convergence results are established assuming that the solutions are in H3. The stability analysis of numerical methods around a numerical solution requires the uniform boundness of such solution. Although such bounds are usually taken as an assumption, in this paper these will be deduced as a corollary of suitable error estimates. As the methods can be simultaneously seen as piecewise linear finite element methods and finite difference methods, the convergence results can be seen simultaneously as superconvergence results and supraconvergence results. Numerical results illustrating the sharpness of the smoothness assumptions and an application to simulation of the solar magnetic field in the umbra (the central zone of a sunspot) are also included.

Ghost Point Diffusion Maps for Solving Elliptic PDEs on Manifolds with Classical Boundary Conditions

In this paper, we extend the class of kernel methods, the so‐called diffusion maps (DM) and its local kernel variants to approximate second‐order differential operators defined on smooth manifolds with boundaries that naturally arise in elliptic PDE models. To achieve this goal, we introduce the ghost point diffusion maps (GPDM) estimator on an extended manifold, identified by the set of point clouds on the unknown original manifold together with a set of ghost points, specified along the estimated tangential direction at the sampled points on the boundary. The resulting GPDM estimator restricts the standard DM matrix to a set of extrapolation equations that estimates the function values at the ghost points. This adjustment is analogous to the classical ghost point method in a finite‐difference scheme for solving PDEs on flat domains. As opposed to the classical DM, which diverges near the boundary, the proposed GPDM estimator converges pointwise even near the boundary. Applying the consistent GPDM estimator to solve well‐posed elliptic PDEs with classical boundary conditions (Dirichlet, Neumann, and Robin), we establish the convergence of the approximate solution under appropriate smoothness assumptions. We numerically validate the proposed mesh‐free PDE solver on various problems defined on simple submanifolds embedded in Euclidean spaces as well as on an unknown manifold. Numerically, we also found that the GPDM is more accurate compared to DM in solving elliptic eigenvalue problems on bounded smooth manifolds. © 2021 Wiley Periodicals LLC.