On the Reconstruction of Static and Dynamic Discrete Structures

We study inverse problems of reconstructing static and dynamic discrete structures from tomographic data (with a special focus on the `classical' task of reconstructing finite point sets in $\mathbb{R}^d$). The main emphasis is on recent mathematical developments and new applications, which emerge in scientific areas such as physics and materials science, but also in inner mathematical fields such as number theory, optimization, and imaging. Along with a concise introduction to the field of discrete tomography, we give pointers to related aspects of computerized tomography in order to contrast the worlds of continuous and discrete inverse problems

Computational aspects of communication amid uncertainty

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2018.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student-submitted PDF version of thesis.Includes bibliographical references (pages 203-215).This thesis focuses on the role of uncertainty in communication and effective (computational) methods to overcome uncertainty. A classical form of uncertainty arises from errors introduced by the communication channel but uncertainty can arise in many other ways if the communicating players do not completely know (or understand) each other. For example, it can occur as mismatches in the shared randomness used by the distributed agents, or as ambiguity in the shared context or goal of the communication. We study many modern models of uncertainty, some of which have been considered in the literature but are not well-understood, while others are introduced in this thesis: Uncertainty in Shared Randomness -- We study common randomness and secret key generation. In common randomness generation, two players are given access to correlated randomness and are required to agree on pure random bits while minimizing communication and maximizing agreement probability. Secret key generation refers to the setup where, in addition, the generated random key is required to be secure against any eavesdropper. These setups are of significant importance in information theory and cryptography. We obtain the first explicit and sample-efficient schemes with the optimal trade-offs between communication, agreement probability and entropy of generated common random bits, in the one-way communication setting. -- We obtain the first decidability result for the computational problem of the noninteractive simulation of joint distributions, which asks whether two parties can convert independent identically distributed samples from a given source of correlation into another desired form of correlation. This class of problems has been well-studied in information theory and its computational complexity has been wide open. Uncertainty in Goal of Communication -- We introduce a model for communication with functional uncertainty. In this setup, we consider the classical model of communication complexity of Yao, and study how this complexity changes if the function being computed is not completely known to both players. This forms a mathematical analogue of a natural situation in human communication: Communicating players do not a priori know what the goal of communication is. We design efficient protocols for dealing with uncertainty in this model in a broad setting. Our solution relies on public random coins being shared by the communicating players. We also study the question of relaxing this requirement and present several results answering different aspects of this question. Uncertainty in Prior Distribution -- We study data compression in a distributed setting where several players observe messages from an unknown distribution, which they wish to encode, communicate and decode. In this setup, we design and analyze a simple, decentralized and efficient protocol. In this thesis, we study these various forms of uncertainty, and provide novel solutions using tools from various areas of theoretical computer science, information theory and mathematics."This research was supported in part by an NSF STC Award CCF 0939370, NSF award numbers CCF-1217423, CCF-1650733 and CCF-1420692, an Irwin and Joan Jacobs Presidential Fellowship and an IBM Ph.D. Fellowship"--Page 7.by Badih Ghazi.Ph. D