Xue Luo

Xue Luo

In this paper, we propose new spectral viscosity methods based on the generalized Hermite functions for the solution of nonlinear scalar conservation laws in the whole line. It is shown rigorously that these schemes converge to the unique entropy solution by using compensated compactness arguments, under some conditions. The numerical experiments of the inviscid Burger's equation support our result, and it verifies the reasonableness of the conditions.

Peng Sun, Ruoyu Wang, Xue Luo

This paper focuses on the state estimation problem in distributed sensor networks, where intermittent packet dropouts, corrupted observations, and unknown noise covariances coexist. To tackle this challenge, we formulate the joint estimation of system states, noise parameters, and network reliability as a Bayesian variational inference problem, and propose a novel variational Bayesian adaptive Kalman filter (VB-AKF) to approximate the joint posterior probability densities of the latent parameters. Unlike existing AKF that separately handle missing data and measurement outliers, the proposed VB-AKF adopts a dual-mask generative model with two independent Bernoulli random variables, explicitly characterizing both observable communication losses and latent data authenticity. Additionally, the VB-AKF integrates multiple concurrent multiple observations into the adaptive filtering framework, which significantly enhances statistical identifiability. Comprehensive numerical experiments verify the effectiveness and asymptotic optimality of the proposed method, showing that both parameter identification and state estimation asymptotically converge to the theoretical optimal lower bound with the increase in the number of sensors.

Ruoyu Wang, Peng Sun, Xue Luo

The feedback particle filter (FPF) is a promising nonlinear filtering (NLF) method, but its practical implementation is hindered by the intractability of the gain function, which satisfies a boundary value problem (BVP). This paper proposes a novel two-step Hermite-Galerkin spectral method to address this challenge. First, the unknown density in the BVP is approximated by a kernel density estimator, whose error bounds are well-established in the literature. Second, rather than directly approximating the gain function, we approximate an auxiliary variable via the Galerkin spectral method using generalized Hermite functions. This auxiliary variable inherits the rapid decay property of the density at infinity, which aligns perfectly with the exponential decay characteristic of generalized Hermite functions, thereby obviating the need for artificial boundary conditions or domain truncation. Furthermore, we rigorously establish two fundamental error estimates: the kernel approximation error decays at the rate $O(N_p^{-\frac{s}{2s+1}})$, while the spectral approximation error converges at $O(M^{-s+1}\log M)$, providing complete theoretical guarantees for the method's accuracy. Comprehensive numerical experiments validate the theoretical results and demonstrate that the proposed method outperforms existing gain approximation schemes in both accuracy and computational efficiency.

Sajib Saha, Fan Gu, Xue Luo et al.

The performance of rigid pavement is greatly affected by the properties of base/subbase as well as subgrade layer. However, the performance predicted by the AASHTOWare Pavement ME design shows low sensitivity to the properties of base and subgrade layers. To improve the sensitivity and better reflect the influence of unbound layers a new set of improved models i.e., resilient modulus (MR) and modulus of subgrade reaction (k-value) are adopted in this study. An Artificial Neural Network (ANN) model is developed to predict the modified k-value based on finite element (FE) analysis. The training and validation datasets in the ANN model consist of 27000 simulation cases with different combinations of pavement layer thickness, layer modulus and slab-base interface bond ratio. To examine the sensitivity of modified MR and k-values on pavement response, eight pavement sections data are collected from the Long-Term Pavement performance (LTPP) database and modeled by using the FE software ISLAB2000. The computational results indicate that the modified MR values have higher sensitivity to water content in base layer on critical stress and deflection response of rigid pavements compared to the results using the Pavement ME design model. It is also observed that the k-values using ANN model has the capability of predicting critical pavement response at any partially bonded conditions whereas the Pavement ME design model can only calculate at two extreme bonding conditions (i.e., fully bonding and no bonding).

Xue Luo, Shing-Tung Yau, Stephen S. -T. Yau

A time-dependent Hermite-Galerkin spectral method (THGSM) is investigated in this paper for the nonlinear convection-diffusion equations in the unbounded domains. The time-dependent scaling factor and translating factor are introduced in the definition of the generalized Hermite functions (GHF). As a consequence, the THGSM based on these GHF has many advantages, not only in theorethical proofs, but also in numerical implementations. The stability and spectral convergence of our proposed method have been established in this paper. The Korteweg-de Vries-Burgers (KdVB) equation and its special cases, including the heat equation and the Burgers' equation, as the examples, have been numerically solved by our method. The numerical results are presented, and it surpasses the existing methods in accuracy. Our theoretical proof of the spectral convergence has been supported by the numerical results.