Numerical methods for random parametric PDEs can greatly benefit from adaptive refinement schemes, in particular when functional approximations are computed as in stochastic Galerkin and stochastic collocations methods. This work is concerned with a non-intrusive generalization of the adaptive Galerkin FEM with residual based error estimation. It combines the non-intrusive character of a randomized least-squares method with the a posteriori error analysis of stochastic Galerkin methods. The proposed approach uses the Variational Monte Carlo method to obtain a quasi-optimal low-rank approximation of the Galerkin projection in a highly efficient hierarchical tensor format. We derive an adaptive refinement algorithm which is steered by a reliable error estimator. Opposite to stochastic Galerkin methods, the approach is easily applicable to a wide range of problems, enabling a fully automated adjustment of all discretization parameters. Benchmark examples with affine and (unbounded) lognormal coefficient fields illustrate the performance of the non-intrusive adaptive algorithm, showing best-in-class performance.
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Function approximation is widely used in reinforcement learning to handle the computational difficulties associated with very large state spaces. However, function approximation introduces errors which may lead to instabilities when using approximate dynamic programming techniques to obtain the optimal policy. Therefore, techniques such as lookahead for policy improvement and m-step rollout for policy evaluation are used in practice to improve the performance of approximate dynamic programming with function approximation. We quantitatively characterize, for the first time, the impact of lookahead and m-step rollout on the performance of approximate dynamic programming (DP) with function approximation: (i) without a sufficient combination of lookahead and m-step rollout, approximate DP may not converge, (ii) both lookahead and m-step rollout improve the convergence rate of approximate DP, and (iii) lookahead helps mitigate the effect of function approximation and the discount factor on the asymptotic performance of the algorithm. Our results are presented for two approximate DP methods: one which uses least-squares regression to perform function approximation and another which performs several steps of gradient descent of the least-squares objective in each iteration.
Non-uniqueness and instability are characteristic features of image reconstruction processes. As a result, it is necessary to develop regularization methods that can be used to compute reliable approximate solutions. A regularization method provides of a family of stable reconstructions that converge to an exact solution of the noise-free problem as the noise level tends to zero. The standard regularization technique is defined by variational image reconstruction, which minimizes a data discrepancy augmented by a regularizer. The actual numerical implementation makes use of iterative methods, often involving proximal mappings of the regularizer. In recent years, plug-and-play image reconstruction (PnP) has been developed as a new powerful generalization of variational methods based on replacing proximal mappings by more general image denoisers. While PnP iterations yield excellent results, neither stability nor convergence in the sense of regularization has been studied so far. In this work, we extend the idea of PnP by considering families of PnP iterations, each being accompanied by its own denoiser. As our main theoretical result, we show that such PnP reconstructions lead to stable and convergent regularization methods. This shows for the first time that PnP is mathematically equally justified for robust image reconstruction as variational methods
Explicitly accounting for uncertainties is paramount to the safety of engineering structures. Optimization which is often carried out at the early stage of the structural design offers an ideal framework for this task. When the uncertainties are mainly affecting the objective function, robust design optimization is traditionally considered. This work further assumes the existence of multiple and competing objective functions that need to be dealt with simultaneously. The optimization problem is formulated by considering quantiles of the objective functions which allows for the combination of both optimality and robustness in a single metric. By introducing the concept of common random numbers, the resulting nested optimization problem may be solved using a general-purpose solver, herein the non-dominated sorting genetic algorithm (NSGA-II). The computational cost of such an approach is however a serious hurdle to its application in real-world problems. We therefore propose a surrogate-assisted approach using Kriging as an inexpensive approximation of the associated computational model. The proposed approach consists of sequentially carrying out NSGA-II while using an adaptively built Kriging model to estimate the quantiles. Finally, the methodology is adapted to account for mixed categorical-continuous parameters as the applications involve the selection of qualitative design parameters as well. The methodology is first applied to two analytical examples showing its efficiency. The third application relates to the selection of optimal renovation scenarios of a building considering both its life cycle cost and environmental impact. It shows that when it comes to renovation, the heating system replacement should be the priority.
In this article we present a numerical analysis for a third-order differential equation with non-periodic boundary conditions and time-dependent coefficients, namely, the linear Korteweg-de Vries Burgers equation. This numerical analysis is motived due to the dispersive and dissipative phenomena that government this kind of equations. This work builds on previous methods for dispersive equations with constant coefficients, expanding the field to include a new class of equations which until now have eluded the time-evolving parameters. More precisely, throughout the Legendre-Petrov-Galerkin method we prove stability and convergence results of the approximation in appropriate weighted Sobolev spaces. These results allow to show the role and trade off of these temporal parameters into the model. Afterwards, we numerically investigate the dispersion-dissipation relation for several profiles, further provide insights into the implementation method, which allow to exhibit the accuracy and efficiency of our numerical algorithms.
We present a novel mathematical framework for computing the number of maintenance cycles in a system with critical and non-critical components, where "critical" (CR) means that the component's failure is fatal for the system's operation and renders any more repairs inapplicable, whereas "noncritical" (NC) means that the component can undergo corrective maintenance (replacement or minimal repair) whenever it fails, provided that the CR component is still in operation. Whenever the NC component fails, the CR component can optionally be preventively replaced. We extend traditional renewal theory (whether classical or generalized) for various maintenance scenarios for a system composed of one CR and one NC component in order to compute the average number of renewals of NC under the restriction ("bound") necessitated by CR. We also develop approximations in closed form for the proposed "bounded" renewal functions. We validate our formulas by simulations on a variety of component lifetime distributions, including actual lifetime distributions of wind turbine components.
Recent developments in in-situ monitoring and process control in Additive Manufacturing (AM), also known as 3D-printing, allows the collection of large amounts of emission data during the build process of the parts being manufactured. This data can be used as input into 3D and 2D representations of the 3D-printed parts. However the analysis and use, as well as the characterization of this data still remains a manual process. The aim of this paper is to propose an adaptive human-in-the-loop approach using Machine Learning techniques that automatically inspect and annotate the emissions data generated during the AM process. More specifically, this paper will look at two scenarios: firstly, using convolutional neural networks (CNNs) to automatically inspect and classify emission data collected by in-situ monitoring and secondly, applying Active Learning techniques to the developed classification model to construct a human-in-the-loop mechanism in order to accelerate the labeling process of the emission data. The CNN-based approach relies on transfer learning and fine-tuning, which makes the approach applicable to other industrial image patterns. The adaptive nature of the approach is enabled by uncertainty sampling strategy to automatic selection of samples to be presented to human experts for annotation.
In this paper, we present a new and effective simulation-based approach to conduct both finite- and large-sample inference for high-dimensional linear regression models. This approach is developed under the so-called repro samples framework, in which we conduct statistical inference by creating and studying the behavior of artificial samples that are obtained by mimicking the sampling mechanism of the data. We obtain confidence sets for (a) the true model corresponding to the nonzero coefficients, (b) a single or any collection of regression coefficients, and (c) both the model and regression coefficients jointly. We also extend our approaches to drawing inferences on functions of the regression coefficients. The proposed approach fills in two major gaps in the high-dimensional regression literature: (1) lack of effective approaches to address model selection uncertainty and provide valid inference for the underlying true model; (2) lack of effective inference approaches that guarantee finite-sample performances. We provide both finite-sample and asymptotic results to theoretically guarantee the performances of the proposed methods. In addition, our numerical results demonstrate that the proposed methods are valid and achieve better coverage with smaller confidence sets than the existing state-of-art approaches, such as debiasing and bootstrap approaches.
We study the sparse high-dimensional Gaussian mixture model when the number of clusters is allowed to grow with the sample size. A minimax lower bound for parameter estimation is established, and we show that a constrained maximum likelihood estimator achieves the minimax lower bound. However, this optimization-based estimator is computationally intractable because the objective function is highly nonconvex and the feasible set involves discrete structures. To address the computational challenge, we propose a Bayesian approach to estimate high-dimensional Gaussian mixtures whose cluster centers exhibit sparsity using a continuous spike-and-slab prior. Posterior inference can be efficiently computed using an easy-to-implement Gibbs sampler. We further prove that the posterior contraction rate of the proposed Bayesian method is minimax optimal. The mis-clustering rate is obtained as a by-product using tools from matrix perturbation theory. The proposed Bayesian sparse Gaussian mixture model does not require pre-specifying the number of clusters, which can be adaptively estimated via the Gibbs sampler. The validity and usefulness of the proposed method is demonstrated through simulation studies and the analysis of a real-world single-cell RNA sequencing dataset.
Machine learning algorithms for three-dimensional mean-curvature computation in the level-set method
We propose a data-driven mean-curvature solver for the level-set method. This work is the natural extension to $\mathbb{R}^3$ of our two-dimensional strategy in [DOI: 10.1007/s10915-022-01952-2][1] and the hybrid inference system of [DOI: 10.1016/j.jcp.2022.111291][2]. However, in contrast to [1,2], which built resolution-dependent neural-network dictionaries, here we develop a pair of models in $\mathbb{R}^3$, regardless of the mesh size. Our feedforward networks ingest transformed level-set, gradient, and curvature data to fix numerical mean-curvature approximations selectively for interface nodes. To reduce the problem's complexity, we have used the Gaussian curvature to classify stencils and fit our models separately to non-saddle and saddle patterns. Non-saddle stencils are easier to handle because they exhibit a curvature error distribution characterized by monotonicity and symmetry. While the latter has allowed us to train only on half the mean-curvature spectrum, the former has helped us blend the data-driven and the baseline estimations seamlessly near flat regions. On the other hand, the saddle-pattern error structure is less clear; thus, we have exploited no latent information beyond what is known. In this regard, we have trained our models on not only spherical but also sinusoidal and hyperbolic paraboloidal patches. Our approach to building their data sets is systematic but gleans samples randomly while ensuring well-balancedness. We have also resorted to standardization and dimensionality reduction and integrated regularization to minimize outliers. In addition, we leverage curvature rotation/reflection invariance to improve precision at inference time. Several experiments confirm that our proposed system can yield more accurate mean-curvature estimations than modern particle-based interface reconstruction and level-set schemes around under-resolved regions.
Engineers and scientists have been collecting and analyzing fatigue data since the 1800s to ensure the reliability of life-critical structures. Applications include (but are not limited to) bridges, building structures, aircraft and spacecraft components, ships, ground-based vehicles, and medical devices. Engineers need to estimate S-N relationships (Stress or Strain versus Number of cycles to failure), typically with a focus on estimating small quantiles of the fatigue-life distribution. Estimates from this kind of model are used as input to models (e.g., cumulative damage models) that predict failure-time distributions under varying stress patterns. Also, design engineers need to estimate lower-tail quantiles of the closely related fatigue-strength distribution. The history of applying incorrect statistical methods is nearly as long and such practices continue to the present. Examples include treating the applied stress (or strain) as the response and the number of cycles to failure as the explanatory variable in regression analyses (because of the need to estimate strength distributions) and ignoring or otherwise mishandling censored observations (known as runouts in the fatigue literature). The first part of the paper reviews the traditional modeling approach where a fatigue-life model is specified. We then show how this specification induces a corresponding fatigue-strength model. The second part of the paper presents a novel alternative modeling approach where a fatigue-strength model is specified and a corresponding fatigue-life model is induced. We explain and illustrate the important advantages of this new modeling approach.
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