Calculation of Densities of States and Spectral Functions by Chebyshev Recursion and Maximum Entropy

We present an efficient algorithm for calculating spectral properties of large sparse Hamiltonian matrices such as densities of states and spectral functions. The combination of Chebyshev recursion and maximum entropy achieves high energy resolution without significant roundoff error, machine precision or numerical instability limitations. If controlled statistical or systematic errors are acceptable, cpu and memory requirements scale linearly in the number of states. The inference of spectral properties from moments is much better conditioned for Chebyshev moments than for power moments. We adapt concepts from the kernel polynomial approximation, a linear Chebyshev approximation with optimized Gibbs damping, to control the accuracy of Fourier integrals of positive non-analytic functions. We compare the performance of kernel polynomial and maximum entropy algorithms for an electronic structure example.Comment: 8 pages RevTex, 3 postscript figure

Optimal Deployments of UAVs With Directional Antennas for a Power-Efficient Coverage

To provide a reliable wireless uplink for users in a given ground area, one can deploy Unmanned Aerial Vehicles (UAVs) as base stations (BSs). In another application, one can use UAVs to collect data from sensors on the ground. For a power-efficient and scalable deployment of such flying BSs, directional antennas can be utilized to efficiently cover arbitrary 2-D ground areas. We consider a large-scale wireless path-loss model with a realistic angle-dependent radiation pattern for the directional antennas. Based on such a model, we determine the optimal 3-D deployment of N UAVs to minimize the average transmit-power consumption of the users in a given target area. The users are assumed to have identical transmitters with ideal omnidirectional antennas and the UAVs have identical directional antennas with given half-power beamwidth (HPBW) and symmetric radiation pattern along the vertical axis. For uniformly distributed ground users, we show that the UAVs have to share a common flight height in an optimal power-efficient deployment. We also derive in closed-form the asymptotic optimal common flight height of $N$ UAVs in terms of the area size, data-rate, bandwidth, HPBW, and path-loss exponent

T-optimal designs formulti-factor polynomial regressionmodelsvia a semidefinite relaxation method

We consider T-optimal experiment design problems for discriminating multi-factor polynomial regression models wherethe design space is defined by polynomial inequalities and the regression parameters are constrained to given convex sets.Our proposed optimality criterion is formulated as a convex optimization problem with a moment cone constraint. When theregression models have one factor, an exact semidefinite representation of the moment cone constraint can be applied to obtainan equivalent semidefinite program.When there are two or more factors in the models, we apply a moment relaxation techniqueand approximate the moment cone constraint by a hierarchy of semidefinite-representable outer approximations. When therelaxation hierarchy converges, an optimal discrimination design can be recovered from the optimal moment matrix, and itsoptimality can be additionally confirmed by an equivalence theorem. The methodology is illustrated with several examples

Compressed sensing reconstruction using Expectation Propagation

Many interesting problems in fields ranging from telecommunications to computational biology can be formalized in terms of large underdetermined systems of linear equations with additional constraints or regularizers. One of the most studied ones, the Compressed Sensing problem (CS), consists in finding the solution with the smallest number of non-zero components of a given system of linear equations $\boldsymbol y = \mathbf{F} \boldsymbol{w}$ for known measurement vector $\boldsymbol{y}$ and sensing matrix $\mathbf{F}$. Here, we will address the compressed sensing problem within a Bayesian inference framework where the sparsity constraint is remapped into a singular prior distribution (called Spike-and-Slab or Bernoulli-Gauss). Solution to the problem is attempted through the computation of marginal distributions via Expectation Propagation (EP), an iterative computational scheme originally developed in Statistical Physics. We will show that this strategy is comparatively more accurate than the alternatives in solving instances of CS generated from statistically correlated measurement matrices. For computational strategies based on the Bayesian framework such as variants of Belief Propagation, this is to be expected, as they implicitly rely on the hypothesis of statistical independence among the entries of the sensing matrix. Perhaps surprisingly, the method outperforms uniformly also all the other state-of-the-art methods in our tests.Comment: 20 pages, 6 figure