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We introduce a filtering technique for Discontinuous Galerkin approximations of hyperbolic problems. Following an approach already proposed for the Hamilton-Jacobi equations by other authors, we aim at reducing the spurious oscillations that arise in presence of discontinuities when high order spatial discretizations are employed. This goal is achieved using a filter function that keeps the high order scheme when the solution is regular and switches to a monotone low order approximation if it is not. The method has been implemented in the framework of the $deal.II$ numerical library, whose mesh adaptation capabilities are also used to reduce the region in which the low order approximation is used. A number of numerical experiments demonstrate the potential of the proposed filtering technique.

Real-time coordination of distributed energy resources (DERs) is crucial for regulating the voltage profile in distribution grids. By capitalizing on a scalable neural network (NN) architecture, one can attain decentralized DER decisions to address the lack of real-time communications. This paper develops an advanced learning-enabled DER coordination scheme by accounting for the potential risks associated with reactive power prediction and voltage deviation. Such risks are quantified by the conditional value-at-risk (CVaR) using the worst-case samples only, and we propose a mini-batch selection algorithm to address the training speed issue in minimizing the CVaR-regularized loss. Numerical tests using real-world data on the IEEE 123-bus test case have demonstrated the computation and safety improvements of the proposed risk-aware learning algorithm for decentralized DER decision making, especially in terms of reducing feeder voltage violations.

This paper makes the first attempt to apply newly developed upwind GFDM for the meshless solution of two-phase porous flow equations. In the presented method, node cloud is used to flexibly discretize the computational domain, instead of complicated mesh generation. Combining with moving least square approximation and local Taylor expansion, spatial derivatives of oil-phase pressure at a node are approximated by generalized difference operators in the local influence domain of the node. By introducing the first-order upwind scheme of phase relative permeability, and combining the discrete boundary conditions, fully-implicit GFDM-based nonlinear discrete equations of the immiscible two-phase porous flow are obtained and solved by the nonlinear solver based on the Newton iteration method with the automatic differentiation, to avoid the additional computational cost and possible computational instability caused by sequentially coupled scheme. Two numerical examples are implemented to test the computational performances of the presented method. Detailed error analysis finds the two sources of the calculation error, roughly studies the convergence order thus find that the low-order error of GFDM makes the convergence order of GFDM lower than that of FDM when node spacing is small, and points out the significant effect of the symmetry or uniformity of the node collocation in the node influence domain on the accuracy of generalized difference operators, and the radius of the node influence domain should be small to achieve high calculation accuracy, which is a significant difference between the studied hyperbolic two-phase porous flow problem and the elliptic problems when GFDM is applied.

Semi-grant-free (SGF) transmission scheme enables grant-free (GF) users to utilize resource blocks allocated for grant-based (GB) users while maintaining the quality of service of GB users. This work investigates the secrecy performance of non-orthogonal multiple access (NOMA)-aided SGF systems. First, analytical expressions for the exact and asymptotic secrecy outage probability (SOP) of NOMA-aided SGF systems with a single GF user are derived. Then, the SGF systems with multiple GF users and a best-user scheduling scheme is considered. By utilizing order statistics theory, closed-form expressions for the exact and asymptotic SOP are derived. Monte Carlo simulation results demonstrate the effects of system parameters on the SOP of the considered system and verify the accuracy of the developed analytical results. The results indicate that both the outage target rate for GB and the secure target rate for GF are the main factors of the secrecy performance of SGF systems.

In this article we suggest two discretization methods based on isogeometric analysis (IGA) for planar linear elasticity. On the one hand, we apply the well-known ansatz of weakly imposed symmetry for the stress tensor and obtain a well-posed mixed formulation. Such modified mixed problems have been already studied by different authors. But we concentrate on the exploitation of IGA results to handle also curved boundary geometries. On the other hand, we consider the more complicated situation of strong symmetry, i.e. we discretize the mixed weak form determined by the so-called Hellinger-Reissner variational principle. We show the existence of suitable approximate fields leading to an inf-sup stable saddle-point problem. For both discretization approaches we prove convergence statements and in case of weak symmetry we illustrate the approximation behavior by means of several numerical experiments.

Multigrid is a powerful solver for large-scale linear systems arising from discretized partial differential equations. The convergence theory of multigrid methods for symmetric positive definite problems has been well developed over the past decades, while, for nonsymmetric problems, such theory is still not mature. As a foundation for multigrid analysis, two-grid convergence theory plays an important role in motivating multigrid algorithms. Regarding two-grid methods for nonsymmetric problems, most previous works focus on the spectral radius of iteration matrix or rely on convergence measures that are typically difficult to compute in practice. Moreover, the existing results are confined to two-grid methods with exact solution of the coarse-grid system. In this paper, we analyze the convergence of a two-grid method for nonsymmetric positive definite problems (e.g., linear systems arising from the discretizations of convection-diffusion equations). In the case of exact coarse solver, we establish an elegant identity for characterizing two-grid convergence factor, which is measured by a smoother-induced norm. The identity can be conveniently used to derive a class of optimal restriction operators and analyze how the convergence factor is influenced by restriction. More generally, we present some convergence estimates for an inexact variant of the two-grid method, in which both linear and nonlinear coarse solvers are considered.

Recently, numerous studies have demonstrated the presence of bias in machine learning powered decision-making systems. Although most definitions of algorithmic bias have solid mathematical foundations, the corresponding bias detection techniques often lack statistical rigor, especially for non-iid data. We fill this gap in the literature by presenting a rigorous non-parametric testing procedure for bias according to Predictive Rate Parity, a commonly considered notion of algorithmic bias. We adapt traditional asymptotic results for non-parametric estimators to test for bias in the presence of dependence commonly seen in user-level data generated by technology industry applications and illustrate how these approaches can be leveraged for mitigation. We further propose modifications of this methodology to address bias measured through marginal outcome disparities in classification settings and extend notions of predictive rate parity to multi-objective models. Experimental results on real data show the efficacy of the proposed detection and mitigation methods.

Bayesian model selection provides a powerful framework for objectively comparing models directly from observed data, without reference to ground truth data. However, Bayesian model selection requires the computation of the marginal likelihood (model evidence), which is computationally challenging, prohibiting its use in many high-dimensional Bayesian inverse problems. With Bayesian imaging applications in mind, in this work we present the proximal nested sampling methodology to objectively compare alternative Bayesian imaging models for applications that use images to inform decisions under uncertainty. The methodology is based on nested sampling, a Monte Carlo approach specialised for model comparison, and exploits proximal Markov chain Monte Carlo techniques to scale efficiently to large problems and to tackle models that are log-concave and not necessarily smooth (e.g., involving l_1 or total-variation priors). The proposed approach can be applied computationally to problems of dimension O(10^6) and beyond, making it suitable for high-dimensional inverse imaging problems. It is validated on large Gaussian models, for which the likelihood is available analytically, and subsequently illustrated on a range of imaging problems where it is used to analyse different choices of dictionary and measurement model.

The minimum energy path (MEP) describes the mechanism of reaction, and the energy barrier along the path can be used to calculate the reaction rate in thermal systems. The nudged elastic band (NEB) method is one of the most commonly used schemes to compute MEPs numerically. It approximates an MEP by a discrete set of configuration images, where the discretization size determines both computational cost and accuracy of the simulations. In this paper, we consider a discrete MEP to be a stationary state of the NEB method and prove an optimal convergence rate of the discrete MEP with respect to the number of images. Numerical simulations for the transitions of some several proto-typical model systems are performed to support the theory.

The numerical solution of singular eigenvalue problems is complicated by the fact that small perturbations of the coefficients may have an arbitrarily bad effect on eigenvalue accuracy. However, it has been known for a long time that such perturbations are exceptional and standard eigenvalue solvers, such as the QZ algorithm, tend to yield good accuracy despite the inevitable presence of roundoff error. Recently, Lotz and Noferini quantified this phenomenon by introducing the concept of $\delta$-weak eigenvalue condition numbers. In this work, we consider singular quadratic eigenvalue problems and two popular linearizations. Our results show that a correctly chosen linearization increases $\delta$-weak eigenvalue condition numbers only marginally, justifying the use of these linearizations in numerical solvers also in the singular case. We propose a very simple but often effective algorithm for computing well-conditioned eigenvalues of a singular quadratic eigenvalue problems by adding small random perturbations to the coefficients. We prove that the eigenvalue condition number is, with high probability, a reliable criterion for detecting and excluding spurious eigenvalues created from the singular part.