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1. Traffic rate network tomography with higher‐order cumulants

   https://doi.org/10.1002/net.22127  

   Lev‐Ari, Hanoch; Ephraim, Yariv; Mark, Brian L. (October 2022, Networks)

   Network tomography aims at estimating source–destination traffic rates from link traffic measurements. This inverse problem was formulated by Vardi in 1996 for Poisson traffic over networks operating under deterministic as well as random routing regimes. In this article, we expand Vardi's second-order moment matching rate estimation approach to higher-order cumulant matching with the goal of increasing the column rank of the mapping and consequently improving the rate estimation accuracy. We develop a systematic set of linear cumulant matching equations and express them compactly in terms of the Khatri–Rao product. Both least squares estimation and iterative minimum I-divergence estimation are considered. We develop an upper bound on the mean squared error (MSE) in least squares rate estimation from empirical cumulants. We demonstrate that supplementing Vardi's approach with the third-order empirical cumulant reduces its minimum averaged normalized MSE in rate estimation by almost 20% when iterative minimum I-divergence estimation was used. 

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2. Mixed Poisson traffic rate network tomography

   Ephraim, Yariv; Coblenz, Joshua; Mark, Brian L.; Lev-Ari, Hanoch (March 2021, 55rd Annual Conference on Information Sciences and Systems (CISS 2021))

   null (Ed.)

   We extend network tomography to traffic flows that are not necessarily Poisson random processes. This assumption has governed the field since its inception in 1996 by Y. Vardi. We allow the distribution of the packet count of each traffic flow in a given time interval to be a mixture of Poisson random variables. Both discrete as well as continuous mixtures are studied. For the latter case, we focus on mixed Poisson distributions with Gamma mixing distribution. As is well known, this mixed Poisson distribution is the negative binomial distribution. Other mixing distributions, such as Wald or the inverse Gaussian distribution can be used. Mixture distributions are overdispersed with variance larger than the mean. Thus, they are more suitable for Internet traffic than the Poisson model. We develop a second-order moment matching approach for estimating the mean traffic rate for each source-destination pair using least squares and the minimum I-divergence iterative procedure. We demonstrate the performance of the proposed approach by several numerical examples. The results show that the averaged normalized mean squared error in rate estimation is of the same order as in the classic Poisson based network tomography. Furthermore, no degradation in performance was observed when traffic rates are Poisson but Poisson mixtures are assumed. 

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3. Fourth-order moments analysis for partially coherent electromagnetic beams in random media

   https://doi.org/10.1080/17455030.2022.2096272  

   Garnier, Josselin; Sølna, Knut (November 2023, Waves in Random and Complex Media)

   A theory for the characterization of the fourth-order moment of electromagnetic wave beams is presented in the case when the source is partially coherent. A Gaussian–Schell model is used for the partially coherent random source. The white-noise paraxial regime is considered, which holds when the wavelength is much smaller than the correlation radius of the source, the beam radius of the source, and the correlation length of the medium, which are themselves much smaller than the propagation distance. The complex wave amplitude field can then be described by the Itô-Schrödinger equation. This equation gives closed evolution equations for the wave field moments at all orders and here the fourth-order moment equations are considered. The general fourth-order moment equations are solved explicitly in the scintillation regime (when the correlation radius of the source is of the same order as the correlation radius of the medium, but the beam radius is much larger) and the result gives a characterization of the intensity covariance function. The form of the intensity covariance function derives from the solution of the transport equation for the Wigner distribution associated with the second-order wave moment. The fourth-order moment results for polarized waves are used in an application for imaging of partially coherent sources. 

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4. A divergence-free and $H\left(div\right)$ -conforming embedded-hybridized DG method for the incompressible resistive MHD equations

   https://doi.org/10.1016/j.cma.2024.117415  

   Chen, Jau-Uei; Horváth, Tamás L; Bui-Thanh, Tan (December 2024, Computer Methods in Applied Mechanics and Engineering)

   We present a divergence-free and $$\Hsp\LRp{div}$$-conforming hybridized discontinuous Galerkin (HDG) method and a computationally efficient variant called embedded-HDG (E-HDG) for solving stationary incompressible viso-resistive magnetohydrodynamic (MHD) equations. The proposed E-HDG approach uses continuous facet unknowns for the vector-valued solutions (velocity and magnetic fields) while it uses discontinuous facet unknowns for the scalar variable (pressure and magnetic pressure). This choice of function spaces makes E-HDG computationally far more advantageous, due to the much smaller number of degrees of freedom, compared to the HDG counterpart. The benefit is even more significant for three-dimensional/high-order/fine mesh scenarios. On simplicial meshes, the proposed methods with a specific choice of approximation spaces are well-posed for linear(ized) MHD equations. For nonlinear MHD problems, we present a simple approach exploiting the proposed linear discretizations by using a Picard iteration. The beauty of this approach is that the divergence-free and $$\Hsp\LRp{div}$$-conforming properties of the velocity and magnetic fields are automatically carried over for nonlinear MHD equations. We study the accuracy and convergence of our E-HDG method for both linear and nonlinear MHD cases through various numerical experiments, including two- and three-dimensional problems with smooth and singular solutions. The numerical examples show that the proposed methods are pressure robust, and the divergence of the resulting velocity and magnetic fields is machine zero for both smooth and singular problems. 

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5. Linear and nonlinear transport equations with coordinate-wise increasing velocity fields

   https://doi.org/10.4171/aihpc/133  

   Lions, Pierre-Louis; Seeger, Benjamin (July 2025, Annales de l'Institut Henri Poincaré C, Analyse non linéaire)

   We consider linear and nonlinear transport equations with irregular velocity fields, motivated by models coming from mean field games. The velocity fields are assumed to increase in each coordinate, and the divergence therefore fails to be absolutely continuous with respect to the Lebesgue measure in general. For such velocity fields, the well-posedness of first- and second-order linear transport equations in Lebesgue spaces is established, as well as the existence and uniqueness of regular ODE and SDE Lagrangian flows. These results are then applied to the study of certain nonconservative, nonlinear systems of transport type, which are used to model mean field games in a finite state space. A notion of weak solution is identified for which unique minimal and maximal solutions exist, which do not coincide in general. A selection-by-noise result is established for a relevant example to demonstrate that different types of noise can select any of the admissible solutions in the vanishing noise limit. 

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