New Large-Scale Optimization Methods with Applications in Communication Networks, Medical Imaging, and System Identification

Abstract

Optimizations with large sizes appear frequently in modern problems of quantitative fields to enhance current solutions. This dissertation is concerned with the development and analysis of large-scale optimization methods for communication networks, magnetic resonance imaging, and dynamical systems. The first chapter concentrates on the algorithm development for the joint problem of mapping virtual network functions to high-volume servers, resource provisioning, and traffic routing where the demand is known. In the next two chapters of this dissertation, we focus on the problem of joint resource reservation in the backhaul and radio access network where user demands and achievable rates of wireless channels are unknown; however, observations from these two randomness sources are available. We propose a novel method to maximize the sum of expected traffic flow rates, subject to link and access point budget constraints, while minimizing the expected outage of wireless channels. We use the proposed resource reservation method to dynamically slice network resources among several slices of different tenants when user demands and achievable rates of wireless connections are uncertain. We propose a two-timescale scheme in which a subset of network slices is activated via a novel sparse optimization framework in the long timescale with the goal of maximizing the expected utilities of tenants while in the short timescale the activated slices arereconfigured according to the time-varying user traffic and channel states.

The next two chapters of this thesis focus on magnetic resonance imaging with single and multiple echoes for fast acquisitions with long readout times, e.g., spiral, when the static magnetic field $B_0$ is largely nonuniform and inhomogeneous. We study the non-convex problem of joint image and $B_0$ field map estimation from a set of distorted images due the $B_0$ inhomogeneity. We propose novel voxel-level decompositions and develop parallel and distributed approaches based on block coordinate descent and golden-section-search methods to iteratively improve the estimates of the field map. Unlike prior works, 2D and 3D spherical harmonics are utilized in proposed algorithms to efficiently regularize field map estimations. The estimated field map is later leveraged to correct different artifacts of distorted images.

The final chapter investigates the problem of estimating the weight matrices of a stable time-invariant linear dynamical system from a single sequence of noisy measurements. Unlike existing methods that identify equivalent systems, we show that if the unknown weight matrices describing the system are in Brunovsky canonical form, we can efficiently estimate the ground truth unknown matrices of the system from a linear system of equations formulated based on the transfer function of the system and iterates of stochastic gradient descent methods.