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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.

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.

This manuscript derives locally weighted ensemble Kalman methods from the point of view of ensemble-based function approximation. This is done by using pointwise evaluations to build up a local linear or quadratic approximation of a function, tapering off the effect of distant particles via local weighting. This introduces a candidate method (the locally weighted Ensemble Kalman method for inversion) with the motivation of combining some of the strengths of the particle filter (ability to cope with nonlinear maps and non-Gaussian distributions) and the Ensemble Kalman filter (no filter degeneracy).

We discuss the asymptotic-preserving properties of a hybridizable discontinuous Galerkin method for the Westervelt model of ultrasound waves. More precisely, we show that the proposed method is robust with respect to small values of the sound diffusivity damping parameter~$\delta$ by deriving low- and high-order energy stability estimates, and \emph{a priori} error bounds that are independent of~$\delta$. Such bounds are then used to show that, when~$\delta \rightarrow 0^+$, the method remains stable and the discrete acoustic velocity potential~$\psi_h^{(\delta)}$ converges to~$\psi_h^{(0)}$, where the latter is the singular vanishing dissipation limit. Moreover, we prove optimal convergence for the approximation of the acoustic particle velocity variable~$\bv = \nabla \psi$. The established theoretical results are illustrated with some numerical experiments.

We study the properties of a family of distances between functions of a single variable. These distances are examples of integral probability metrics, and have been used previously for comparing probability measures on the line; special cases include the Earth Mover's Distance and the Kolmogorov Metric. We examine their properties for general signals, proving that they are robust to a broad class of deformations. We also establish corresponding robustness results for the induced sliced distances between multivariate functions. Finally, we establish error bounds for approximating the univariate metrics from finite samples, and prove that these approximations are robust to additive Gaussian noise. The results are illustrated in numerical experiments, which include comparisons with Wasserstein distances.

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.

We present an approach for the efficient implementation of self-adjusting multi-rate Runge-Kutta methods and we extend the previously available stability analyses of these methods to the case of an arbitrary number of sub-steps for the active components. We propose a physically motivated model problem that can be used to assess the stability of different multi-rate versions of standard Runge-Kutta methods and the impact of different interpolation methods for the latent variables. Finally, we present the results of several numerical experiments, performed with implementations of the proposed methods in the framework of the \textit{OpenModelica} open-source modelling and simulation software, which demonstrate the efficiency gains deriving from the use of the proposed multi-rate approach for physical modelling problems with multiple time scales.

This thesis is a corpus-based, quantitative, and typological analysis of the functions of Early Slavic participle constructions and their finite competitors ($jegda$-'when'-clauses). The first part leverages detailed linguistic annotation on Early Slavic corpora at the morphosyntactic, dependency, information-structural, and lexical levels to obtain indirect evidence for different potential functions of participle clauses and their main finite competitor and understand the roles of compositionality and default discourse reasoning as explanations for the distribution of participle constructions and $jegda$-clauses in the corpus. The second part uses massively parallel data to analyze typological variation in how languages express the semantic space of English $when$, whose scope encompasses that of Early Slavic participle constructions and $jegda$-clauses. Probabilistic semantic maps are generated and statistical methods (including Kriging, Gaussian Mixture Modelling, precision and recall analysis) are used to induce cross-linguistically salient dimensions from the parallel corpus and to study conceptual variation within the semantic space of the hypothetical concept WHEN.

Semi-Lagrangian (SL) schemes are highly efficient for simulating transport equations and are widely used across various applications. Despite their success, designing genuinely multi-dimensional and conservative SL schemes remains a significant challenge. Building on our previous work [Chen et al., J. Comput. Phys., V490 112329, (2023)], we introduce a conservative machine-learning-based SL finite difference (FD) method that allows for extra-large time step evolution. At the core of our approach is a novel dynamical graph neural network designed to handle the complexities associated with tracking accurately upstream points along characteristics. This proposed neural transport solver learns the conservative SL FD discretization directly from data, improving accuracy and efficiency compared to traditional numerical schemes, while significantly simplifying algorithm implementation. We validate the method' s effectiveness and efficiency through numerical tests on benchmark transport equations in both one and two dimensions, as well as the nonlinear Vlasov-Poisson system.