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Plug-and-Play (PnP) methods solve ill-posed inverse problems through iterative proximal algorithms by replacing a proximal operator by a denoising operation. When applied with deep neural network denoisers, these methods have shown state-of-the-art visual performance for image restoration problems. However, their theoretical convergence analysis is still incomplete. Most of the existing convergence results consider nonexpansive denoisers, which is non-realistic, or limit their analysis to strongly convex data-fidelity terms in the inverse problem to solve. Recently, it was proposed to train the denoiser as a gradient descent step on a functional parameterized by a deep neural network. Using such a denoiser guarantees the convergence of the PnP version of the Half-Quadratic-Splitting (PnP-HQS) iterative algorithm. In this paper, we show that this gradient denoiser can actually correspond to the proximal operator of another scalar function. Given this new result, we exploit the convergence theory of proximal algorithms in the nonconvex setting to obtain convergence results for PnP-PGD (Proximal Gradient Descent) and PnP-ADMM (Alternating Direction Method of Multipliers). When built on top of a smooth gradient denoiser, we show that PnP-PGD and PnP-ADMM are convergent and target stationary points of an explicit functional. These convergence results are confirmed with numerical experiments on deblurring, super-resolution and inpainting.

A mass-preserving two-step Lagrange-Galerkin scheme of second order in time for convection-diffusion problems is presented, and convergence with optimal error estimates is proved in the framework of $L^2$-theory. The introduced scheme maintains the advantages of the Lagrange-Galerkin method, i.e., CFL-free robustness for convection-dominated problems and a symmetric and positive coefficient matrix resulting from the discretization. In addition, the scheme conserves the mass on the discrete level if the involved integrals are computed exactly. Unconditional stability and error estimates of second order in time are proved by employing two new key lemmas on the truncation error of the material derivative in conservative form and on a discrete Gronwall inequality for multistep methods. The mass-preserving property is achieved by the Jacobian multiplication technique introduced by Rui and Tabata in 2010, and the accuracy of second order in time is obtained based on the idea of the multistep Galerkin method along characteristics originally introduced by Ewing and Russel in 1981. For the first time step, the mass-preserving scheme of first order in time by Rui and Tabata in 2010 is employed, which is efficient and does not cause any loss of convergence order in the $\ell^\infty(L^2)$- and $\ell^2(H^1_0)$-norms. For the time increment $\Delta t$, the mesh size $h$ and a conforming finite element space of polynomial degree $k$, the convergence order is of $O(\Delta t^2 + h^k)$ in the $\ell^\infty(L^2)\cap \ell^2(H^1_0)$-norm and of $O(\Delta t^2 + h^{k+1})$ in the $\ell^\infty(L^2)$-norm if the duality argument can be employed. Error estimates of $O(\Delta t^{3/2}+h^k)$ in discrete versions of the $L^\infty(H^1_0)$- and $H^1(L^2)$-norm are additionally proved. Numerical results confirm the theoretical convergence orders in one, two and three dimensions.

This paper addresses the transmission-aware transceiver allocation problem of flexible optical networks for a multi-period planning. The proposed approach aims at assigning the best configuration of bandwidth variable transceivers (BVTRX) considering the amplifier noise and nonlinear channel interferences using the incoherent Extended Gaussian Noise (EGN) model. The proposed solution improves the network throughput and spectrum utilization in the early planning periods and allocates lower number of BV-TRXs in later periods in comparison to algorithms presented recently. A heuristic approach to regenerator placement has also been applied achieving up to 25% transceiver and 50% spectrum utilization savings in comparison to configurations without regenerators.

We study the complexity of approximating the multimarginal optimal transport (MOT) distance, a generalization of the classical optimal transport distance, considered here between $m$ discrete probability distributions supported each on $n$ support points. First, we show that the standard linear programming (LP) representation of the MOT problem is not a minimum-cost flow problem when $m \geq 3$. This negative result implies that some combinatorial algorithms, e.g., network simplex method, are not suitable for approximating the MOT problem, while the worst-case complexity bound for the deterministic interior-point algorithm remains a quantity of $\tilde{O}(n^{3m})$. We then propose two simple and \textit{deterministic} algorithms for approximating the MOT problem. The first algorithm, which we refer to as \textit{multimarginal Sinkhorn} algorithm, is a provably efficient multimarginal generalization of the Sinkhorn algorithm. We show that it achieves a complexity bound of $\tilde{O}(m^3n^m\varepsilon^{-2})$ for a tolerance $\varepsilon \in (0, 1)$. This provides a first \textit{near-linear time} complexity bound guarantee for approximating the MOT problem and matches the best known complexity bound for the Sinkhorn algorithm in the classical OT setting when $m = 2$. The second algorithm, which we refer to as \textit{accelerated multimarginal Sinkhorn} algorithm, achieves the acceleration by incorporating an estimate sequence and the complexity bound is $\tilde{O}(m^3n^{m+1/3}\varepsilon^{-4/3})$. This bound is better than that of the first algorithm in terms of $1/\varepsilon$, and accelerated alternating minimization algorithm~\citep{Tupitsa-2020-Multimarginal} in terms of $n$. Finally, we compare our new algorithms with the commercial LP solver \textsc{Gurobi}. Preliminary results on synthetic data and real images demonstrate the effectiveness and efficiency of our algorithms.

Time efficiency is one of the more critical concerns in computational fluid dynamics simulations of industrial applications. Extensive research has been conducted to improve the underlying numerical schemes to achieve time process reduction. Within this context, this paper presents a new time discretization method based on the Adomian decomposition technique for Euler equations. The obtained scheme is time-order adaptive; the order is automatically adjusted at each time step and over the space domain, leading to significant processing time reduction. The scheme is formulated in an appropriate recursive formula, and its efficiency is demonstrated through numerical tests by comparison to exact solutions and the popular Runge-Kutta-DG method.

Base stations in 5G and beyond use advanced antenna systems (AASs), where multiple passive antenna elements and radio units are integrated into a single box. A critical bottleneck of such a system is the digital fronthaul between the AAS and baseband unit (BBU), which has limited capacity. In this paper, we study an AAS used for precoded downlink transmission over a multi-user multiple-input multiple-output (MU-MIMO) channel. First, we present the baseline quantization-unaware precoding scheme created when a precoder is computed at the BBU and then quantized to be sent over the fronthaul. We propose a new precoding design that is aware of the fronthaul quantization. We formulate an optimization problem to minimize the mean squared error at the receiver side. We rewrite the problem to utilize mixed-integer programming to solve it. The numerical results manifest that our proposed precoding greatly outperforms quantization-unaware precoding in terms of sum rate.

We consider a hierarchical edge-cloud architecture in which services are provided to mobile users as chains of virtual network functions. Each service has specific computation requirements and target delay performance, which require placing the corresponding chain properly and allocating a suitable amount of computing resources. Furthermore, chain migration may be necessary to meet the services' target delay, or convenient to keep the service provisioning cost low. We tackle such issues by formalizing the problem of optimal chain placement and resource allocation in the edge-cloud continuum, taking into account migration, bandwidth, and computation costs. Specifically, we first envision an algorithm that, leveraging resource augmentation, addresses the above problem and provides an upper bound to the amount of resources required to find a feasible solution. We use this algorithm as a building block to devise an efficient approach targeting the minimum-cost solution, while minimizing the required resource augmentation. Our results, obtained through trace-driven, large-scale simulations, show that our solution can provide a feasible solution by using half the amount of resources required by state-of-the-art alternatives.

We consider the on-time transmissions of a sequence of packets over a fading channel.Different from traditional in-time communications, we investigate how many packets can be received $\delta$-on-time, meaning that the packet is received with a deviation no larger than $\delta$ slots. In this framework, we first derive the on-time reception rate of the random transmissions over the fading channel when no controlling is used. To improve the on-time reception rate, we further propose to schedule the transmissions by delaying, dropping, or repeating the packets. Specifically, we model the scheduling over the fading channel as a Markov decision process (MDP) and then obtain the optimal scheduling policy using an efficient iterative algorithm. For a given sequence of packet transmissions, we analyze the on-time reception rate for the random transmissions and the optimal scheduling. Our analytical and simulation results show that the on-time reception rate of random transmissions decreases (to zero) with the sequence length.By using the optimal packet scheduling, the on-time reception rate converges to a much larger constant. Moreover, we show that the on-time reception rate increases if the target reception interval and/or the deviation tolerance $\delta$ is increased, or the randomness of the fading channel is reduced.

Achieving an end-to-end low-latency for computations offloading, in Mobile Edge Computing (MEC) systems, is still a critical design problem. This is because the offloading of computational tasks via the MEC servers entails the use of uplink (UL) and downlink (DL) radio links that are usually assumed to be coupled to a single base station (BS). However, for heterogeneous networks, a new architectural paradigm whereby UL and DL are not associated with the same BS is proposed and seen to provide gains in network throughput due to the improved UL performance. Motivated by such gains, and by using typical results from stochastic geometry, we formulate the offloading latency for the MEC-based scheme with decoupled UL/DL association, or decoupled access, and compare its performance to the conventional coupled access scheme. Despite the backhaul delay necessary for the communication between the two serving BSs in UL and DL, the offloading scheme with decoupled access is still capable of providing a fairly lower offloading latency compared to the conventional offloading scheme with coupled access.

In this paper, we study the optimal convergence rate for distributed convex optimization problems in networks. We model the communication restrictions imposed by the network as a set of affine constraints and provide optimal complexity bounds for four different setups, namely: the function $F(\xb) \triangleq \sum_{i=1}^{m}f_i(\xb)$ is strongly convex and smooth, either strongly convex or smooth or just convex. Our results show that Nesterov's accelerated gradient descent on the dual problem can be executed in a distributed manner and obtains the same optimal rates as in the centralized version of the problem (up to constant or logarithmic factors) with an additional cost related to the spectral gap of the interaction matrix. Finally, we discuss some extensions to the proposed setup such as proximal friendly functions, time-varying graphs, improvement of the condition numbers.