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We present novel Monte Carlo (MC) and multilevel Monte Carlo (MLMC) methods to determine the unbiased covariance of random variables using h-statistics. The advantage of this procedure lies in the unbiased construction of the estimator's mean square error in a closed form. This is in contrast to conventional MC and MLMC covariance estimators, which are based on biased mean square errors defined solely by upper bounds, particularly within the MLMC. The numerical results of the algorithms are demonstrated by estimating the covariance of the stochastic response of a simple 1D stochastic elliptic PDE such as Poisson's model.

Randomized iterative methods, such as the Kaczmarz method and its variants, have gained growing attention due to their simplicity and efficiency in solving large-scale linear systems. Meanwhile, absolute value equations (AVE) have attracted increasing interest due to their connection with the linear complementarity problem. In this paper, we investigate the application of randomized iterative methods to generalized AVE (GAVE). Our approach differs from most existing works in that we tackle GAVE with non-square coefficient matrices. We establish more comprehensive sufficient and necessary conditions for characterizing the solvability of GAVE and propose precise error bound conditions. Furthermore, we introduce a flexible and efficient randomized iterative algorithmic framework for solving GAVE, which employs sampling matrices drawn from user-specified distributions. This framework is capable of encompassing many well-known methods, including the Picard iteration method and the randomized Kaczmarz method. Leveraging our findings on solvability and error bounds, we establish both almost sure convergence and linear convergence rates for this versatile algorithmic framework. Finally, we present numerical examples to illustrate the advantages of the new algorithms.

Satellite imagery has played an increasingly important role in post-disaster building damage assessment. Unfortunately, current methods still rely on manual visual interpretation, which is often time-consuming and can cause very low accuracy. To address the limitations of manual interpretation, there has been a significant increase in efforts to automate the process. We present a solution that performs the two most important tasks in building damage assessment, segmentation and classification, through deep-learning models. We show our results submitted as part of the xView2 Challenge, a competition to design better models for identifying buildings and their damage level after exposure to multiple kinds of natural disasters. Our best model couples a building identification semantic segmentation convolutional neural network (CNN) to a building damage classification CNN, with a combined F1 score of 0.66, surpassing the xView2 challenge baseline F1 score of 0.28. We find that though our model was able to identify buildings with relatively high accuracy, building damage classification across various disaster types is a difficult task due to the visual similarity between different damage levels and different damage distribution between disaster types, highlighting the fact that it may be important to have a probabilistic prior estimate regarding disaster damage in order to obtain accurate predictions.

An additive Runge-Kutta method is used for the time stepping, which integrates the linear stiff terms by an explicit singly diagonally implicit Runge-Kutta (ESDIRK) method and the nonlinear terms by an explicit Runge-Kutta (ERK) method. In each time step, the implicit solve is performed by the recently developed Hierarchical Poincar\'e-Steklov (HPS) method. This is a fast direct solver for elliptic equations that decomposes the space domain into a hierarchical tree of subdomains and builds spectral collocation solvers locally on the subdomains. These ideas are naturally combined in the presented method since the singly diagonal coefficient in ESDIRK and a fixed time-step ensures that the coefficient matrix in the implicit solve of HPS remains the same for all time stages. This means that the precomputed inverse can be efficiently reused, leading to a scheme with complexity (in two dimensions) $\mathcal{O}(N^{1.5})$ for the precomputation where the solution operator to the elliptic problems is built, and then $\mathcal{O}(N \log N)$ for the solve in each time step. The stability of the method is proved for first order in time and any order in space, and numerical evidence substantiates a claim of stability for a much broader class of time discretization methods. Numerical experiments supporting the accuracy of efficiency of the method in one and two dimensions are presented.

Discretization of the uniform norm of functions from a given finite dimensional subspace of continuous functions is studied. Previous known results show that for any $N$-dimensional subspace of the space of continuous functions it is sufficient to use $e^{CN}$ sample points for an accurate upper bound for the uniform norm by the discrete norm and that one cannot improve on the exponential growth of the number of sampling points for a good discretization theorem in the uniform norm. In this paper we focus on two types of results, which allow us to obtain good discretization of the uniform norm with polynomial in $N$ number of points. In the first way we weaken the discretization inequality by allowing a bound of the uniform norm by the discrete norm multiplied by an extra factor, which may depend on $N$. In the second way we impose restrictions on the finite dimensional subspace under consideration. In particular, we prove a general result, which connects the upper bound on the number of sampling points in the discretization theorem for the uniform norm with the best $m$-term bilinear approximation of the Dirichlet kernel associated with the given subspace.

We prove that throughout the satisfiable phase, the logarithm of the number of satisfying assignments of a random 2-SAT formula satisfies a central limit theorem. This implies that the log of the number of satisfying assignments exhibits fluctuations of order $\sqrt n$, with $n$ the number of variables. The formula for the variance can be evaluated effectively. By contrast, for numerous other random constraint satisfaction problems the typical fluctuations of the logarithm of the number of solutions are {\em bounded} throughout all or most of the satisfiable regime.

A novel fourth-order finite difference formula coupling the Crank-Nicolson explicit linearized method is proposed to solve Riesz space fractional nonlinear reaction-diffusion equations in two dimensions. Theoretically, under the Lipschitz assumption on the nonlinear term, the proposed high-order scheme is proved to be unconditionally stable and convergent in the discrete $L_2$-norm. Moreover, a $\tau$-matrix based preconditioner is developed to speed up the convergence of the conjugate gradient method with an optimal convergence rate (a convergence rate independent of mesh sizes) for solving the symmetric discrete linear system. Theoretical analysis shows that the spectra of the preconditioned matrices are uniformly bounded in the open interval $(3/8,2)$. To the best of our knowledge, this is the first attempt to develop a preconditioned iterative solver with a mesh-independent convergence rate for the linearized high-order scheme. Numerical examples are given to validate the accuracy of the scheme and the effectiveness of the proposed preconditioned solver.

The evaluation of text-generative vision-language models is a challenging yet crucial endeavor. By addressing the limitations of existing Visual Question Answering (VQA) benchmarks and proposing innovative evaluation methodologies, our research seeks to advance our understanding of these models' capabilities. We propose a novel VQA benchmark based on well-known visual classification datasets which allows a granular evaluation of text-generative vision-language models and their comparison with discriminative vision-language models. To improve the assessment of coarse answers on fine-grained classification tasks, we suggest using the semantic hierarchy of the label space to ask automatically generated follow-up questions about the ground-truth category. Finally, we compare traditional NLP and LLM-based metrics for the problem of evaluating model predictions given ground-truth answers. We perform a human evaluation study upon which we base our decision on the final metric. We apply our benchmark to a suite of vision-language models and show a detailed comparison of their abilities on object, action, and attribute classification. Our contributions aim to lay the foundation for more precise and meaningful assessments, facilitating targeted progress in the exciting field of vision-language modeling.

We propose an algorithm to construct optimal exact designs (EDs). Most of the work in the optimal regression design literature focuses on the approximate design (AD) paradigm due to its desired properties, including the optimality verification conditions derived by Kiefer (1959, 1974). ADs may have unbalanced weights, and practitioners may have difficulty implementing them with a designated run size $n$. Some EDs are constructed using rounding methods to get an integer number of runs at each support point of an AD, but this approach may not yield optimal results. To construct EDs, one may need to perform new combinatorial constructions for each $n$, and there is no unified approach to construct them. Therefore, we develop a systematic way to construct EDs for any given $n$. Our method can transform ADs into EDs while retaining high statistical efficiency in two steps. The first step involves constructing an AD by utilizing the convex nature of many design criteria. The second step employs a simulated annealing algorithm to search for the ED stochastically. Through several applications, we demonstrate the utility of our method for various design problems. Additionally, we show that the design efficiency approaches unity as the number of design points increases.

We propose a high-performance glass-plastic hybrid minimalist aspheric panoramic annular lens (ASPAL) to solve several major limitations of the traditional panoramic annular lens (PAL), such as large size, high weight, and complex system. The field of view (FoV) of the ASPAL is 360{\deg}x(35{\deg}~110{\deg}) and the imaging quality is close to the diffraction limit. This large FoV ASPAL is composed of only 4 lenses. Moreover, we establish a physical structure model of PAL using the ray tracing method and study the influence of its physical parameters on compactness ratio. In addition, for the evaluation of local tolerances of annular surfaces, we propose a tolerance analysis method suitable for ASPAL. This analytical method can effectively analyze surface irregularities on annular surfaces and provide clear guidance on manufacturing tolerances for ASPAL. Benefiting from high-precision glass molding and injection molding aspheric lens manufacturing techniques, we finally manufactured 20 ASPALs in small batches. The weight of an ASPAL prototype is only 8.5 g. Our framework provides promising insights for the application of panoramic systems in space and weight-constrained environmental sensing scenarios such as intelligent security, micro-UAVs, and micro-robots.