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In this work a general semi-parametric multivariate model where the first two conditional moments are assumed to be multivariate time series is introduced. The focus of the estimation is the conditional mean parameter vector for discrete-valued distributions. Quasi-Maximum Likelihood Estimators (QMLEs) based on the linear exponential family are typically employed for such estimation problems when the true multivariate conditional probability distribution is unknown or too complex. Although QMLEs provide consistent estimates they may be inefficient. In this paper novel two-stage Multivariate Weighted Least Square Estimators (MWLSEs) are introduced which enjoy the same consistency property as the QMLEs but can provide improved efficiency with suitable choice of the covariance matrix of the observations. The proposed method allows for a more accurate estimation of model parameters in particular for count and categorical data when maximum likelihood estimation is unfeasible. Moreover, consistency and asymptotic normality of MWLSEs are derived. The estimation performance of QMLEs and MWLSEs is compared through simulation experiments and a real data application, showing superior accuracy of the proposed methodology.

Models of complex technological systems inherently contain interactions and dependencies among their input variables that affect their joint influence on the output. Such models are often computationally expensive and few sensitivity analysis methods can effectively process such complexities. Moreover, the sensitivity analysis field as a whole pays limited attention to the nature of interaction effects, whose understanding can prove to be critical for the design of safe and reliable systems. In this paper, we introduce and extensively test a simple binning approach for computing sensitivity indices and demonstrate how complementing it with the smart visualization method, simulation decomposition (SimDec), can permit important insights into the behavior of complex engineering models. The simple binning approach computes first-, second-order effects, and a combined sensitivity index, and is considerably more computationally efficient than Sobol' indices. The totality of the sensitivity analysis framework provides an efficient and intuitive way to analyze the behavior of complex systems containing interactions and dependencies.

Differential geometric approaches are ubiquitous in several fields of mathematics, physics and engineering, and their discretizations enable the development of network-based mathematical and computational frameworks, which are essential for large-scale data science. The Forman-Ricci curvature (FRC) - a statistical measure based on Riemannian geometry and designed for networks - is known for its high capacity for extracting geometric information from complex networks. However, extracting information from dense networks is still challenging due to the combinatorial explosion of high-order network structures. Motivated by this challenge we sought a set-theoretic representation theory for high-order network cells and FRC, as well as their associated concepts and properties, which together provide an alternative and efficient formulation for computing high-order FRC in complex networks. We provide a pseudo-code, a software implementation coined FastForman, as well as a benchmark comparison with alternative implementations. Crucially, our representation theory reveals previous computational bottlenecks and also accelerates the computation of FRC. As a consequence, our findings open new research possibilities in complex systems where higher-order geometric computations are required.

High-dimensional central limit theorems have been intensively studied with most focus being on the case where the data is sub-Gaussian or sub-exponential. However, heavier tails are omnipresent in practice. In this article, we study the critical growth rates of dimension $d$ below which Gaussian approximations are asymptotically valid but beyond which they are not. We are particularly interested in how these thresholds depend on the number of moments $m$ that the observations possess. For every $m\in(2,\infty)$, we construct i.i.d. random vectors $\textbf{X}_1,...,\textbf{X}_n$ in $\mathbb{R}^d$, the entries of which are independent and have a common distribution (independent of $n$ and $d$) with finite $m$th absolute moment, and such that the following holds: if there exists an $\varepsilon\in(0,\infty)$ such that $d/n^{m/2-1+\varepsilon}\not\to 0$, then the Gaussian approximation error (GAE) satisfies $$ \limsup_{n\to\infty}\sup_{t\in\mathbb{R}}\left[\mathbb{P}\left(\max_{1\leq j\leq d}\frac{1}{\sqrt{n}}\sum_{i=1}^n\textbf{X}_{ij}\leq t\right)-\mathbb{P}\left(\max_{1\leq j\leq d}\textbf{Z}_j\leq t\right)\right]=1,$$ where $\textbf{Z} \sim \mathsf{N}_d(\textbf{0}_d,\mathbf{I}_d)$. On the other hand, a result in Chernozhukov et al. (2023a) implies that the left-hand side above is zero if just $d/n^{m/2-1-\varepsilon}\to 0$ for some $\varepsilon\in(0,\infty)$. In this sense, there is a moment-dependent phase transition at the threshold $d=n^{m/2-1}$ above which the limiting GAE jumps from zero to one.

A new mechanical model on noncircular shallow tunnelling considering initial stress field is proposed in this paper by constraining far-field ground surface to eliminate displacement singularity at infinity, and the originally unbalanced tunnel excavation problem in existing solutions is turned to an equilibrium one of mixed boundaries. By applying analytic continuation, the mixed boundaries are transformed to a homogenerous Riemann-Hilbert problem, which is subsequently solved via an efficient and accurate iterative method with boundary conditions of static equilibrium, displacement single-valuedness, and traction along tunnel periphery. The Lanczos filtering technique is used in the final stress and displacement solution to reduce the Gibbs phenomena caused by the constrained far-field ground surface for more accurte results. Several numerical cases are conducted to intensively verify the proposed solution by examining boundary conditions and comparing with existing solutions, and all the results are in good agreements. Then more numerical cases are conducted to investigate the stress and deformation distribution along ground surface and tunnel periphery, and several engineering advices are given. Further discussions on the defects of the proposed solution are also conducted for objectivity.

The proximal Galerkin finite element method is a high-order, low iteration complexity, nonlinear numerical method that preserves the geometric and algebraic structure of pointwise bound constraints in infinite-dimensional function spaces. This paper introduces the proximal Galerkin method and applies it to solve free boundary problems, enforce discrete maximum principles, and develop a scalable, mesh-independent algorithm for optimal design problems with pointwise bound constraints. This paper also provides a derivation of the latent variable proximal point (LVPP) algorithm, an unconditionally stable alternative to the interior point method. LVPP is an infinite-dimensional optimization algorithm that may be viewed as having an adaptive barrier function that is updated with a new informative prior at each (outer loop) optimization iteration. One of its main benefits is witnessed when analyzing the classical obstacle problem. Therein, we find that the original variational inequality can be replaced by a sequence of partial differential equations (PDEs) that are readily discretized and solved with, e.g., high-order finite elements. Throughout this work, we arrive at several unexpected contributions that may be of independent interest. These include (1) a semilinear PDE we refer to as the entropic Poisson equation; (2) an algebraic/geometric connection between high-order positivity-preserving discretizations and certain infinite-dimensional Lie groups; and (3) a gradient-based, bound-preserving algorithm for two-field density-based topology optimization. The complete latent variable proximal Galerkin methodology combines ideas from nonlinear programming, functional analysis, tropical algebra, and differential geometry and can potentially lead to new synergies among these areas as well as within variational and numerical analysis.

Rational function approximations provide a simple but flexible alternative to polynomial approximation, allowing one to capture complex non-linearities without oscillatory artifacts. However, there have been few attempts to use rational functions on noisy data due to the likelihood of creating spurious singularities. To avoid the creation of singularities, we use Bernstein polynomials and appropriate conditions on their coefficients to force the denominator to be strictly positive. While this reduces the range of rational polynomials that can be expressed, it keeps all the benefits of rational functions while maintaining the robustness of polynomial approximation in noisy data scenarios. Our numerical experiments on noisy data show that existing rational approximation methods continually produce spurious poles inside the approximation domain. This contrasts our method, which cannot create poles in the approximation domain and provides better fits than a polynomial approximation and even penalized splines on functions with multiple variables. Moreover, guaranteeing pole-free in an interval is critical for estimating non-constant coefficients when numerically solving differential equations using spectral methods. This provides a compact representation of the original differential equation, allowing numeric solvers to achieve high accuracy quickly, as seen in our experiments.

Friction drag from a turbulent fluid moving past or inside an object plays a crucial role in domains as diverse as transportation, public utility infrastructure, energy technology, and human health. As a direct measure of the shear-induced friction forces, an accurate prediction of the wall-shear stress can contribute to sustainability, conservation of resources, and carbon neutrality in civil aviation as well as enhanced medical treatment of vascular diseases and cancer. Despite such importance for our modern society, we still lack adequate experimental methods to capture the instantaneous wall-shear stress dynamics. In this contribution, we present a holistic approach that derives velocity and wall-shear stress fields with impressive spatial and temporal resolution from flow measurements using a deep optical flow estimator with physical knowledge. The validity and physical correctness of the derived flow quantities is demonstrated with synthetic and real-world experimental data covering a range of relevant fluid flows.

Nonignorable missing outcomes are common in real world datasets and often require strong parametric assumptions to achieve identification. These assumptions can be implausible or untestable, and so we may forgo them in favour of partially identified models that narrow the set of a priori possible values to an identification region. Here we propose a new nonparametric Bayes method that allows for the incorporation of multiple clinically relevant restrictions of the parameter space simultaneously. We focus on two common restrictions, instrumental variables and the direction of missing data bias, and investigate how these restrictions narrow the identification region for parameters of interest. Additionally, we propose a rejection sampling algorithm that allows us to quantify the evidence for these assumptions in the data. We compare our method to a standard Heckman selection model in both simulation studies and in an applied problem examining the effectiveness of cash-transfers for people experiencing homelessness.

Hashing has been widely used in approximate nearest search for large-scale database retrieval for its computation and storage efficiency. Deep hashing, which devises convolutional neural network architecture to exploit and extract the semantic information or feature of images, has received increasing attention recently. In this survey, several deep supervised hashing methods for image retrieval are evaluated and I conclude three main different directions for deep supervised hashing methods. Several comments are made at the end. Moreover, to break through the bottleneck of the existing hashing methods, I propose a Shadow Recurrent Hashing(SRH) method as a try. Specifically, I devise a CNN architecture to extract the semantic features of images and design a loss function to encourage similar images projected close. To this end, I propose a concept: shadow of the CNN output. During optimization process, the CNN output and its shadow are guiding each other so as to achieve the optimal solution as much as possible. Several experiments on dataset CIFAR-10 show the satisfying performance of SRH.