HAL - L’archive ouverte pluridisciplinaire

Articles

• [hal-04152611] On the variable bandwidth kernel estimation of conditional U -statistics at optimal rates in sup-norm

  9 juillet, par ano.nymous@ccsd.cnrs.fr.invalid (Salim Bouzebda), Salim Bouzebda
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• [hal-05136705] Singularities and Regular Correction for Elliptic Problems with Non-Constant Coefficients and Dirac Sources on the Boundary

  30 juin, par ano.nymous@ccsd.cnrs.fr.invalid (Ameni Béjaoui), Ameni Béjaoui

  We focus on the singularity of the potentials generated by Dirac sources located on the boundary. The diffusivity parameters of the medium are non-constant. We present and prove a singular/regular expansion of these potentials, following a prediction-correction approach. The singularity is made explicit using the fundamental Green's kernel of the Laplace operator. The regular correction problem can be efficiently solved using classical finite element methods. A numerical discussion highlights the relevance of this approach in achieving significant accuracy.

• [hal-05128905] A Rocq Formalization of Simplicial Lagrange Finite Elements

  26 juin, par ano.nymous@ccsd.cnrs.fr.invalid (Sylvie Boldo), Sylvie Boldo
  The finite element method is a popular method to numerically solve partial differential equations. In the long-term goal of proving its correctness, we focus here on the formal definition of what is a finite element: a record in the Rocq proof assistant with both values and proofs of validity, including the main one called unisolvence. We then instantiate this record with the most popular and useful, the simplicial Lagrange finite elements for evenly distributed nodes, for any dimension and any polynomial degree. These proofs require many results (definitions, lemmas, canonical structures) about finite families, affine spaces, multidimensional polynomials, in the context of finite or infinite-dimensional spaces.

• [hal-05128954] A Rocq Formalization of Simplicial Lagrange Finite Elements

  25 juin, par ano.nymous@ccsd.cnrs.fr.invalid (Sylvie Boldo), Sylvie Boldo
  The finite elements method is a popular method to numerically solve partial differential equations. In the long-term goal of proving its correctness, we focus here on the formal definition of what is a finite element: a record in the Rocq proof assistant with both values and proofs of validity, including the main one called unisolvence. We then instantiate this record with the most popular and useful, the simplicial Lagrange finite elements for evenly distributed nodes, for any dimension and any polynomial degree. These proofs require many results (definitions, lemmas, canonical structures) about finite families, affine spaces, multidimensional polynomials, in the context of finite or infinite-dimensional spaces.

• [hal-05027791] PatchTrAD : A Patch-Based Transformer focusing on Patch-Wise Reconstruction Error for Time Series Anomaly Detection

  8 juin, par ano.nymous@ccsd.cnrs.fr.invalid (Samy-Melwan Vilhes), Samy-Melwan Vilhes
  Time series anomaly detection (TSAD) focuses on identifying whether observations in streaming data deviate significantly from normal patterns. With the prevalence of connected devices, anomaly detection on time series has become paramount, as it enables real-time monitoring and early detection of irregular behaviors across various application domains. In this work, we introduce PatchTrAD, a Patch-based Transformer model for time series anomaly detection. Our approach leverages a Transformer encoder along with the use of patches under a reconstructionbased framework for anomaly detection. Empirical evaluations on multiple benchmark datasets show that PatchTrAD is on par, in terms of detection performance, with state-of-the-art deep learning models for anomaly detection while being time efficient during inference.

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