Tensor Completion for Multidimensional Inverse Problems with Applications to Magnetic Resonance Relaxometry

This thesis deals with tensor completion for the solution of multidimensional inverse problems. We study the problem of reconstructing an approximately low rank tensor from a small number of noisy linear measurements. New recovery guarantees, numerical algorithms, non-uniform sampling strategies, and parameter selection algorithms are developed. We derive a fixed point continuation algorithm for tensor completion and prove its convergence. A restricted isometry property (RIP) based tensor recovery guarantee is proved. Probabilistic recovery guarantees are obtained for sub-Gaussian measurement operators and for measurements obtained by non-uniform sampling from a Parseval tight frame. We show how tensor completion can be used to solve multidimensional inverse problems arising in NMR relaxometry. Algorithms are developed for regularization parameter selection, including accelerated k-fold cross-validation and generalized cross-validation. These methods are validated on experimental and simulated data. We also derive condition number estimates for nonnegative least squares problems. Tensor recovery promises to significantly accelerate N-dimensional NMR relaxometry and related experiments, enabling previously impractical experiments. Our methods could also be applied to other inverse problems arising in machine learning, image processing, signal processing, computer vision, and other fields.

A Ride-Sharing System with High Capacity Vehicles and Flexible Meeting Points

In many major cities, fixed route transit systems such as bus and rail serve millions of trips per day. These systems have people collect at common locations (the station or stop), and board at common times (for example according to a predetermined schedule or headway). By using common service locations and times, these modes can consolidate many trips that have similar origins and destinations or overlapping routes. However, the routes are not sensitive to changing travel patterns, and have no way of identifying which trips are going unserved, or are poorly served, by the existing routes. On the opposite end of the spectrum, personal modes of transportation, such as a private vehicle or taxi, offer service to and from the exact origin and destination of a rider, at close to exactly the time they desire to travel. Despite the apparent increased convenience to users, the presence of a large number of small vehicles results in a disorganized, and potentially congested road network during high demand periods. The focus of the research presented in this paper is to develop a system that possesses both the on-demand nature of a personal mode, with the efficiency of shared modes. In this system, users submit their request for travel, but are asked to make small compromises in their origin and destination location by walking to a nearby meeting point, as well as slightly modifying their time of travel, in order to accommodate other passengers. Because the origin and destination location of the request can be adjusted, this is a more general case of the Dial-a-Ride problem with time windows. The solution methodology uses a graph clustering algorithm coupled with a greedy insertion technique. A case study is presented using actual requests for taxi trips in Washington DC, and shows a significant decrease in the number of vehicles required to serve the demand.