Relating thermodynamics to information theory : the equality of free energy and mutual information

In this thesis we uncover a new relation which links thermodynamics and information theory. We consider time as a channel and the detailed state of a physical system as a message. As the system evolves with time, ever present noise insures that the "message" is corrupted. Thermodynamic free energy measures the approach of the system toward equilibrium. Information theoretical mutual information measures the loss of memory of initial state. We regard the free energy and the mutual information as operators which map probability distributions over state space to real numbers. In the limit of long times, we show how the free energy operator and the mutual information operator asymptotically attain a very simple relationship to one another. This relationship is founded on the common appearance of entropy in the two operators and on an identity between internal energy and conditional entropy. The use of conditional entropy is what distinguishes our approach from previous efforts to relate thermodynamics and information theory.

Some modified bifurcation problems with application to imperfection sensitivity in buckling

The branching theory of solutions of certain nonlinear elliptic partial differential equations is developed, when the nonlinear term is perturbed from unforced to forced. We find families of branching points and the associated nonisolated solutions which emanate from a bifurcation point of the unforced problem. Nontrivial solution branches are constructed which contain the nonisolated solutions, and the branching is exhibited. An iteration procedure is used to establish the existence of these solutions, and a formal perturbation theory is shown to give asymptotically valid results. The stability of the solutions is examined and certain solution branches are shown to consist of minimal positive solutions. Other solution branches which do not contain branching points are also found in a neighborhood of the bifurcation point.

The qualitative features of branching points and their associated nonisolated solutions are used to obtain useful information about buckling of columns and arches. Global stability characteristics for the buckled equilibrium states of imperfect columns and arches are discussed. Asymptotic expansions for the imperfection sensitive buckling load of a column on a nonlinearly elastic foundation are found and rigorously justified.

On erasure coding for distributed storage and streaming communications

The work presented in this thesis revolves around erasure correction coding, as applied to distributed data storage and real-time streaming communications.

First, we examine the problem of allocating a given storage budget over a set of nodes for maximum reliability. The objective is to find an allocation of the budget that maximizes the probability of successful recovery by a data collector accessing a random subset of the nodes. This optimization problem is challenging in general because of its combinatorial nature, despite its simple formulation. We study several variations of the problem, assuming different allocation models and access models, and determine the optimal allocation and the optimal symmetric allocation (in which all nonempty nodes store the same amount of data) for a variety of cases. Although the optimal allocation can have nonintuitive structure and can be difficult to find in general, our results suggest that, as a simple heuristic, reliable storage can be achieved by spreading the budget maximally over all nodes when the budget is large, and spreading it minimally over a few nodes when it is small. Coding would therefore be beneficial in the former case, while uncoded replication would suffice in the latter case.

Second, we study how distributed storage allocations affect the recovery delay in a mobile setting. Specifically, two recovery delay optimization problems are considered for a network of mobile storage nodes: the maximization of the probability of successful recovery by a given deadline, and the minimization of the expected recovery delay. We show that the first problem is closely related to the earlier allocation problem, and solve the second problem completely for the case of symmetric allocations. It turns out that the optimal allocations for the two problems can be quite different. In a simulation study, we evaluated the performance of a simple data dissemination and storage protocol for mobile delay-tolerant networks, and observed that the choice of allocation can have a significant impact on the recovery delay under a variety of scenarios.

Third, we consider a real-time streaming system where messages created at regular time intervals at a source are encoded for transmission to a receiver over a packet erasure link; the receiver must subsequently decode each message within a given delay from its creation time. For erasure models containing a limited number of erasures per coding window, per sliding window, and containing erasure bursts whose maximum length is sufficiently short or long, we show that a time-invariant intrasession code asymptotically achieves the maximum message size among all codes that allow decoding under all admissible erasure patterns. For the bursty erasure model, we also show that diagonally interleaved codes derived from specific systematic block codes are asymptotically optimal over all codes in certain cases. We also study an i.i.d. erasure model in which each transmitted packet is erased independently with the same probability; the objective is to maximize the decoding probability for a given message size. We derive an upper bound on the decoding probability for any time-invariant code, and show that the gap between this bound and the performance of a family of time-invariant intrasession codes is small when the message size and packet erasure probability are small. In a simulation study, these codes performed well against a family of random time-invariant convolutional codes under a number of scenarios.

Finally, we consider the joint problems of routing and caching for named data networking. We propose a backpressure-based policy that employs virtual interest packets to make routing and caching decisions. In a packet-level simulation, the proposed policy outperformed a basic protocol that combines shortest-path routing with least-recently-used (LRU) cache replacement.

Singularities and horizons in the collisions of gravitational waves

This thesis presents a study of the dynamical, nonlinear interaction of colliding gravitational waves, as described by classical general relativity. It is focused mainly on two fundamental questions: First, what is the general structure of the singularities and Killing-Cauchy horizons produced in the collisions of exactly plane-symmetric gravitational waves? Second, under what conditions will the collisions of almost-plane gravitational waves (waves with large but finite transverse sizes) produce singularities?

In the work on the collisions of exactly-plane waves, it is shown that Killing horizons in any plane-symmetric spacetime are unstable against small plane-symmetric perturbations. It is thus concluded that the Killing-Cauchy horizons produced by the collisions of some exactly plane gravitational waves are nongeneric, and that generic initial data for the colliding plane waves always produce "pure" spacetime singularities without such horizons. This conclusion is later proved rigorously (using the full nonlinear theory rather than perturbation theory), in connection with an analysis of the asymptotic singularity structure of a general colliding plane-wave spacetime. This analysis also proves that asymptotically the singularities created by colliding plane waves are of inhomogeneous-Kasner type; the asymptotic Kasner axes and exponents of these singularities in general depend on the spatial coordinate that runs tangentially to the singularity in the non-plane-symmetric direction.

In the work on collisions of almost-plane gravitational waves, first some general properties of single almost-plane gravitational-wave spacetimes are explored. It is shown that, by contrast with an exact plane wave, an almost-plane gravitational wave cannot have a propagation direction that is Killing; i.e., it must diffract and disperse as it propagates. It is also shown that an almost-plane wave cannot be precisely sandwiched between two null wavefronts; i.e., it must leave behind tails in the spacetime region through which it passes. Next, the occurrence of spacetime singularities in the collisions of almost-plane waves is investigated. It is proved that if two colliding, almost-plane gravitational waves are initially exactly plane-symmetric across a central region of sufficiently large but finite transverse dimensions, then their collision produces a spacetime singularity with the same local structure as in the exact-plane-wave collision. Finally, it is shown that a singularity still forms when the central regions are only approximately plane-symmetric initially. Stated more precisely, it is proved that if the colliding almost-plane waves are initially sufficiently close to being exactly plane-symmetric across a bounded central region of sufficiently large transverse dimensions, then their collision necessarily produces spacetime singularities. In this case, nothing is now known about the local and global structures of the singularities.

A probabilistic treatment of uncertainty in nonlinear dynamical systems

In this work, computationally efficient approximate methods are developed for analyzing uncertain dynamical systems. Uncertainties in both the excitation and the modeling are considered and examples are presented illustrating the accuracy of the proposed approximations.

For nonlinear systems under uncertain excitation, methods are developed to approximate the stationary probability density function and statistical quantities of interest. The methods are based on approximating solutions to the Fokker-Planck equation for the system and differ from traditional methods in which approximate solutions to stochastic differential equations are found. The new methods require little computational effort and examples are presented for which the accuracy of the proposed approximations compare favorably to results obtained by existing methods. The most significant improvements are made in approximating quantities related to the extreme values of the response, such as expected outcrossing rates, which are crucial for evaluating the reliability of the system.

Laplace's method of asymptotic approximation is applied to approximate the probability integrals which arise when analyzing systems with modeling uncertainty. The asymptotic approximation reduces the problem of evaluating a multidimensional integral to solving a minimization problem and the results become asymptotically exact as the uncertainty in the modeling goes to zero. The method is found to provide good approximations for the moments and outcrossing rates for systems with uncertain parameters under stochastic excitation, even when there is a large amount of uncertainty in the parameters. The method is also applied to classical reliability integrals, providing approximations in both the transformed (independently, normally distributed) variables and the original variables. In the transformed variables, the asymptotic approximation yields a very simple formula for approximating the value of SORM integrals. In many cases, it may be computationally expensive to transform the variables, and an approximation is also developed in the original variables. Examples are presented illustrating the accuracy of the approximations and results are compared with existing approximations.

Efficient methods for empirical tests of behavioral economics theories in laboratory and field experiments

In the quest for a descriptive theory of decision-making, the rational actor model in economics imposes rather unrealistic expectations and abilities on human decision makers. The further we move from idealized scenarios, such as perfectly competitive markets, and ambitiously extend the reach of the theory to describe everyday decision making situations, the less sense these assumptions make. Behavioural economics has instead proposed models based on assumptions that are more psychologically realistic, with the aim of gaining more precision and descriptive power. Increased psychological realism, however, comes at the cost of a greater number of parameters and model complexity. Now there are a plethora of models, based on different assumptions, applicable in differing contextual settings, and selecting the right model to use tends to be an ad-hoc process. In this thesis, we develop optimal experimental design methods and evaluate different behavioral theories against evidence from lab and field experiments.

We look at evidence from controlled laboratory experiments. Subjects are presented with choices between monetary gambles or lotteries. Different decision-making theories evaluate the choices differently and would make distinct predictions about the subjects' choices. Theories whose predictions are inconsistent with the actual choices can be systematically eliminated. Behavioural theories can have multiple parameters requiring complex experimental designs with a very large number of possible choice tests. This imposes computational and economic constraints on using classical experimental design methods. We develop a methodology of adaptive tests: Bayesian Rapid Optimal Adaptive Designs (BROAD) that sequentially chooses the "most informative" test at each stage, and based on the response updates its posterior beliefs over the theories, which informs the next most informative test to run. BROAD utilizes the Equivalent Class Edge Cutting (EC²) criteria to select tests. We prove that the EC² criteria is adaptively submodular, which allows us to prove theoretical guarantees against the Bayes-optimal testing sequence even in the presence of noisy responses. In simulated ground-truth experiments, we find that the EC² criteria recovers the true hypotheses with significantly fewer tests than more widely used criteria such as Information Gain and Generalized Binary Search. We show, theoretically as well as experimentally, that surprisingly these popular criteria can perform poorly in the presence of noise, or subject errors. Furthermore, we use the adaptive submodular property of EC² to implement an accelerated greedy version of BROAD which leads to orders of magnitude speedup over other methods.

We use BROAD to perform two experiments. First, we compare the main classes of theories for decision-making under risk, namely: expected value, prospect theory, constant relative risk aversion (CRRA) and moments models. Subjects are given an initial endowment, and sequentially presented choices between two lotteries, with the possibility of losses. The lotteries are selected using BROAD, and 57 subjects from Caltech and UCLA are incentivized by randomly realizing one of the lotteries chosen. Aggregate posterior probabilities over the theories show limited evidence in favour of CRRA and moments' models. Classifying the subjects into types showed that most subjects are described by prospect theory, followed by expected value. Adaptive experimental design raises the possibility that subjects could engage in strategic manipulation, i.e. subjects could mask their true preferences and choose differently in order to obtain more favourable tests in later rounds thereby increasing their payoffs. We pay close attention to this problem; strategic manipulation is ruled out since it is infeasible in practice, and also since we do not find any signatures of it in our data.

In the second experiment, we compare the main theories of time preference: exponential discounting, hyperbolic discounting, "present bias" models: quasi-hyperbolic (α, β) discounting and fixed cost discounting, and generalized-hyperbolic discounting. 40 subjects from UCLA were given choices between 2 options: a smaller but more immediate payoff versus a larger but later payoff. We found very limited evidence for present bias models and hyperbolic discounting, and most subjects were classified as generalized hyperbolic discounting types, followed by exponential discounting.

In these models the passage of time is linear. We instead consider a psychological model where the perception of time is subjective. We prove that when the biological (subjective) time is positively dependent, it gives rise to hyperbolic discounting and temporal choice inconsistency.

We also test the predictions of behavioral theories in the "wild". We pay attention to prospect theory, which emerged as the dominant theory in our lab experiments of risky choice. Loss aversion and reference dependence predicts that consumers will behave in a uniquely distinct way than the standard rational model predicts. Specifically, loss aversion predicts that when an item is being offered at a discount, the demand for it will be greater than that explained by its price elasticity. Even more importantly, when the item is no longer discounted, demand for its close substitute would increase excessively. We tested this prediction using a discrete choice model with loss-averse utility function on data from a large eCommerce retailer. Not only did we identify loss aversion, but we also found that the effect decreased with consumers' experience. We outline the policy implications that consumer loss aversion entails, and strategies for competitive pricing.

In future work, BROAD can be widely applicable for testing different behavioural models, e.g. in social preference and game theory, and in different contextual settings. Additional measurements beyond choice data, including biological measurements such as skin conductance, can be used to more rapidly eliminate hypothesis and speed up model comparison. Discrete choice models also provide a framework for testing behavioural models with field data, and encourage combined lab-field experiments.

Rewriting Schemes for Flash Memory

Flash memory is a leading storage media with excellent features such as random access and high storage density. However, it also faces significant reliability and endurance challenges. In flash memory, the charge level in the cells can be easily increased, but removing charge requires an expensive erasure operation. In this thesis we study rewriting schemes that enable the data stored in a set of cells to be rewritten by only increasing the charge level in the cells. We consider two types of modulation scheme; a convectional modulation based on the absolute levels of the cells, and a recently-proposed scheme based on the relative cell levels, called rank modulation. The contributions of this thesis to the study of rewriting schemes for rank modulation include the following: we

•propose a new method of rewriting in rank modulation, beyond the previously proposed method of “push-to-the-top”;

•study the limits of rewriting with the newly proposed method, and derive a tight upper bound of 1 bit per cell;

•extend the rank-modulation scheme to support rankings with repetitions, in order to improve the storage density;

•derive a tight upper bound of 2 bits per cell for rewriting in rank modulation with repetitions;

•construct an efficient rewriting scheme that asymptotically approaches the upper bound of 2 bit per cell.

The next part of this thesis studies rewriting schemes for a conventional absolute-levels modulation. The considered model is called “write-once memory” (WOM). We focus on WOM schemes that achieve the capacity of the model. In recent years several capacity-achieving WOM schemes were proposed, based on polar codes and randomness extractors. The contributions of this thesis to the study of WOM scheme include the following: we

•propose a new capacity-achievingWOM scheme based on sparse-graph codes, and show its attractive properties for practical implementation;

•improve the design of polarWOMschemes to remove the reliance on shared randomness and include an error-correction capability.

The last part of the thesis studies the local rank-modulation (LRM) scheme, in which a sliding window going over a sequence of real-valued variables induces a sequence of permutations. The LRM scheme is used to simulate a single conventional multi-level flash cell. The simulated cell is realized by a Gray code traversing all the relative-value states where, physically, the transition between two adjacent states in the Gray code is achieved by using a single “push-to-the-top” operation. The main results of the last part of the thesis are two constructions of Gray codes with asymptotically-optimal rate.

Some central limit theorems for doubly restricted partitions

Let P_{K, L}(N) be the number of unordered partitions of a positive integer N into K or fewer positive integer parts, each part not exceeding L. A distribution of the form

Ʃ/N≤x P_K,L(N)

is considered first. For any fixed K, this distribution approaches a piecewise polynomial function as L increases to infinity. As both K and L approach infinity, this distribution is asymptotically normal. These results are proved by studying the convergence of the characteristic function.

The main result is the asymptotic behavior of P_K,K(N) itself, for certain large K and N. This is obtained by studying a contour integral of the generating function taken along the unit circle. The bulk of the estimate comes from integrating along a small arc near the point 1. Diophantine approximation is used to show that the integral along the rest of the circle is much smaller.

Advancements in jet turbulence and noise modeling: accurate one-way solutions and empirical evaluation of the nonlinear forcing of wavepackets

Jet noise reduction is an important goal within both commercial and military aviation. Although large-scale numerical simulations are now able to simultaneously compute turbulent jets and their radiated sound, lost-cost, physically-motivated models are needed to guide noise-reduction efforts. A particularly promising modeling approach centers around certain large-scale coherent structures, called wavepackets, that are observed in jets and their radiated sound. The typical approach to modeling wavepackets is to approximate them as linear modal solutions of the Euler or Navier-Stokes equations linearized about the long-time mean of the turbulent flow field. The near-field wavepackets obtained from these models show compelling agreement with those educed from experimental and simulation data for both subsonic and supersonic jets, but the acoustic radiation is severely under-predicted in the subsonic case. This thesis contributes to two aspects of these models. First, two new solution methods are developed that can be used to efficiently compute wavepackets and their acoustic radiation, reducing the computational cost of the model by more than an order of magnitude. The new techniques are spatial integration methods and constitute a well-posed, convergent alternative to the frequently used parabolized stability equations. Using concepts related to well-posed boundary conditions, the methods are formulated for general hyperbolic equations and thus have potential applications in many fields of physics and engineering. Second, the nonlinear and stochastic forcing of wavepackets is investigated with the goal of identifying and characterizing the missing dynamics responsible for the under-prediction of acoustic radiation by linear wavepacket models for subsonic jets. Specifically, we use ensembles of large-eddy-simulation flow and force data along with two data decomposition techniques to educe the actual nonlinear forcing experienced by wavepackets in a Mach 0.9 turbulent jet. Modes with high energy are extracted using proper orthogonal decomposition, while high gain modes are identified using a novel technique called empirical resolvent-mode decomposition. In contrast to the flow and acoustic fields, the forcing field is characterized by a lack of energetic coherent structures. Furthermore, the structures that do exist are largely uncorrelated with the acoustic field. Instead, the forces that most efficiently excite an acoustic response appear to take the form of random turbulent fluctuations, implying that direct feedback from nonlinear interactions amongst wavepackets is not an essential noise source mechanism. This suggests that the essential ingredients of sound generation in high Reynolds number jets are contained within the linearized Navier-Stokes operator rather than in the nonlinear forcing terms, a conclusion that has important implications for jet noise modeling.

Large plane deformations of thin elastic sheets of neo-hookean material

Large plane deformations of thin elastic sheets of neo-Hookean material are considered and a method of successive substitutions is developed to solve problems within the two-dimensional theory of finite plane stress. The first approximation is determined by linear boundary value problems on two harmonic functions, and it is approached asymptotically at very large extensions in the plane of the sheet. The second and higher approximations are obtained by solving Poisson equations. The method requires modification when the membrane has a traction-free edge.

Several problems are treated involving infinite sheets under uniform biaxial stretching at infinity. First approximations are obtained when a circular or elliptic inclusion is present and when the sheet has a circular or elliptic hole, including the limiting cases of a line inclusion and a straight crack or slit. Good agreement with exact solutions is found for circularly symmetric deformations. Other examples discuss the stretching of a short wide strip, the deformation near a boundary corner which is traction-free, and the application of a concentrated load to a boundary point.

On the uniqueness of singular solutions to boundary-initial value problems in linear elastodynamics

This investigation deals with certain generalizations of the classical uniqueness theorem for the second boundary-initial value problem in the linearized dynamical theory of not necessarily homogeneous nor isotropic elastic solids. First, the regularity assumptions underlying the foregoing theorem are relaxed by admitting stress fields with suitably restricted finite jump discontinuities. Such singularities are familiar from known solutions to dynamical elasticity problems involving discontinuous surface tractions or non-matching boundary and initial conditions. The proof of the appropriate uniqueness theorem given here rests on a generalization of the usual energy identity to the class of singular elastodynamic fields under consideration.

Following this extension of the conventional uniqueness theorem, we turn to a further relaxation of the customary smoothness hypotheses and allow the displacement field to be differentiable merely in a generalized sense, thereby admitting stress fields with square-integrable unbounded local singularities, such as those encountered in the presence of focusing of elastic waves. A statement of the traction problem applicable in these pathological circumstances necessitates the introduction of "weak solutions'' to the field equations that are accompanied by correspondingly weakened boundary and initial conditions. A uniqueness theorem pertaining to this weak formulation is then proved through an adaptation of an argument used by O. Ladyzhenskaya in connection with the first boundary-initial value problem for a second-order hyperbolic equation in a single dependent variable. Moreover, the second uniqueness theorem thus obtained contains, as a special case, a slight modification of the previously established uniqueness theorem covering solutions that exhibit only finite stress-discontinuities.

Dynamic response of hysteretic systems with application to a system containing limited slip

A general class of single degree of freedom systems possessing rate-independent hysteresis is defined. The hysteretic behavior in a system belonging to this class is depicted as a sequence of single-valued functions; at any given time, the current function is determined by some set of mathematical rules concerning the entire previous response of the system. Existence and uniqueness of solutions are established and boundedness of solutions is examined.

An asymptotic solution procedure is used to derive an approximation to the response of viscously damped systems with a small hysteretic nonlinearity and trigonometric excitation. Two properties of the hysteresis loops associated with any given system completely determine this approximation to the response: the area enclosed by each loop, and the average of the ascending and descending branches of each loop.

The approximation, supplemented by numerical calculations, is applied to investigate the steady-state response of a system with limited slip. Such features as disconnected response curves and jumps in response exist for a certain range of system parameters for any finite amount of slip.

To further understand the response of this system, solutions of the initial-value problem are examined. The boundedness of solutions is investigated first. Then the relationship between initial conditions and resulting steady-state solution is examined when multiple steady-state solutions exist. Using the approximate analysis and numerical calculations, it is found that significant regions of initial conditions in the initial condition plane lead to the different asymptotically stable steady-state solutions.