Periodic solutions of integro-differential equations which arise in population dynamics

The problem of the existence and stability of periodic solutions of infinite-lag integra-differential equations is considered. Specifically, the integrals involved are of the convolution type with the dependent variable being integrated over the range (- ∞,t), as occur in models of population growth. It is shown that Hopf bifurcation of periodic solutions from a steady state can occur, when a pair of eigenvalues crosses the imaginary axis. Also considered is the existence of traveling wave solutions of a model population equation allowing spatial diffusion in addition to the usual temporal variation. Lastly, the stability of the periodic solutions resulting from Hopf bifurcation is determined with aid of a Floquet theory.

The first chapter is devoted to linear integro-differential equations with constant coefficients utilizing the method of semi-groups of operators. The second chapter analyzes the Hopf bifurcation providing an existence theorem. Also, the two-timing perturbation procedure is applied to construct the periodic solutions. The third chapter uses two-timing to obtain traveling wave solutions of the diffusive model, as well as providing an existence theorem. The fourth chapter develops a Floquet theory for linear integro-differential equations with periodic coefficients again using the semi-group approach. The fifth chapter gives sufficient conditions for the stability or instability of a periodic solution in terms of the linearization of the equations. These results are then applied to the Hopf bifurcation problem and to a certain population equation modeling periodically fluctuating environments to deduce the stability of the corresponding periodic solutions.

Asymptotic scaling in the two-dimensional O(3) Nonlinear sigma model: a Monte Carlo study on parallel computers

We investigate the 2d O(3) model with the standard action by Monte Carlo simulation at couplings β up to 2.05. We measure the energy density, mass gap and susceptibility of the model, and gather high statistics on lattices of size L ≤ 1024 using the Floating Point Systems T-series vector hypercube and the Thinking Machines Corp.'s Connection Machine 2. Asymptotic scaling does not appear to set in for this action, even at β = 2.10, where the correlation length is 420. We observe a 20% difference between our estimate m/Λ^─_(Ms) = 3.52(6) at this β and the recent exact analytical result . We use the overrelaxation algorithm interleaved with Metropolis updates and show that decorrelation time scales with the correlation length and the number of overrelaxation steps per sweep. We determine its effective dynamical critical exponent to be z' = 1.079(10); thus critical slowing down is reduced significantly for this local algorithm that is vectorizable and parallelizable.

We also use the cluster Monte Carlo algorithms, which are non-local Monte Carlo update schemes which can greatly increase the efficiency of computer simulations of spin models. The major computational task in these algorithms is connected component labeling, to identify clusters of connected sites on a lattice. We have devised some new SIMD component labeling algorithms, and implemented them on the Connection Machine. We investigate their performance when applied to the cluster update of the two dimensional Ising spin model.

Finally we use a Monte Carlo Renormalization Group method to directly measure the couplings of block Hamiltonians at different blocking levels. For the usual averaging block transformation we confirm the renormalized trajectory (RT) observed by Okawa. For another improved probabilistic block transformation we find the RT, showing that it is much closer to the Standard Action. We then use this block transformation to obtain the discrete β-function of the model which we compare to the perturbative result. We do not see convergence, except when using a rescaled coupling β_E to effectively resum the series. For the latter case we see agreement for m/ Λ^─_(Ms) at , β = 2.14, 2.26, 2.38 and 2.50. To three loops m/Λ^─_(Ms) = 3.047(35) at β = 2.50, which is very close to the exact value m/ Λ^─_(Ms) = 2.943. Our last point at β = 2.62 disagrees with this estimate however.

Earthquakes of the Nepal Himalaya : towards a physical model of the seismic cycle

Home to hundreds of millions of souls and land of excessiveness, the Himalaya is also the locus of a unique seismicity whose scope and peculiarities still remain to this day somewhat mysterious. Having claimed the lives of kings, or turned ancient timeworn cities into heaps of rubbles and ruins, earthquakes eerily inhabit Nepalese folk tales with the fatalistic message that nothing lasts forever. From a scientific point of view as much as from a human perspective, solving the mysteries of Himalayan seismicity thus represents a challenge of prime importance. Documenting geodetic strain across the Nepal Himalaya with various GPS and leveling data, we show that unlike other subduction zones that exhibit a heterogeneous and patchy coupling pattern along strike, the last hundred kilometers of the Main Himalayan Thrust fault, or MHT, appear to be uniformly locked, devoid of any of the “creeping barriers” that traditionally ward off the propagation of large events. The approximately 20 mm/yr of reckoned convergence across the Himalaya matching previously established estimates of the secular deformation at the front of the arc, the slip accumulated at depth has to somehow elastically propagate all the way to the surface at some point. And yet, neither large events from the past nor currently recorded microseismicity nearly compensate for the massive moment deficit that quietly builds up under the giant mountains. Along with this large unbalanced moment deficit, the uncommonly homogeneous coupling pattern on the MHT raises the question of whether or not the locked portion of the MHT can rupture all at once in a giant earthquake. Univocally answering this question appears contingent on the still elusive estimate of the magnitude of the largest possible earthquake in the Himalaya, and requires tight constraints on local fault properties. What makes the Himalaya enigmatic also makes it the potential source of an incredible wealth of information, and we exploit some of the oddities of Himalayan seismicity in an effort to improve the understanding of earthquake physics and cipher out the properties of the MHT. Thanks to the Himalaya, the Indo-Gangetic plain is deluged each year under a tremendous amount of water during the annual summer monsoon that collects and bears down on the Indian plate enough to pull it away from the Eurasian plate slightly, temporarily relieving a small portion of the stress mounting on the MHT. As the rainwater evaporates in the dry winter season, the plate rebounds and tension is increased back on the fault. Interestingly, the mild waggle of stress induced by the monsoon rains is about the same size as that from solid-Earth tides which gently tug at the planets solid layers, but whereas changes in earthquake frequency correspond with the annually occurring monsoon, there is no such correlation with Earth tides, which oscillate back-and-forth twice a day. We therefore investigate the general response of the creeping and seismogenic parts of MHT to periodic stresses in order to link these observations to physical parameters. First, the response of the creeping part of the MHT is analyzed with a simple spring-and-slider system bearing rate-strengthening rheology, and we show that at the transition with the locked zone, where the friction becomes near velocity neutral, the response of the slip rate may be amplified at some periods, which values are analytically related to the physical parameters of the problem. Such predictions therefore hold the potential of constraining fault properties on the MHT, but still await observational counterparts to be applied, as nothing indicates that the variations of seismicity rate on the locked part of the MHT are the direct expressions of variations of the slip rate on its creeping part, and no variations of the slip rate have been singled out from the GPS measurements to this day. When shifting to the locked seismogenic part of the MHT, spring-and-slider models with rate-weakening rheology are insufficient to explain the contrasted responses of the seismicity to the periodic loads that tides and monsoon both place on the MHT. Instead, we resort to numerical simulations using the Boundary Integral CYCLes of Earthquakes algorithm and examine the response of a 2D finite fault embedded with a rate-weakening patch to harmonic stress perturbations of various periods. We show that such simulations are able to reproduce results consistent with a gradual amplification of sensitivity as the perturbing period get larger, up to a critical period corresponding to the characteristic time of evolution of the seismicity in response to a step-like perturbation of stress. This increase of sensitivity was not reproduced by simple 1D-spring-slider systems, probably because of the complexity of the nucleation process, reproduced only by 2D-fault models. When the nucleation zone is close to its critical unstable size, its growth becomes highly sensitive to any external perturbations and the timings of produced events may therefore find themselves highly affected. A fully analytical framework has yet to be developed and further work is needed to fully describe the behavior of the fault in terms of physical parameters, which will likely provide the keys to deduce constitutive properties of the MHT from seismological observations.

Multiscale model reduction methods for deterministic and stochastic partial differential equations

Partial differential equations (PDEs) with multiscale coefficients are very difficult to solve due to the wide range of scales in the solutions. In the thesis, we propose some efficient numerical methods for both deterministic and stochastic PDEs based on the model reduction technique.

For the deterministic PDEs, the main purpose of our method is to derive an effective equation for the multiscale problem. An essential ingredient is to decompose the harmonic coordinate into a smooth part and a highly oscillatory part of which the magnitude is small. Such a decomposition plays a key role in our construction of the effective equation. We show that the solution to the effective equation is smooth, and could be resolved on a regular coarse mesh grid. Furthermore, we provide error analysis and show that the solution to the effective equation plus a correction term is close to the original multiscale solution.

For the stochastic PDEs, we propose the model reduction based data-driven stochastic method and multilevel Monte Carlo method. In the multiquery, setting and on the assumption that the ratio of the smallest scale and largest scale is not too small, we propose the multiscale data-driven stochastic method. We construct a data-driven stochastic basis and solve the coupled deterministic PDEs to obtain the solutions. For the tougher problems, we propose the multiscale multilevel Monte Carlo method. We apply the multilevel scheme to the effective equations and assemble the stiffness matrices efficiently on each coarse mesh grid. In both methods, the $\KL$ expansion plays an important role in extracting the main parts of some stochastic quantities.

For both the deterministic and stochastic PDEs, numerical results are presented to demonstrate the accuracy and robustness of the methods. We also show the computational time cost reduction in the numerical examples.

Computational modeling and psychophysics in low- and mid-level vision

This thesis addresses a series of topics related to the question of how people find the foreground objects from complex scenes. With both computer vision modeling, as well as psychophysical analyses, we explore the computational principles for low- and mid-level vision.

We first explore the computational methods of generating saliency maps from images and image sequences. We propose an extremely fast algorithm called Image Signature that detects the locations in the image that attract human eye gazes. With a series of experimental validations based on human behavioral data collected from various psychophysical experiments, we conclude that the Image Signature and its spatial-temporal extension, the Phase Discrepancy, are among the most accurate algorithms for saliency detection under various conditions.

In the second part, we bridge the gap between fixation prediction and salient object segmentation with two efforts. First, we propose a new dataset that contains both fixation and object segmentation information. By simultaneously presenting the two types of human data in the same dataset, we are able to analyze their intrinsic connection, as well as understanding the drawbacks of today’s “standard” but inappropriately labeled salient object segmentation dataset. Second, we also propose an algorithm of salient object segmentation. Based on our novel discoveries on the connections of fixation data and salient object segmentation data, our model significantly outperforms all existing models on all 3 datasets with large margins.

In the third part of the thesis, we discuss topics around the human factors of boundary analysis. Closely related to salient object segmentation, boundary analysis focuses on delimiting the local contours of an object. We identify the potential pitfalls of algorithm evaluation for the problem of boundary detection. Our analysis indicates that today’s popular boundary detection datasets contain significant level of noise, which may severely influence the benchmarking results. To give further insights on the labeling process, we propose a model to characterize the principles of the human factors during the labeling process.

The analyses reported in this thesis offer new perspectives to a series of interrelating issues in low- and mid-level vision. It gives warning signs to some of today’s “standard” procedures, while proposing new directions to encourage future research.

Wave interactions in periodic structures and periodic dielectric waveguides

This work is concerned with a general analysis of wave interactions in periodic structures and particularly periodic thin film dielectric waveguides.

The electromagnetic wave propagation in an asymmetric dielectric waveguide with a periodically perturbed surface is analyzed in terms of a Floquet mode solution. First order approximate analytical expressions for the space harmonics are obtained. The solution is used to analyze various applications: (1) phase matched second harmonic generation in periodically perturbed optical waveguides; (2) grating couplers and thin film filters; (3) Bragg reflection devices; (4) the calculation of the traveling wave interaction impedance for solid state and vacuum tube optical traveling wave amplifiers which utilize periodic dielectric waveguides. Some of these applications are of interest in the field of integrated optics.

A special emphasis is put on the analysis of traveling wave interaction between electrons and electromagnetic waves in various operation regimes. Interactions with a finite temperature electron beam at the collision-dominated, collisionless, and quantum regimes are analyzed in detail assuming a one-dimensional model and longitudinal coupling.

The analysis is used to examine the possibility of solid state traveling wave devices (amplifiers, modulators), and some monolithic structures of these devices are suggested, designed to operate at the submillimeter-far infrared frequency regime. The estimates of attainable traveling wave interaction gain are quite low (on the order of a few inverse centimeters). However, the possibility of attaining net gain with different materials, structures and operation condition is not ruled out.

The developed model is used to discuss the possibility and the theoretical limitations of high frequency (optical) operation of vacuum electron beam tube; and the relation to other electron-electromagnetic wave interaction effects (Smith-Purcell and Cerenkov radiation and the free electron laser) are pointed out. Finally, the case where the periodic structure is the natural crystal lattice is briefly discussed. The longitudinal component of optical space harmonics in the crystal is calculated and found to be of the order of magnitude of the macroscopic wave, and some comments are made on the possibility of coherent bremsstrahlung and distributed feedback lasers in single crystals.

Neural and computational representations of decision variables

These studies explore how, where, and when representations of variables critical to decision-making are represented in the brain. In order to produce a decision, humans must first determine the relevant stimuli, actions, and possible outcomes before applying an algorithm that will select an action from those available. When choosing amongst alternative stimuli, the framework of value-based decision-making proposes that values are assigned to the stimuli and that these values are then compared in an abstract “value space” in order to produce a decision. Despite much progress, in particular regarding the pinpointing of ventromedial prefrontal cortex (vmPFC) as a region that encodes the value, many basic questions remain. In Chapter 2, I show that distributed BOLD signaling in vmPFC represents the value of stimuli under consideration in a manner that is independent of the type of stimulus it is. Thus the open question of whether value is represented in abstraction, a key tenet of value-based decision-making, is confirmed. However, I also show that stimulus-dependent value representations are also present in the brain during decision-making and suggest a potential neural pathway for stimulus-to-value transformations that integrates these two results.

More broadly speaking, there is both neural and behavioral evidence that two distinct control systems are at work during action selection. These two systems compose the “goal-directed system”, which selects actions based on an internal model of the environment, and the “habitual” system, which generates responses based on antecedent stimuli only. Computational characterizations of these two systems imply that they have different informational requirements in terms of input stimuli, actions, and possible outcomes. Associative learning theory predicts that the habitual system should utilize stimulus and action information only, while goal-directed behavior requires that outcomes as well as stimuli and actions be processed. In Chapter 3, I test whether areas of the brain hypothesized to be involved in habitual versus goal-directed control represent the corresponding theorized variables.

The question of whether one or both of these neural systems drives Pavlovian conditioning is less well-studied. Chapter 4 describes an experiment in which subjects were scanned while engaged in a Pavlovian task with a simple non-trivial structure. After comparing a variety of model-based and model-free learning algorithms (thought to underpin goal-directed and habitual decision-making, respectively), it was found that subjects’ reaction times were better explained by a model-based system. In addition, neural signaling of precision, a variable based on a representation of a world model, was found in the amygdala. These data indicate that the influence of model-based representations of the environment can extend even to the most basic learning processes.

Knowledge of the state of hidden variables in an environment is required for optimal inference regarding the abstract decision structure of a given environment and therefore can be crucial to decision-making in a wide range of situations. Inferring the state of an abstract variable requires the generation and manipulation of an internal representation of beliefs over the values of the hidden variable. In Chapter 5, I describe behavioral and neural results regarding the learning strategies employed by human subjects in a hierarchical state-estimation task. In particular, a comprehensive model fit and comparison process pointed to the use of "belief thresholding". This implies that subjects tended to eliminate low-probability hypotheses regarding the state of the environment from their internal model and ceased to update the corresponding variables. Thus, in concert with incremental Bayesian learning, humans explicitly manipulate their internal model of the generative process during hierarchical inference consistent with a serial hypothesis testing strategy.

Collective behavior of asperities as a model for friction and adhesion

Understanding friction and adhesion in static and sliding contact of surfaces is important in numerous physical phenomena and technological applications. Most surfaces are rough at the microscale, and thus the real area of contact is only a fraction of the nominal area. The macroscopic frictional and adhesive response is determined by the collective behavior of the population of evolving and interacting microscopic contacts. This collective behavior can be very different from the behavior of individual contacts. It is thus important to understand how the macroscopic response emerges from the microscopic one. In this thesis, we develop a theoretical and computational framework to study the collective behavior. Our philosophy is to assume a simple behavior of a single asperity and study the collective response of an ensemble. Our work bridges the existing well-developed studies of single asperities with phenomenological laws that describe macroscopic rate-and-state behavior of frictional interfaces. We find that many aspects of the macroscopic behavior are robust with respect to the microscopic response. This explains why qualitatively similar frictional features are seen for a diverse range of materials. We first show that the collective response of an ensemble of one-dimensional independent viscoelastic elements interacting through a mean field reproduces many qualitative features of static and sliding friction evolution. The resulting macroscopic behavior is different from the microscopic one: for example, even if each contact is velocity-strengthening, the macroscopic behavior can be velocity-weakening. The framework is then extended to incorporate three-dimensional rough surfaces, long- range elastic interactions between contacts, and time-dependent material behaviors such as viscoelasticity and viscoplasticity. Interestingly, the mean field behavior dominates and the elastic interactions, though important from a quantitative perspective, do not change the qualitative macroscopic response. Finally, we examine the effect of adhesion on the frictional response as well as develop a force threshold model for adhesion and mode I interfacial cracks.