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We study the phase synchronization problem with noisy measurements $Y=z^*z^{*H}+\sigma W\in\mathbb{C}^{n\times n}$, where $z^*$ is an $n$-dimensional complex unit-modulus vector and $W$ is a complex-valued Gaussian random matrix. It is assumed that each entry $Y_{jk}$ is observed with probability $p$. We prove that an SDP relaxation of the MLE achieves the error bound $(1+o(1))\frac{\sigma^2}{2np}$ under a normalized squared $\ell_2$ loss. This result matches the minimax lower bound of the problem, and even the leading constant is sharp. The analysis of the SDP is based on an equivalent non-convex programming whose solution can be characterized as a fixed point of the generalized power iteration lifted to a higher dimensional space. This viewpoint unifies the proofs of the statistical optimality of three different methods: MLE, SDP, and generalized power method. The technique is also applied to the analysis of the SDP for $\mathbb{Z}_2$ synchronization, and we achieve the minimax optimal error $\exp\left(-(1-o(1))\frac{np}{2\sigma^2}\right)$ with a sharp constant in the exponent.

This letter studies the ergodic mutual information (EMI) of keyhole multiple-input multiple-output (MIMO) channels having finite input signals. At first, the EMI of single-stream transmission is investigated depending on whether the channel state information at the transmitter (CSIT) is available or not. Then, the derived results are extended to the case of multi-stream transmission. For the sake of providing more system insights, asymptotic analyses are performed in the regime of high signal-to-noise ratio (SNR), which suggests that the high-SNR EMI converges to some constant with its rate of convergence (ROC) determined by the diversity order. All the results are validated by numerical simulations and are in excellent agreement.

This paper deals with a special type of Lyapunov functions, namely the solution of Zubov's equation. Such a function can be used to characterize the domain of attraction for systems of ordinary differential equations. We derive and prove an integral form solution to Zubov's equation. For numerical computation, we develop two data-driven methods. One is based on the integration of an augmented system of differential equations; and the other one is based on deep learning. The former is effective for systems with a relatively low state space dimension and the latter is developed for high dimensional problems. The deep learning method is applied to a New England 10-generator power system model. We prove that a neural network approximation exists for the Lyapunov function of power systems such that the approximation error is a cubic polynomial of the number of generators. The error convergence rate as a function of n, the number of neurons, is proved.

Analysis and use of stochastic models represented by a discrete-time Markov Chain require evaluation of performance measures and characterization of its stationary distribution. Analytical solutions are often unavailable when the system states are continuous or mixed. This paper presents a new method for computing the stationary distribution and performance measures for stochastic systems represented by continuous-, or mixed-state Markov chains. We show the asymptotic convergence and provide deterministic non-asymptotic error bounds for our method under the supremum norm. Our finite approximation method is near-optimal among all discrete approximate distributions, including empirical distributions obtained from Markov chain Monte Carlo (MCMC). Numerical experiments validate the accuracy and efficiency of our method and show that it significantly outperforms MCMC based approach.

We propose an enriched finite element formulation to address the computational modeling of contact problems and the coupling of non-conforming discretizations in the small deformation setting. The displacement field is augmented by enriched terms that are associated with generalized degrees of freedom collocated along non-conforming interfaces or contact surfaces. The enrichment strategy effectively produces an enriched node-to-node discretization that can be used with any constraint enforcement criterion; this is demonstrated with both multiple-point constraints and Lagrange multipliers, the latter in a generalized Newton implementation where both primal and Lagrange multiplier fields are updated simultaneously. The method's ability to ensure continuity of the displacement field -- without locking -- in mesh coupling problems, and to transfer fairly accurate tractions at contact interfaces -- without the need for contact stabilization -- is demonstrated by means of several examples. In addition, we show that the formulation is stable with respect to the condition number of the stiffness matrix by using a simple Jacobi-like diagonal preconditioner.

The mutual information (MI) of Gaussian multi-input multi-output (MIMO) channels has been evaluated by utilizing random matrix theory (RMT) and shown to asymptotically follow Gaussian distribution, where the ergodic mutual information (EMI) converges to a deterministic quantity. However, with non-Gaussian channels, there is a bias between the EMI and its deterministic equivalent (DE), whose evaluation is not available in the literature. This bias of the EMI is related to the bias for the trace of the resolvent in large RMT. In this paper, we first derive the bias for the trace of the resolvent, which is further extended to compute the bias for the linear spectral statistics (LSS). Then, we apply the above results on non-Gaussian MIMO channels to determine the bias for the EMI. It is also proved that the bias for the EMI is $-0.5$ times of that for the variance of the MI. Finally, the derived bias is utilized to modify the central limit theory (CLT) and calculate the outage probability. Numerical results show that the modified CLT significantly outperforms previous methods in approximating the distribution of the MI and improves the accuracy for the outage probability evaluation.

Driven by B5G and 6G technologies, multi-network fusion is an indispensable tendency for future communications. In this paper, we focus on and analyze the \emph{security performance} (SP) of the \emph{satellite-terrestrial downlink transmission} (STDT). Here, the STDT is composed of a satellite network and a vehicular network with a legitimate mobile receiver and an mobile eavesdropper distributing. To theoretically analyze the SP of this system from the perspective of mobile terminals better, the random geometry theory is adopted, which assumes that both terrestrial vehicles are distributed stochastically in one beam of the satellite. Furthermore, based on this theory, the closed-form analytical expressions for two crucial and specific indicators in the STDT are derived, respectively, the secrecy outage probability and the ergodic secrecy capacity. Additionally, several related variables restricting the SP of the STDT are discussed, and specific schemes are presented to enhance the SP. Then, the asymptotic property is investigated in the high signal-to-noise ratio scenario, and accurate and asymptotic closed-form expressions are given. Finally, simulation results show that, under the precondition of guaranteeing the reliability of the STDT, the asymptotic solutions outperform the corresponding accurate results significantly in the effectiveness.

Intelligent reflecting surface (IRS) has emerged as a promising technique to enhance wireless communication performance cost effectively. The existing literature has mainly considered IRS being deployed near user terminals to improve their performance. However, this approach may incur a high cost if IRSs need to be densely deployed in the network to cater to random user locations. To avoid such high deployment cost, in this paper we consider a new IRS aided wireless network architecture, where IRSs are deployed in the vicinity of each base station (BS) to assist in its communications with distributed users regardless of their locations. Besides significantly enhancing IRSs' signal coverage, this scheme helps reduce the IRS associated channel estimation overhead as compared to conventional user-side IRSs, by exploiting the nearly static BS-IRS channels over short distance. For this scheme, we propose a new two stage transmission protocol to achieve IRS channel estimation and reflection optimization for uplink data transmission efficiently. In addition, we propose effective methods for solving the user IRS association problem based on long term channel knowledge and the selected user IRS BS cascaded channel estimation problem. Finally, all IRSs' passive reflections are jointly optimized with the BS's multi-antenna receive combining to maximize the minimum achievable rate among all users for data transmission. Numerical results show that the proposed co site IRS empowered BS scheme can achieve significant performance gains over the conventional BS without co site IRS and existing schemes for IRS channel estimation and reflection optimization, thus enabling an appealing low cost and high performance BS design for future wireless networks.

We investigate data-driven forward-inverse problems for Yajima-Oikawa (YO) system by employing two technologies which improve the performance of neural network in deep physics-informed neural network (PINN), namely neuron-wise locally adaptive activation functions and $L^2$ norm parameter regularization. Indeed, we not only recover three different forms of vector rogue waves (RWs) by means of three distinct initial-boundary value conditions in the forward problem of YO system, including bright-bright RWs, intermediate-bright RWs and dark-bright RWs, but also study the inverse problem of YO system by using training data with different noise intensity. In order to deal with the problem that the capacity of learning unknown parameters is not ideal when the PINN with only locally adaptive activation functions utilizes training data with noise interference in the inverse problem of YO system, thus we introduce $L^2$ norm regularization, which can drive the weights closer to origin, into PINN with locally adaptive activation functions, then find that the PINN model with two strategies shows amazing training effect by using training data with noise interference to investigate the inverse problem of YO system.

Various 3D reconstruction methods have enabled civil engineers to detect damage on a road surface. To achieve the millimetre accuracy required for road condition assessment, a disparity map with subpixel resolution needs to be used. However, none of the existing stereo matching algorithms are specially suitable for the reconstruction of the road surface. Hence in this paper, we propose a novel dense subpixel disparity estimation algorithm with high computational efficiency and robustness. This is achieved by first transforming the perspective view of the target frame into the reference view, which not only increases the accuracy of the block matching for the road surface but also improves the processing speed. The disparities are then estimated iteratively using our previously published algorithm where the search range is propagated from three estimated neighbouring disparities. Since the search range is obtained from the previous iteration, errors may occur when the propagated search range is not sufficient. Therefore, a correlation maxima verification is performed to rectify this issue, and the subpixel resolution is achieved by conducting a parabola interpolation enhancement. Furthermore, a novel disparity global refinement approach developed from the Markov Random Fields and Fast Bilateral Stereo is introduced to further improve the accuracy of the estimated disparity map, where disparities are updated iteratively by minimising the energy function that is related to their interpolated correlation polynomials. The algorithm is implemented in C language with a near real-time performance. The experimental results illustrate that the absolute error of the reconstruction varies from 0.1 mm to 3 mm.