Model Order Reduction

An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This three-volume handbook covers methods as well as applications. This third volume focuses on applications in engineering, biomedical engineering, computational physics and computer science

Nonlocal games and their device-independent quantum applications

Device-independence is a property of certain protocols that allows one to ensure their proper execution given only classical interaction with devices and assuming the correctness of the laws of physics. This scenario describes the most general form of cryptographic security, in which no trust is placed in the hardware involved; indeed, one may even take it to have been prepared by an adversary. Many quantum tasks have been shown to admit device-independent protocols by augmentation with "nonlocal games". These are games in which noncommunicating parties jointly attempt to fulfil some conditions imposed by a referee. We introduce examples of such games and examine the optimal strategies of players who are allowed access to different possible shared resources, such as entangled quantum states. We then study their role in self-testing, private random number generation, and secure delegated quantum computation. Hardware imperfections are naturally incorporated in the device-independent scenario as adversarial, and we thus also perform noise robustness analysis where feasible. We first study a generalization of the Mermin–Peres magic square game to arbitrary rectangular dimensions. After exhibiting some general properties, these "magic rectangle" games are fully characterized in terms of their optimal win probabilities for quantum strategies. We find that for m×n magic rectangle games with dimensions m,n≥3, there are quantum strategies that win with certainty, while for dimensions 1×n quantum strategies do not outperform classical strategies. The final case of dimensions 2×n is richer, and we give upper and lower bounds that both outperform the classical strategies. As an initial usage scenario, we apply our findings to quantum certified randomness expansion to find noise tolerances and rates for all magic rectangle games. To do this, we use our previous results to obtain the winning probabilities of games with a distinguished input for which the devices give a deterministic outcome and follow the analysis of C. A. Miller and Y. Shi [SIAM J. Comput. 46, 1304 (2017)]. Self-testing is a method to verify that one has a particular quantum state from purely classical statistics. For practical applications, such as device-independent delegated verifiable quantum computation, it is crucial that one self-tests multiple Bell states in parallel while keeping the quantum capabilities required of one side to a minimum. We use our 3×n magic rectangle games to obtain a self-test for n Bell states where one side needs only to measure single-qubit Pauli observables. The protocol requires small input sizes [constant for Alice and O(log n) bits for Bob] and is robust with robustness O(n⁵/²√ε), where ε is the closeness of the ideal (perfect) correlations to those observed. To achieve the desired self-test, we introduce a one-side-local quantum strategy for the magic square game that wins with certainty, we generalize this strategy to the family of 3×n magic rectangle games, and we supplement these nonlocal games with extra check rounds (of single and pairs of observables). Finally, we introduce a device-independent two-prover scheme in which a classical verifier can use a simple untrusted quantum measurement device (the client device) to securely delegate a quantum computation to an untrusted quantum server. To do this, we construct a parallel self-testing protocol to perform device-independent remote state preparation of n qubits and compose this with the unconditionally secure universal verifiable blind quantum computation (VBQC) scheme of J. F. Fitzsimons and E. Kashefi [Phys. Rev. A 96, 012303 (2017)]. Our self-test achieves a multitude of desirable properties for the application we consider, giving rise to practical and fully device-independent VBQC. It certifies parallel measurements of all cardinal and intercardinal directions in the XY-plane as well as the computational basis, uses few input questions (of size logarithmic in n for the client and a constant number communicated to the server), and requires only single-qubit measurements to be performed by the client device

Application of non-linear system identification approaches to modelling, analysis, and control of fluid flows.

Flow control has become a topic of great importance for several applications, ranging from commercial aircraft, to intercontinental pipes and skyscrapers. In these applications, and many more, the interaction with a fluid flow can have a significant influence on the performance of the system. In many cases the fluids encountered are turbulent and detrimental to the latter. Several attempts have been made to solve this problem. However, due to the non-linearity and infinite dimensionality of fluid flows and their governing equations, a complete understanding of turbulent behaviour and a feasible control approach has not been obtained. In this thesis, model reduction approaches that exploit non-linear system identification are applied using data obtained from numerical simulations of turbulent three-dimensional channel flow, and two-dimensional flow over the backward facing step. A multiple-input multiple-output model, consisting of 27 sub-structures, is obtained for the fluctuations of the velocity components of the channel flow. A single-input single-output model for fluctuations of the pressure coefficient, and two multiple-input single-output models for fluctuations of the velocity magnitude are obtained in flow over the BFS. A non-linear model predictive control strategy is designed using identified one- and multi-step ahead predictors, with the inclusion of integral action for robustness. The proposed control approach incorporates a non-linear model without the need for expensive non-linear optimizations. Finally, a frequency domain analysis of unmanipulated turbulent flow is perfumed using five systems. Higher order generalized frequency response functions (GFRF) are computed to study the non-linear energy transfer phenomena. A more detailed investigation is performed using the output FRF (OFRF), which can elucidate the contribution of the n-th order frequency response to the output frequency response

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