Achieving Global Optimality for Weighted Sum-Rate Maximization in the K-User Gaussian Interference Channel with Multiple Antennas

Characterizing the global maximum of weighted sum-rate (WSR) for the K-user Gaussian interference channel (GIC), with the interference treated as Gaussian noise, is a key problem in wireless communication. However, due to the users' mutual interference, this problem is in general non-convex and thus cannot be solved directly by conventional convex optimization techniques. In this paper, by jointly utilizing the monotonic optimization and rate profile techniques, we develop a new framework to obtain the globally optimal power control and/or beamforming solutions to the WSR maximization problems for the GICs with single-antenna transmitters and single-antenna receivers (SISO), single-antenna transmitters and multi-antenna receivers (SIMO), or multi-antenna transmitters and single-antenna receivers (MISO). Different from prior work, this paper proposes to maximize the WSR in the achievable rate region of the GIC directly by exploiting the facts that the achievable rate region is a "normal" set and the users' WSR is a "strictly increasing" function over the rate region. Consequently, the WSR maximization is shown to be in the form of monotonic optimization over a normal set and thus can be solved globally optimally by the existing outer polyblock approximation algorithm. However, an essential step in the algorithm hinges on how to efficiently characterize the intersection point on the Pareto boundary of the achievable rate region with any prescribed "rate profile" vector. This paper shows that such a problem can be transformed into a sequence of signal-to-interference-plus-noise ratio (SINR) feasibility problems, which can be solved efficiently by existing techniques. Numerical results validate that the proposed algorithms can achieve the global WSR maximum for the SISO, SIMO or MISO GIC.Comment: This is the longer version of a paper to appear in IEEE Transactions on Wireless Communication

A volume-preserving sharpening approach for the propagation of sharp phase boundaries in multiphase lattice Boltzmann simulations

Lattice Boltzmann models that recover a macroscopic description of multiphase flow of immiscible liquids typically represent the boundaries between phases using a scalar function, the phase field, that varies smoothly over several grid points. Attempts to tune the model parameters to minimise the thicknesses of these interfaces typically lead to the interfaces becoming fixed to the underlying grid instead of advecting with the fluid velocity. This phenomenon, known as lattice pinning, is strikingly similar to that associated with the numerical simulation of conservation laws coupled to stiff algebraic source terms. We present a lattice Boltzmann formulation of the model problem proposed by LeVeque and Yee [J. Comput. Phys. 86, 187] to study the latter phenomenon in the context of computational combustion, and offer a volume-conserving extension in multiple space dimensions. Inspired by the random projection method of Bao and Jin [J. Comput. Phys. 163, 216] we further generalise this formulation by introducing a uniformly distributed quasi-random variable into the term responsible for the sharpening of phase boundaries. This method is mass conserving and the statistical average of this method is shown to significantly delay the onset of pinning

Asymptotic behaviour and numerical approximation of optimal eigenvalues of the Robin Laplacian

We consider the problem of minimising the $n^{th}-$eigenvalue of the Robin Laplacian in $\mathbb{R}^{N}$. Although for $n=1,2$ and a positive boundary parameter $\alpha$ it is known that the minimisers do not depend on $\alpha$, we demonstrate numerically that this will not always be the case and illustrate how the optimiser will depend on $\alpha$. We derive a Wolf-Keller type result for this problem and show that optimal eigenvalues grow at most with $n^{1/N}$, which is in sharp contrast with the Weyl asymptotics for a fixed domain. We further show that the gap between consecutive eigenvalues does go to zero as $n$ goes to infinity. Numerical results then support the conjecture that for each $n$ there exists a positive value of $\alpha_{n}$ such that the $n^{\rm th}$ eigenvalue is minimised by $n$ disks for all $0<\alpha<\alpha_{n}$ and, combined with analytic estimates, that this value is expected to grow with $n^{1/N}$

Lorenz-Mie theory for 2D scattering and resonance calculations

This PhD tutorial is concerned with a description of the two-dimensional generalized Lorenz-Mie theory (2D-GLMT), a well-established numerical method used to compute the interaction of light with arrays of cylindrical scatterers. This theory is based on the method of separation of variables and the application of an addition theorem for cylindrical functions. The purpose of this tutorial is to assemble the practical tools necessary to implement the 2D-GLMT method for the computation of scattering by passive scatterers or of resonances in optically active media. The first part contains a derivation of the vector and scalar Helmholtz equations for 2D geometries, starting from Maxwell's equations. Optically active media are included in 2D-GLMT using a recent stationary formulation of the Maxwell-Bloch equations called steady-state ab initio laser theory (SALT), which introduces new classes of solutions useful for resonance computations. Following these preliminaries, a detailed description of 2D-GLMT is presented. The emphasis is placed on the derivation of beam-shape coefficients for scattering computations, as well as the computation of resonant modes using a combination of 2D-GLMT and SALT. The final section contains several numerical examples illustrating the full potential of 2D-GLMT for scattering and resonance computations. These examples, drawn from the literature, include the design of integrated polarization filters and the computation of optical modes of photonic crystal cavities and random lasers.Comment: This is an author-created, un-copyedited version of an article published in Journal of Optics. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from i