Computation of Approximate Welfare-Maximizing Correlated Equilibria and Pareto-Optima with Applications to Wireless Communication

In a wireless application with multiple communication links, the data rate of each link is subject to degradation due to transmitting interference from other links. A competitive wireless game then arises as each link acts as a player maximizing its own data rate. The game outcome can be evaluated using the solution concept of game equilibria. However, when significant interference among the links arises, uniqueness of equilibrium is not guaranteed. To select among multiple equilibria, the sum of network rate or social welfare is used as the selection criterion. This thesis aims to offer the theoretical foundation and the computational tool for determining approximate correlated equilibria with global maximum expected social welfare in polynomial games. Using sum of utilities as the global objective, we give two theoretical and two wireless-specific contributions. 1. We give a problem formulation for computing near-exact ε -correlated equilibria with highest possible expected social welfare. We then give a sequential Semidefinite Programming (SDP) algorithm that computes the solution. The solution consists of bounds information on the social welfare. 2. We give a novel reformulation to arrive at a leaner problem for computing near-exact ε -correlated equilibria using Kantorovich polynomials with sparsity. 3. Forgoing near-exactness, we consider approximate correlated equilibria. To account for the loss in precision, we introduce the notion of regret. We give theoretical bounds on the regrets at any iteration of the sequential SDP algorithm. Moreover, we give a heuristic procedure for extracting a discrete probability distribution. Subject to players’ acceptance of the regrets, the computed distributions can be used to implement central arbitrators to facilitate real-life implementation of the correlated equilibrium concept. 4. We demonstrate how to compute Pareto-optimal solutions by dropping the correlated equilibria constraints. For demonstration purpose, we focus only on Pareto-optima with equal weights among the players

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

New Directions for Contact Integrators

Contact integrators are a family of geometric numerical schemes which guarantee the conservation of the contact structure. In this work we review the construction of both the variational and Hamiltonian versions of these methods. We illustrate some of the advantages of geometric integration in the dissipative setting by focusing on models inspired by recent studies in celestial mechanics and cosmology.Comment: To appear as Chapter 24 in GSI 2021, Springer LNCS 1282