HAL - L’archive ouverte pluridisciplinaire

Articles

• [hal-05504439] Convergence of a scheme for a two dimensional nonlocal system of transport equations

  11 février, par ano.nymous@ccsd.cnrs.fr.invalid (Diana Al Zareef), Diana Al Zareef
  In this paper, we numerically study a two-dimensional system modeling the dynamics of dislocation densities. This system is hyperbolic, but not strictly hyperbolic, and couples two non-local transport equations. It is characterized by weak regularity in both the velocity and the initial data. We propose a semi-explicit finite difference (IMEX) numerical scheme for the discretization of this system, after regularizing the singular velocity using a Fejér kernel. We show that this scheme preserves, at the discrete level, an entropy estimate on the gradient, which then allows us to establish the convergence of the discrete solution to the continuous solution. To our knowledge, this is the first convergence result obtained for this type of system. We conclude with some numerical illustrations highlighting the performance of the proposed scheme.

• [hal-05430728] Embedding Birth-Death Processes within a Dynamic Stochastic Block Model

  26 janvier, par ano.nymous@ccsd.cnrs.fr.invalid (Gabriela Bayolo Soler), Gabriela Bayolo Soler
  Statistical clustering in dynamic networks aims to identify groups of nodes with similar or distinct internal connectivity patterns as the network evolves over time. While early research primarily focused on static Stochastic Block Models (SBMs), recent advancements have extended these models to handle dynamic and weighted networks, allowing for a more accurate representation of temporal variations in structure. Additional developments have introduced methods for detecting structural changes, such as shifts in community membership. However, limited attention has been paid to dynamic networks with variable population sizes, where nodes may enter or exit the network. To address this gap, we propose an extension of dynamic SBMs (dSBMs) that incorporates a birth-death process, enabling the statistical clustering of nodes in dynamic networks with evolving population sizes. This work makes three main contributions: (1) the introduction of a novel model for dSBMs with birth-death processes, (2) a framework for parameter inference and prediction of latent communities in this model, and (3) the development of an adapted Variational Expectation-Maximization (VEM) algorithm for efficient inference within this extended framework.

• [hal-05472046] Comparison of two iterative schemes to solve the Chemotaxis - Biodegradation System

  22 janvier, par ano.nymous@ccsd.cnrs.fr.invalid (Mostafa Abaali), Mostafa Abaali
  The mathematical model describing the biodegradation process in porous media by bacteria-divided into planktonic and adherent types-incorporates the chemotaxis effect, wherein the flow velocity depends on the concentration of adherent bacteria. This model consists of a system of five strongly coupled nonlinear parabolic equations, including a nonlinear advection term. Using the fixed-point theorem, we establish the existence, uniqueness, and non-negativity of the solution. To approximate the solution, we employ the finite element (FE) method. To linearize the system at each time step, we compare two iterative schemes. The first, called the Coupled Prediction Scheme (CPS), is shown to converge, while the second is the conventional Fixed-Point Method (FPM). Theoretical and numerical results show that CPS offers a faster and more efficient alternative to the conventional FPM.

• [hal-05411519] Local Linear Regression for Functional Ergodic Data with Missing at Random Responses

  19 janvier, par ano.nymous@ccsd.cnrs.fr.invalid (Yassine Baghli), Yassine Baghli
  In this article, we develop a novel kernel-based estimation framework for functional regression models in the presence of missing responses, with particular emphasis on the Missing At Random (MAR) mechanism. The analysis is carried out in the setting of stationary and ergodic functional data, where we introduce apparently for the first time a local linear estimator of the regression operator. The principal theoretical contributions of the paper may be summarized as follows. First, we establish almost sure uniform rates of convergence for the proposed estimator, thereby quantifying its asymptotic accuracy in a strong sense. Second, we prove its asymptotic normality, which provides the foundation for distributional approximations and subsequent inference. Third, we derive explicit closed-form expressions for the associated asymptotic variance, yielding a precise characterization of the limiting law. These results are obtained under standard structural assumptions on the relevant functional classes and under mild regularity conditions on the underlying model, ensuring broad applicability of the theory. On the methodological side, the asymptotic analysis is exploited to construct pointwise confidence regions for the regression operator, thereby enabling valid statistical inference. Furthermore, a comprehensive set of simulation experiments is conducted, demonstrating that the proposed estimator exhibits superior finite-sample predictive performance when compared to existing procedures, while simultaneously retaining robustness in the presence of missingness governed by MAR mechanisms.

• [hal-05084108] Scalar-on-Function Mode Estimation Using Entropy and Ergodic Properties of Functional Time Series Data

  15 janvier, par ano.nymous@ccsd.cnrs.fr.invalid (Mohammed Alamari), Mohammed Alamari
  In this paper, we investigate the recursive L1 estimator of the conditional mode when the input variable takes values in a pseudo-metric space. The new proposed estimator is constructed under an ergodicity assumption, which provides a robust alternative to the standard mixing processes in various practical settings. The particular interest of this contribution arises from the difficulty in incorporating the mathematical properties of a functional mixing process. In contrast, ergodicity is characterized by the Kolmogorov–Sinai entropy, which measures the dynamics, the sparsity, and the microscopic fluctuations of the functional process. Using an observation sampled from ergodic functional time series (fts), we establish the asymptotic properties of this estimator. In particular, we derive its convergence rate and show Borel–Cantelli (BC) consistency. The general expression for the convergence rate is then specialized to several notable scenarios, including the independence case, the classical kernel method, and the vector-valued case. Finally, numerical experiments on both simulated and real-world datasets demonstrate the superiority of the L1-recursive estimator compared to existing competitors.

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