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In this work, we study and extend a class of semi-Lagrangian exponential methods, which combine exponential time integration techniques, suitable for integrating stiff linear terms, with a semi-Lagrangian treatment of nonlinear advection terms. Partial differential equations involving both processes arise for instance in atmospheric circulation models. Through a truncation error analysis, we first show that previously formulated semi-Lagrangian exponential schemes are limited to first-order accuracy due to the discretization of the linear term; we then formulate a new discretization leading to a second-order accurate method. Also, a detailed stability study, both considering a linear stability analysis and an empirical simulation-based one, is conducted to compare several Eulerian and semi-Lagrangian exponential schemes, as well as a well-established semi-Lagrangian semi-implicit method, which is used in operational atmospheric models. Numerical simulations of the shallow-water equations on the rotating sphere, considering standard and challenging benchmark test cases, are performed to assess the orders of convergence, stability properties, and computational cost of each method. The proposed second-order semi-Lagrangian exponential method was shown to be more stable and accurate than the previously formulated schemes of the same class at the expense of larger wall-clock times; however, the method is more stable and has a similar cost compared to the well-established semi-Lagrangian semi-implicit; therefore, it is a competitive candidate for potential operational applications in atmospheric circulation modeling.

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.

In this study, we develop a novel multi-fidelity deep learning approach that transforms low-fidelity solution maps into high-fidelity ones by incorporating parametric space information into a standard autoencoder architecture. It is shown that, due to the integration of parametric space data, this method requires significantly less training data to achieve effective performance in predicting high-fidelity solution from the low-fidelity one. In this study, our focus is on a 2D steady-state heat transfer analysis in highly heterogeneous materials microstructure, where the spatial distribution of heat conductivity coefficients for two distinct materials is condensed. Subsequently, the boundary value problem is solved on the coarsest grid using a pre-trained physics-informed neural operator network. Afterward, the calculated low-fidelity result is upscaled using the newly designed enhanced autoencoder. The novelty of the developed enhanced autoencoder lies in the concatenation of heat conductivity maps of different resolutions to the decoder segment in distinct steps. We then compare the outcomes of developed algorithm with the corresponding finite element results, standard U-Net architecture as well as other upscaling approaches such as interpolation functions of varying orders and feedforward neural networks (FFNN). The analysis of the results based on the new approach demonstrates superior performance compared to other approaches in terms of computational cost and error on the test cases. Therefore, as a potential supplement to neural operators networks, our architecture upscales low-fidelity solutions to high-fidelity ones while preserving critical details that are often lost in conventional upscaling methods, especially at sharp interfaces, such as those encountered with interpolation methods.

Linkage methods are among the most popular algorithms for hierarchical clustering. Despite their relevance the current knowledge regarding the quality of the clustering produced by these methods is limited. Here, we improve the currently available bounds on the maximum diameter of the clustering obtained by complete-link for metric spaces. One of our new bounds, in contrast to the existing ones, allows us to separate complete-link from single-link in terms of approximation for the diameter, which corroborates the common perception that the former is more suitable than the latter when the goal is producing compact clusters. We also show that our techniques can be employed to derive upper bounds on the cohesion of a class of linkage methods that includes the quite popular average-link.

We present a novel approach for modeling bounded count time series data, by deriving accurate upper and lower bounds for the variance of a bounded count random variable while maintaining a fixed mean. Leveraging these bounds, we propose semiparametric mean and variance joint (MVJ) models utilizing a clipped-Laplace link function. These models offer a flexible and feasible structure for both mean and variance, accommodating various scenarios of under-dispersion, equi-dispersion, or over-dispersion in bounded time series. The proposed MVJ models feature a linear mean structure with positive regression coefficients summing to one and allow for negative regression cefficients and autocorrelations. We demonstrate that the autocorrelation structure of MVJ models mirrors that of an autoregressive moving-average (ARMA) process, provided the proposed clipped-Laplace link functions with nonnegative regression coefficients summing to one are utilized. We establish conditions ensuring the stationarity and ergodicity properties of the MVJ process, along with demonstrating the consistency and asymptotic normality of the conditional least squares estimators. To aid model selection and diagnostics, we introduce two model selection criteria and apply two model diagnostics statistics. Finally, we conduct simulations and real data analyses to investigate the finite-sample properties of the proposed MVJ models, providing insights into their efficacy and applicability in practical scenarios.

Single-particle entangled states (SPES) can offer a more secure way of encoding and processing quantum information than their multi-particle counterparts. The SPES generated via a 2D alternate quantum-walk setup from initially separable states can be either 3-way or 2-way entangled. This letter shows that the generated genuine three-way and nonlocal two-way SPES can be used as cryptographic keys to securely encode two distinct messages simultaneously. We detail the message encryption-decryption steps and show the resilience of the 3-way and 2-way SPES-based cryptographic protocols against eavesdropper attacks like intercept-and-resend and man-in-the-middle. We also detail how these protocols can be experimentally realized using single photons, with the three degrees of freedom being OAM, path, and polarization. These have unparalleled security for quantum communication tasks. The ability to simultaneously encode two distinct messages using the generated SPES showcases the versatility and efficiency of the proposed cryptographic protocol. This capability could significantly improve the throughput of quantum communication systems.

In this work, we introduce a novel approach to regularization in multivariable regression problems. Our regularizer, called DLoss, penalises differences between the model's derivatives and derivatives of the data generating function as estimated from the training data. We call these estimated derivatives data derivatives. The goal of our method is to align the model to the data, not only in terms of target values but also in terms of the derivatives involved. To estimate data derivatives, we select (from the training data) 2-tuples of input-value pairs, using either nearest neighbour or random, selection. On synthetic and real datasets, we evaluate the effectiveness of adding DLoss, with different weights, to the standard mean squared error loss. The experimental results show that with DLoss (using nearest neighbour selection) we obtain, on average, the best rank with respect to MSE on validation data sets, compared to no regularization, L2 regularization, and Dropout.

We develop an inferential toolkit for analyzing object-valued responses, which correspond to data situated in general metric spaces, paired with Euclidean predictors within the conformal framework. To this end we introduce conditional profile average transport costs, where we compare distance profiles that correspond to one-dimensional distributions of probability mass falling into balls of increasing radius through the optimal transport cost when moving from one distance profile to another. The average transport cost to transport a given distance profile to all others is crucial for statistical inference in metric spaces and underpins the proposed conditional profile scores. A key feature of the proposed approach is to utilize the distribution of conditional profile average transport costs as conformity score for general metric space-valued responses, which facilitates the construction of prediction sets by the split conformal algorithm. We derive the uniform convergence rate of the proposed conformity score estimators and establish asymptotic conditional validity for the prediction sets. The finite sample performance for synthetic data in various metric spaces demonstrates that the proposed conditional profile score outperforms existing methods in terms of both coverage level and size of the resulting prediction sets, even in the special case of scalar and thus Euclidean responses. We also demonstrate the practical utility of conditional profile scores for network data from New York taxi trips and for compositional data reflecting energy sourcing of U.S. states.

Contraction coefficients give a quantitative strengthening of the data processing inequality. As such, they have many natural applications whenever closer analysis of information processing is required. However, it is often challenging to calculate these coefficients. As a remedy we discuss a quantum generalization of Doeblin coefficients. These give an efficiently computable upper bound on many contraction coefficients. We prove several properties and discuss generalizations and applications. In particular, we give additional stronger bounds for PPT channels and introduce reverse Doeblin coefficients that bound certain expansion coefficients.

In recent years, object detection has experienced impressive progress. Despite these improvements, there is still a significant gap in the performance between the detection of small and large objects. We analyze the current state-of-the-art model, Mask-RCNN, on a challenging dataset, MS COCO. We show that the overlap between small ground-truth objects and the predicted anchors is much lower than the expected IoU threshold. We conjecture this is due to two factors; (1) only a few images are containing small objects, and (2) small objects do not appear enough even within each image containing them. We thus propose to oversample those images with small objects and augment each of those images by copy-pasting small objects many times. It allows us to trade off the quality of the detector on large objects with that on small objects. We evaluate different pasting augmentation strategies, and ultimately, we achieve 9.7\% relative improvement on the instance segmentation and 7.1\% on the object detection of small objects, compared to the current state of the art method on MS COCO.