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We consider a one-dimensional singularly perturbed 4th order problem with the additional feature of a shift term. An expansion into a smooth term, boundary layers and an inner layer yields a formal solution decomposition, and together with a stability result we have estimates for the subsequent numerical analysis. With classical layer adapted meshes we present a numerical method, that achieves supercloseness and optimal convergence orders in the associated energy norm. We also consider coarser meshes in view of the weak layers. Some numerical examples conclude the paper and support the theory.

We present a novel stabilized isogeometric formulation for the Stokes problem, where the geometry of interest is obtained via overlapping NURBS (non-uniform rational B-spline) patches, i.e., one patch on top of another in an arbitrary but predefined hierarchical order. All the visible regions constitute the computational domain, whereas independent patches are coupled through visible interfaces using Nitsche's formulation. Such a geometric representation inevitably involves trimming, which may yield trimmed elements of extremely small measures (referred to as bad elements) and thus lead to the instability issue. Motivated by the minimal stabilization method that rigorously guarantees stability for trimmed geometries [1], in this work we generalize it to the Stokes problem on overlapping patches. Central to our method is the distinct treatments for the pressure and velocity spaces: Stabilization for velocity is carried out for the flux terms on interfaces, whereas pressure is stabilized in all the bad elements. We provide a priori error estimates with a comprehensive theoretical study. Through a suite of numerical tests, we first show that optimal convergence rates are achieved, which consistently agrees with our theoretical findings. Second, we show that the accuracy of pressure is significantly improved by several orders using the proposed stabilization method, compared to the results without stabilization. Finally, we also demonstrate the flexibility and efficiency of the proposed method in capturing local features in the solution field.

Information inequalities appear in many database applications such as query output size bounds, query containment, and implication between data dependencies. Recently Khamis et al. proposed to study the algorithmic aspects of information inequalities, including the information inequality problem: decide whether a linear inequality over entropies of random variables is valid. While the decidability of this problem is a major open question, applications often involve only inequalities that adhere to specific syntactic forms linked to useful semantic invariance properties. This paper studies the information inequality problem in different syntactic and semantic scenarios that arise from database applications. Focusing on the boundary between tractability and intractability, we show that the information inequality problem is coNP-complete if restricted to normal polymatroids, and in polynomial time if relaxed to monotone functions. We also examine syntactic restrictions related to query output size bounds, and provide an alternative proof, through monotone functions, for the polynomial-time computability of the entropic bound over simple sets of degree constraints.

We give a short survey of recent results on sparse-grid linear algorithms of approximate recovery and integration of functions possessing a unweighted or weighted Sobolev mixed smoothness based on their sampled values at a certain finite set. Some of them are extended to more general cases.

Traditional applications of latent class models (LCMs) often focus on scenarios where a set of unobserved classes are well-defined and easily distinguishable. However, in numerous real-world applications, these classes are weakly separated and difficult to distinguish, creating significant numerical challenges. To address these issues, we have developed an R package ddtlcm that provides comprehensive analysis and visualization tools designed to enhance the robustness and interpretability of LCMs in the presence of weak class separation, particularly useful for small sample sizes. This package implements a tree-regularized Bayesian LCM that leverages statistical strength between latent classes to make better estimates using limited data. A Shiny app has also been developed to improve user interactivity. In this paper, we showcase a typical analysis pipeline with simulated data using ddtlcm. All software has been made publicly available on CRAN and GitHub.

Ordinary state-based peridynamic (OSB-PD) models have an unparalleled capability to simulate crack propagation phenomena in solids with arbitrary Poisson's ratio. However, their non-locality also leads to prohibitively high computational cost. In this paper, a fast solution scheme for OSB-PD models based on matrix operation is introduced, with which, the graphics processing units (GPUs) are used to accelerate the computation. For the purpose of comparison and verification, a commonly used solution scheme based on loop operation is also presented. An in-house software is developed in MATLAB. Firstly, the vibration of a cantilever beam is solved for validating the loop- and matrix-based schemes by comparing the numerical solutions to those produced by a FEM software. Subsequently, two typical dynamic crack propagation problems are simulated to illustrate the effectiveness of the proposed schemes in solving dynamic fracture problems. Finally, the simulation of the Brokenshire torsion experiment is carried out by using the matrix-based scheme, and the similarity in the shapes of the experimental and numerical broken specimens further demonstrates the ability of the proposed approach to deal with 3D non-planar fracture problems. In addition, the speed-up of the matrix-based scheme with respect to the loop-based scheme and the performance of the GPU acceleration are investigated. The results emphasize the high computational efficiency of the matrix-based implementation scheme.

Parametric mathematical models such as parameterizations of partial differential equations with random coefficients have received a lot of attention within the field of uncertainty quantification. The model uncertainties are often represented via a series expansion in terms of the parametric variables. In practice, this series expansion needs to be truncated to a finite number of terms, introducing a dimension truncation error to the numerical simulation of a parametric mathematical model. There have been several studies of the dimension truncation error corresponding to different models of the input random field in recent years, but many of these analyses have been carried out within the context of numerical integration. In this paper, we study the $L^2$ dimension truncation error of the parametric model problem. Estimates of this kind arise in the assessment of the dimension truncation error for function approximation in high dimensions. In addition, we show that the dimension truncation error rate is invariant with respect to certain transformations of the parametric variables. Numerical results are presented which showcase the sharpness of the theoretical results.

An extension of Cencov's categorical description of classical inference theory to the domain of quantum systems is presented. It provides a novel categorical foundation to the theory of quantum information that embraces both classical and quantum information theory in a natural way, while also allowing to formalise the notion of quantum environment. A first application of these ideas is provided by extending the notion of statistical manifold to incorporate categories, and investigating a possible, uniparametric Cramer-Rao inequality in this setting.

We propose an approach to compute inner and outer-approximations of the sets of values satisfying constraints expressed as arbitrarily quantified formulas. Such formulas arise for instance when specifying important problems in control such as robustness, motion planning or controllers comparison. We propose an interval-based method which allows for tractable but tight approximations. We demonstrate its applicability through a series of examples and benchmarks using a prototype implementation.

Since the invention of word2vec, the skip-gram model has significantly advanced the research of network embedding, such as the recent emergence of the DeepWalk, LINE, PTE, and node2vec approaches. In this work, we show that all of the aforementioned models with negative sampling can be unified into the matrix factorization framework with closed forms. Our analysis and proofs reveal that: (1) DeepWalk empirically produces a low-rank transformation of a network's normalized Laplacian matrix; (2) LINE, in theory, is a special case of DeepWalk when the size of vertices' context is set to one; (3) As an extension of LINE, PTE can be viewed as the joint factorization of multiple networks' Laplacians; (4) node2vec is factorizing a matrix related to the stationary distribution and transition probability tensor of a 2nd-order random walk. We further provide the theoretical connections between skip-gram based network embedding algorithms and the theory of graph Laplacian. Finally, we present the NetMF method as well as its approximation algorithm for computing network embedding. Our method offers significant improvements over DeepWalk and LINE for conventional network mining tasks. This work lays the theoretical foundation for skip-gram based network embedding methods, leading to a better understanding of latent network representation learning.