Numerical Solution of Quantum-Mechanical Pair Equations

We discuss and illustrate the numerical solution of the differential equation satisfied by the first‐order pair functions of Sinanoğlu. An expansion of the pair function in spherical harmonics and the use of finite difference methods convert the differential equation into a set of simultaneous equations. Large systems of such equations can be solved economically. The method is simple and straightforward, and we have applied it to the first‐order pair function for helium with 1 / r_(12) as the perturbation. The results are accurate and encouraging, and since the method is numerical they are indicative of its potential for obtaining atomic‐pair functions in general

Decentralized Delay Optimal Control for Interference Networks with Limited Renewable Energy Storage

In this paper, we consider delay minimization for interference networks with renewable energy source, where the transmission power of a node comes from both the conventional utility power (AC power) and the renewable energy source. We assume the transmission power of each node is a function of the local channel state, local data queue state and local energy queue state only. In turn, we consider two delay optimization formulations, namely the decentralized partially observable Markov decision process (DEC-POMDP) and Non-cooperative partially observable stochastic game (POSG). In DEC-POMDP formulation, we derive a decentralized online learning algorithm to determine the control actions and Lagrangian multipliers (LMs) simultaneously, based on the policy gradient approach. Under some mild technical conditions, the proposed decentralized policy gradient algorithm converges almost surely to a local optimal solution. On the other hand, in the non-cooperative POSG formulation, the transmitter nodes are non-cooperative. We extend the decentralized policy gradient solution and establish the technical proof for almost-sure convergence of the learning algorithms. In both cases, the solutions are very robust to model variations. Finally, the delay performance of the proposed solutions are compared with conventional baseline schemes for interference networks and it is illustrated that substantial delay performance gain and energy savings can be achieved

Delay-Optimal User Scheduling and Inter-Cell Interference Management in Cellular Network via Distributive Stochastic Learning

In this paper, we propose a distributive queueaware intra-cell user scheduling and inter-cell interference (ICI) management control design for a delay-optimal celluar downlink system with M base stations (BSs), and K users in each cell. Each BS has K downlink queues for K users respectively with heterogeneous arrivals and delay requirements. The ICI management control is adaptive to joint queue state information (QSI) over a slow time scale, while the user scheduling control is adaptive to both the joint QSI and the joint channel state information (CSI) over a faster time scale. We show that the problem can be modeled as an infinite horizon average cost Partially Observed Markov Decision Problem (POMDP), which is NP-hard in general. By exploiting the special structure of the problem, we shall derive an equivalent Bellman equation to solve the POMDP problem. To address the distributive requirement and the issue of dimensionality and computation complexity, we derive a distributive online stochastic learning algorithm, which only requires local QSI and local CSI at each of the M BSs. We show that the proposed learning algorithm converges almost surely (with probability 1) and has significant gain compared with various baselines. The proposed solution only has linear complexity order O(MK)

Dynamic Interference Mitigation for Generalized Partially Connected Quasi-static MIMO Interference Channel

Recent works on MIMO interference channels have shown that interference alignment can significantly increase the achievable degrees of freedom (DoF) of the network. However, most of these works have assumed a fully connected interference graph. In this paper, we investigate how the partial connectivity can be exploited to enhance system performance in MIMO interference networks. We propose a novel interference mitigation scheme which introduces constraints for the signal subspaces of the precoders and decorrelators to mitigate "many" interference nulling constraints at a cost of "little" freedoms in precoder and decorrelator design so as to extend the feasibility region of the interference alignment scheme. Our analysis shows that the proposed algorithm can significantly increase system DoF in symmetric partially connected MIMO interference networks. We also compare the performance of the proposed scheme with various baselines and show via simulations that the proposed algorithms could achieve significant gain in the system performance of randomly connected interference networks.Comment: 30 pages, 10 figures, accepted by IEEE Transaction on Signal Processin