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A detailed numerical study of solutions to the Serre-Green-Naghdi (SGN) equations in 2D with vanishing curl of the velocity field is presented. The transverse stability of line solitary waves, 1D solitary waves being exact solutions of the 2D equations independent of the second variable, is established numerically. The study of localized initial data as well as crossing 1D solitary waves does not give an indication of existence of stable structures in SGN solutions localized in two spatial dimensions. For the numerical experiments, an approach based on a Fourier spectral method with a Krylov subspace technique is applied.

We consider arbitrary bounded discrete time series. From its statistical feature, without any use of the Fourier transform, we find an almost periodic function which suitably characterizes the corresponding time series.

This paper investigates extremal quantiles under two-way cluster dependence. We demonstrate that the limiting distribution of the unconditional intermediate order quantiles in the tails converges to a Gaussian distribution. This is remarkable as two-way cluster dependence entails potential non-Gaussianity in general, but extremal quantiles do not suffer from this issue. Building upon this result, we extend our analysis to extremal quantile regressions of intermediate order.

This paper addresses the problem of segmenting a stream of graph signals: we aim to detect changes in the mean of a multivariate signal defined over the nodes of a known graph. We propose an offline method that relies on the concept of graph signal stationarity and allows the convenient translation of the problem from the original vertex domain to the spectral domain (Graph Fourier Transform), where it is much easier to solve. Although the obtained spectral representation is sparse in real applications, to the best of our knowledge this property has not been sufficiently exploited in the existing related literature. Our change-point detection method adopts a model selection approach that takes into account the sparsity of the spectral representation and determines automatically the number of change-points. Our detector comes with a proof of a non-asymptotic oracle inequality. Numerical experiments demonstrate the performance of the proposed method.

We consider the statistical linear inverse problem of making inference on an unknown source function in an elliptic partial differential equation from noisy observations of its solution. We employ nonparametric Bayesian procedures based on Gaussian priors, leading to convenient conjugate formulae for posterior inference. We review recent results providing theoretical guarantees on the quality of the resulting posterior-based estimation and uncertainty quantification, and we discuss the application of the theory to the important classes of Gaussian series priors defined on the Dirichlet-Laplacian eigenbasis and Mat\'ern process priors. We provide an implementation of posterior inference for both classes of priors, and investigate its performance in a numerical simulation study.

We propose and analyse boundary-preserving schemes for the strong approximations of some scalar SDEs with non-globally Lipschitz drift and diffusion coefficients whose state-space is bounded. The schemes consists of a Lamperti transform followed by a Lie--Trotter splitting. We prove $L^{p}(\Omega)$-convergence of order $1$, for every $p \geq 1$, of the schemes and exploit the Lamperti transform to confine the numerical approximations to the state-space of the considered SDE. We provide numerical experiments that confirm the theoretical results and compare the proposed Lamperti-splitting schemes to other numerical schemes for SDEs.

In an error estimation of finite element solutions to the Poisson equation, we usually impose the shape regularity assumption on the meshes to be used. In this paper, we show that even if the shape regularity condition is violated, the standard error estimation can be obtained if "bad" elements (elements that violate the shape regularity or maximum angle condition) are covered virtually by "good" simplices. A numerical experiment confirms the theoretical result.

This paper introduces an efficient high-order numerical method for solving the 1D stationary Schr\"odinger equation in the highly oscillatory regime. Building upon the ideas from [2], we first analytically transform the given equation into a smoother (i.e. less oscillatory) equation. By developing sufficiently accurate quadratures for several (iterated) oscillatory integrals occurring in the Picard approximation of the solution, we obtain a one-step method that is third order w.r.t. the step size. The accuracy and efficiency of the method are illustrated through several numerical examples.

Fully decentralized learning is gaining momentum for training AI models at the Internet's edge, addressing infrastructure challenges and privacy concerns. In a decentralized machine learning system, data is distributed across multiple nodes, with each node training a local model based on its respective dataset. The local models are then shared and combined to form a global model capable of making accurate predictions on new data. Our exploration focuses on how different types of network structures influence the spreading of knowledge - the process by which nodes incorporate insights gained from learning patterns in data available on other nodes across the network. Specifically, this study investigates the intricate interplay between network structure and learning performance using three network topologies and six data distribution methods. These methods consider different vertex properties, including degree centrality, betweenness centrality, and clustering coefficient, along with whether nodes exhibit high or low values of these metrics. Our findings underscore the significance of global centrality metrics (degree, betweenness) in correlating with learning performance, while local clustering proves less predictive. We highlight the challenges in transferring knowledge from peripheral to central nodes, attributed to a dilution effect during model aggregation. Additionally, we observe that central nodes exert a pull effect, facilitating the spread of knowledge. In examining degree distribution, hubs in Barabasi-Albert networks positively impact learning for central nodes but exacerbate dilution when knowledge originates from peripheral nodes. Finally, we demonstrate the formidable challenge of knowledge circulation outside of segregated communities.

With advances in scientific computing and mathematical modeling, complex scientific phenomena such as galaxy formations and rocket propulsion can now be reliably simulated. Such simulations can however be very time-intensive, requiring millions of CPU hours to perform. One solution is multi-fidelity emulation, which uses data of different fidelities to train an efficient predictive model which emulates the expensive simulator. For complex scientific problems and with careful elicitation from scientists, such multi-fidelity data may often be linked by a directed acyclic graph (DAG) representing its scientific model dependencies. We thus propose a new Graphical Multi-fidelity Gaussian Process (GMGP) model, which embeds this DAG structure (capturing scientific dependencies) within a Gaussian process framework. We show that the GMGP has desirable modeling traits via two Markov properties, and admits a scalable algorithm for recursive computation of the posterior mean and variance along at each depth level of the DAG. We also present a novel experimental design methodology over the DAG given an experimental budget, and propose a nonlinear extension of the GMGP via deep Gaussian processes. The advantages of the GMGP are then demonstrated via a suite of numerical experiments and an application to emulation of heavy-ion collisions, which can be used to study the conditions of matter in the Universe shortly after the Big Bang. The proposed model has broader uses in data fusion applications with graphical structure, which we further discuss.