Convex Relaxation for Low-Dimensional Representation: Phase Transitions and Limitations

There is a growing interest in taking advantage of possible patterns and structures in data so as to extract the desired information and overcome the curse of dimensionality. In a wide range of applications, including computer vision, machine learning, medical imaging, and social networks, the signal that gives rise to the observations can be modeled to be approximately sparse and exploiting this fact can be very beneficial. This has led to an immense interest in the problem of efficiently reconstructing a sparse signal from limited linear observations. More recently, low-rank approximation techniques have become prominent tools to approach problems arising in machine learning, system identification and quantum tomography.

In sparse and low-rank estimation problems, the challenge is the inherent intractability of the objective function, and one needs efficient methods to capture the low-dimensionality of these models. Convex optimization is often a promising tool to attack such problems. An intractable problem with a combinatorial objective can often be "relaxed" to obtain a tractable but almost as powerful convex optimization problem. This dissertation studies convex optimization techniques that can take advantage of low-dimensional representations of the underlying high-dimensional data. We provide provable guarantees that ensure that the proposed algorithms will succeed under reasonable conditions, and answer questions of the following flavor:

More specifically, i) Focusing on linear inverse problems, we generalize the classical error bounds known for the least-squares technique to the lasso formulation, which incorporates the signal model. ii) We show that intuitive convex approaches do not perform as well as expected when it comes to signals that have multiple low-dimensional structures simultaneously. iii) Finally, we propose convex relaxations for the graph clustering problem and give sharp performance guarantees for a family of graphs arising from the so-called stochastic block model. We pay particular attention to the following aspects. For i) and ii), we aim to provide a general geometric framework, in which the results on sparse and low-rank estimation can be obtained as special cases. For i) and iii), we investigate the precise performance characterization, which yields the right constants in our bounds and the true dependence between the problem parameters.

Single cell proteomics microchip to profile immune function, with applications in stem cell biology, translational disease mechanism study and clinical therapeutics monitoring

In response to infection or tissue dysfunction, immune cells develop into highly heterogeneous repertoires with diverse functions. Capturing the full spectrum of these functions requires analysis of large numbers of effector molecules from single cells. However, currently only 3-5 functional proteins can be measured from single cells. We developed a single cell functional proteomics approach that integrates a microchip platform with multiplex cell purification. This approach can quantitate 20 proteins from >5,000 phenotypically pure single cells simultaneously. With a 1-million fold miniaturization, the system can detect down to ~100 molecules and requires only ~104 cells. Single cell functional proteomic analysis finds broad applications in basic, translational and clinical studies. In the three studies conducted, it yielded critical insights for understanding clinical cancer immunotherapy, inflammatory bowel disease (IBD) mechanism and hematopoietic stem cell (HSC) biology.

To study phenotypically defined cell populations, single cell barcode microchips were coupled with upstream multiplex cell purification based on up to 11 parameters. Statistical algorithms were developed to process and model the high dimensional readouts. This analysis evaluates rare cells and is versatile for various cells and proteins. (1) We conducted an immune monitoring study of a phase 2 cancer cellular immunotherapy clinical trial that used T-cell receptor (TCR) transgenic T cells as major therapeutics to treat metastatic melanoma. We evaluated the functional proteome of 4 antigen-specific, phenotypically defined T cell populations from peripheral blood of 3 patients across 8 time points. (2) Natural killer (NK) cells can play a protective role in chronic inflammation and their surface receptor – killer immunoglobulin-like receptor (KIR) – has been identified as a risk factor of IBD. We compared the functional behavior of NK cells that had differential KIR expressions. These NK cells were retrieved from the blood of 12 patients with different genetic backgrounds. (3) HSCs are the progenitors of immune cells and are thought to have no immediate functional capacity against pathogen. However, recent studies identified expression of Toll-like receptors (TLRs) on HSCs. We studied the functional capacity of HSCs upon TLR activation. The comparison of HSCs from wild-type mice against those from genetics knock-out mouse models elucidates the responding signaling pathway.

In all three cases, we observed profound functional heterogeneity within phenotypically defined cells. Polyfunctional cells that conduct multiple functions also produce those proteins in large amounts. They dominate the immune response. In the cancer immunotherapy, the strong cytotoxic and antitumor functions from transgenic TCR T cells contributed to a ~30% tumor reduction immediately after the therapy. However, this infused immune response disappeared within 2-3 weeks. Later on, some patients gained a second antitumor response, consisted of the emergence of endogenous antitumor cytotoxic T cells and their production of multiple antitumor functions. These patients showed more effective long-term tumor control. In the IBD mechanism study, we noticed that, compared with others, NK cells expressing KIR2DL3 receptor secreted a large array of effector proteins, such as TNF-α, CCLs and CXCLs. The functions from these cells regulated disease-contributing cells and protected host tissues. Their existence correlated with IBD disease susceptibility. In the HSC study, the HSCs exhibited functional capacity by producing TNF-α, IL-6 and GM-CSF. TLR stimulation activated the NF-κB signaling in HSCs. Single cell functional proteome contains rich information that is independent from the genome and transcriptome. In all three cases, functional proteomic evaluation uncovered critical biological insights that would not be resolved otherwise. The integrated single cell functional proteomic analysis constructed a detail kinetic picture of the immune response that took place during the clinical cancer immunotherapy. It revealed concrete functional evidence that connected genetics to IBD disease susceptibility. Further, it provided predictors that correlated with clinical responses and pathogenic outcomes.

Control of dynamical systems with temporal logic specifications

This thesis is motivated by safety-critical applications involving autonomous air, ground, and space vehicles carrying out complex tasks in uncertain and adversarial environments. We use temporal logic as a language to formally specify complex tasks and system properties. Temporal logic specifications generalize the classical notions of stability and reachability that are studied in the control and hybrid systems communities. Given a system model and a formal task specification, the goal is to automatically synthesize a control policy for the system that ensures that the system satisfies the specification. This thesis presents novel control policy synthesis algorithms for optimal and robust control of dynamical systems with temporal logic specifications. Furthermore, it introduces algorithms that are efficient and extend to high-dimensional dynamical systems.

The first contribution of this thesis is the generalization of a classical linear temporal logic (LTL) control synthesis approach to optimal and robust control. We show how we can extend automata-based synthesis techniques for discrete abstractions of dynamical systems to create optimal and robust controllers that are guaranteed to satisfy an LTL specification. Such optimal and robust controllers can be computed at little extra computational cost compared to computing a feasible controller.

The second contribution of this thesis addresses the scalability of control synthesis with LTL specifications. A major limitation of the standard automaton-based approach for control with LTL specifications is that the automaton might be doubly-exponential in the size of the LTL specification. We introduce a fragment of LTL for which one can compute feasible control policies in time polynomial in the size of the system and specification. Additionally, we show how to compute optimal control policies for a variety of cost functions, and identify interesting cases when this can be done in polynomial time. These techniques are particularly relevant for online control, as one can guarantee that a feasible solution can be found quickly, and then iteratively improve on the quality as time permits.

The final contribution of this thesis is a set of algorithms for computing feasible trajectories for high-dimensional, nonlinear systems with LTL specifications. These algorithms avoid a potentially computationally-expensive process of computing a discrete abstraction, and instead compute directly on the system's continuous state space. The first method uses an automaton representing the specification to directly encode a series of constrained-reachability subproblems, which can be solved in a modular fashion by using standard techniques. The second method encodes an LTL formula as mixed-integer linear programming constraints on the dynamical system. We demonstrate these approaches with numerical experiments on temporal logic motion planning problems with high-dimensional (10+ states) continuous systems.

Optimal design of building structures using genetic algorithms

A general framework for multi-criteria optimal design is presented which is well-suited for automated design of structural systems. A systematic computer-aided optimal design decision process is developed which allows the designer to rapidly evaluate and improve a proposed design by taking into account the major factors of interest related to different aspects such as design, construction, and operation.

The proposed optimal design process requires the selection of the most promising choice of design parameters taken from a large design space, based on an evaluation using specified criteria. The design parameters specify a particular design, and so they relate to member sizes, structural configuration, etc. The evaluation of the design uses performance parameters which may include structural response parameters, risks due to uncertain loads and modeling errors, construction and operating costs, etc. Preference functions are used to implement the design criteria in a "soft" form. These preference functions give a measure of the degree of satisfaction of each design criterion. The overall evaluation measure for a design is built up from the individual measures for each criterion through a preference combination rule. The goal of the optimal design process is to obtain a design that has the highest overall evaluation measure - an optimization problem.

Genetic algorithms are stochastic optimization methods that are based on evolutionary theory. They provide the exploration power necessary to explore high-dimensional search spaces to seek these optimal solutions. Two special genetic algorithms, hGA and vGA, are presented here for continuous and discrete optimization problems, respectively.

The methodology is demonstrated with several examples involving the design of truss and frame systems. These examples are solved by using the proposed hGA and vGA.

Macroscopically Dissipative Systems with Underlying Microscopic Dynamics : Properties and Limits of Measurement

While some of the deepest results in nature are those that give explicit bounds between important physical quantities, some of the most intriguing and celebrated of such bounds come from fields where there is still a great deal of disagreement and confusion regarding even the most fundamental aspects of the theories. For example, in quantum mechanics, there is still no complete consensus as to whether the limitations associated with Heisenberg's Uncertainty Principle derive from an inherent randomness in physics, or rather from limitations in the measurement process itself, resulting from phenomena like back action. Likewise, the second law of thermodynamics makes a statement regarding the increase in entropy of closed systems, yet the theory itself has neither a universally-accepted definition of equilibrium, nor an adequate explanation of how a system with underlying microscopically Hamiltonian dynamics (reversible) settles into a fixed distribution.

Motivated by these physical theories, and perhaps their inconsistencies, in this thesis we use dynamical systems theory to investigate how the very simplest of systems, even with no physical constraints, are characterized by bounds that give limits to the ability to make measurements on them. Using an existing interpretation, we start by examining how dissipative systems can be viewed as high-dimensional lossless systems, and how taking this view necessarily implies the existence of a noise process that results from the uncertainty in the initial system state. This fluctuation-dissipation result plays a central role in a measurement model that we examine, in particular describing how noise is inevitably injected into a system during a measurement, noise that can be viewed as originating either from the randomness of the many degrees of freedom of the measurement device, or of the environment. This noise constitutes one component of measurement back action, and ultimately imposes limits on measurement uncertainty. Depending on the assumptions we make about active devices, and their limitations, this back action can be offset to varying degrees via control. It turns out that using active devices to reduce measurement back action leads to estimation problems that have non-zero uncertainty lower bounds, the most interesting of which arise when the observed system is lossless. One such lower bound, a main contribution of this work, can be viewed as a classical version of a Heisenberg uncertainty relation between the system's position and momentum. We finally also revisit the murky question of how macroscopic dissipation appears from lossless dynamics, and propose alternative approaches for framing the question using existing systematic methods of model reduction.

Advanced Monte Carlo Simulation and Machine Learning for Frequency Domain Optical Coherence Tomography

Optical Coherence Tomography(OCT) is a popular, rapidly growing imaging technique with an increasing number of bio-medical applications due to its noninvasive nature. However, there are three major challenges in understanding and improving an OCT system: (1) Obtaining an OCT image is not easy. It either takes a real medical experiment or requires days of computer simulation. Without much data, it is difficult to study the physical processes underlying OCT imaging of different objects simply because there aren't many imaged objects. (2) Interpretation of an OCT image is also hard. This challenge is more profound than it appears. For instance, it would require a trained expert to tell from an OCT image of human skin whether there is a lesion or not. This is expensive in its own right, but even the expert cannot be sure about the exact size of the lesion or the width of the various skin layers. The take-away message is that analyzing an OCT image even from a high level would usually require a trained expert, and pixel-level interpretation is simply unrealistic. The reason is simple: we have OCT images but not their underlying ground-truth structure, so there is nothing to learn from. (3) The imaging depth of OCT is very limited (millimeter or sub-millimeter on human tissues). While OCT utilizes infrared light for illumination to stay noninvasive, the downside of this is that photons at such long wavelengths can only penetrate a limited depth into the tissue before getting back-scattered. To image a particular region of a tissue, photons first need to reach that region. As a result, OCT signals from deeper regions of the tissue are both weak (since few photons reached there) and distorted (due to multiple scatterings of the contributing photons). This fact alone makes OCT images very hard to interpret.

This thesis addresses the above challenges by successfully developing an advanced Monte Carlo simulation platform which is 10000 times faster than the state-of-the-art simulator in the literature, bringing down the simulation time from 360 hours to a single minute. This powerful simulation tool not only enables us to efficiently generate as many OCT images of objects with arbitrary structure and shape as we want on a common desktop computer, but it also provides us the underlying ground-truth of the simulated images at the same time because we dictate them at the beginning of the simulation. This is one of the key contributions of this thesis. What allows us to build such a powerful simulation tool includes a thorough understanding of the signal formation process, clever implementation of the importance sampling/photon splitting procedure, efficient use of a voxel-based mesh system in determining photon-mesh interception, and a parallel computation of different A-scans that consist a full OCT image, among other programming and mathematical tricks, which will be explained in detail later in the thesis.

Next we aim at the inverse problem: given an OCT image, predict/reconstruct its ground-truth structure on a pixel level. By solving this problem we would be able to interpret an OCT image completely and precisely without the help from a trained expert. It turns out that we can do much better. For simple structures we are able to reconstruct the ground-truth of an OCT image more than 98% correctly, and for more complicated structures (e.g., a multi-layered brain structure) we are looking at 93%. We achieved this through extensive uses of Machine Learning. The success of the Monte Carlo simulation already puts us in a great position by providing us with a great deal of data (effectively unlimited), in the form of (image, truth) pairs. Through a transformation of the high-dimensional response variable, we convert the learning task into a multi-output multi-class classification problem and a multi-output regression problem. We then build a hierarchy architecture of machine learning models (committee of experts) and train different parts of the architecture with specifically designed data sets. In prediction, an unseen OCT image first goes through a classification model to determine its structure (e.g., the number and the types of layers present in the image); then the image is handed to a regression model that is trained specifically for that particular structure to predict the length of the different layers and by doing so reconstruct the ground-truth of the image. We also demonstrate that ideas from Deep Learning can be useful to further improve the performance.

It is worth pointing out that solving the inverse problem automatically improves the imaging depth, since previously the lower half of an OCT image (i.e., greater depth) can be hardly seen but now becomes fully resolved. Interestingly, although OCT signals consisting the lower half of the image are weak, messy, and uninterpretable to human eyes, they still carry enough information which when fed into a well-trained machine learning model spits out precisely the true structure of the object being imaged. This is just another case where Artificial Intelligence (AI) outperforms human. To the best knowledge of the author, this thesis is not only a success but also the first attempt to reconstruct an OCT image at a pixel level. To even give a try on this kind of task, it would require fully annotated OCT images and a lot of them (hundreds or even thousands). This is clearly impossible without a powerful simulation tool like the one developed in this thesis.

A new high-order Fourier continuation-based elasticity solver for complex three-dimensional geometries

This thesis presents a new approach for the numerical solution of three-dimensional problems in elastodynamics. The new methodology, which is based on a recently introduced Fourier continuation (FC) algorithm for the solution of Partial Differential Equations on the basis of accurate Fourier expansions of possibly non-periodic functions, enables fast, high-order solutions of the time-dependent elastic wave equation in a nearly dispersionless manner, and it requires use of CFL constraints that scale only linearly with spatial discretizations. A new FC operator is introduced to treat Neumann and traction boundary conditions, and a block-decomposed (sub-patch) overset strategy is presented for implementation of general, complex geometries in distributed-memory parallel computing environments. Our treatment of the elastic wave equation, which is formulated as a complex system of variable-coefficient PDEs that includes possibly heterogeneous and spatially varying material constants, represents the first fully-realized three-dimensional extension of FC-based solvers to date. Challenges for three-dimensional elastodynamics simulations such as treatment of corners and edges in three-dimensional geometries, the existence of variable coefficients arising from physical configurations and/or use of curvilinear coordinate systems and treatment of boundary conditions, are all addressed. The broad applicability of our new FC elasticity solver is demonstrated through application to realistic problems concerning seismic wave motion on three-dimensional topographies as well as applications to non-destructive evaluation where, for the first time, we present three-dimensional simulations for comparison to experimental studies of guided-wave scattering by through-thickness holes in thin plates.

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.

Scanning tunneling spectroscopic studies on high-temperature superconductors and Dirac materials

This thesis details the investigations of the unconventional low-energy quasiparticle excitations in electron-type cuprate superconductors and electron-type ferrous superconductors as well as the electronic properties of Dirac fermions in graphene and three-dimensional strong topological insulators through experimental studies using spatially resolved scanning tunneling spectroscopy (STS) experiments.

Magnetic-field- and temperature-dependent evolution of the spatially resolved quasiparticle spectra in the electron-type cuprate La_0.1Sr_0.9CuO₂ (La-112) T_C = 43 K, are investigated experimentally. For temperature (T) less than the superconducting transition temperature (TC), and in zero field, the quasiparticle spectra of La-112 exhibits gapped behavior with two coherence peaks and no satellite features. For magnetic field measurements at T < TC, first ever observation of vortices in La-112 are reported. Moreover, pseudogap-like spectra are revealed inside the core of vortices, where superconductivity is suppressed. The intra-vortex pseudogap-like spectra are characterized by an energy gap of V_PG = 8.5 ± 0.6 meV, while the inter-vortex quasiparticle spectra shows larger peak-to-peak gap values characterized by Δ_pk-pk(H) >V_PG, and Δ_pk-pk (0)=12.2 ± 0.8 meV > Δ_pk-pk (H > 0). The quasiparticle spectra are found to be gapped at all locations up to the highest magnetic field examined (H = 6T) and reveal an apparent low-energy cutoff at the V_PG energy scale.

Magnetic-field- and temperature-dependent evolution of the spatially resolved quasiparticle spectra in the electron-type "122" iron-based Ba(Fe_1-xCo_x)₂A_s2 are investigated for multiple doping levels (x = 0.06, 0.08, 0.12 with T_C= 14 K, 24 K, and 20 K). For all doping levels and the T < T_C, two-gap superconductivity is observed. Both superconducting gaps decrease monotonically in size with increasing temperature and disappear for temperatures above the superconducting transition temperature, T_C. Magnetic resonant modes that follow the temperature dependence of the superconducting gaps have been identified in the tunneling quasiparticle spectra. Together with quasiparticle interference (QPI) analysis and magnetic field studies, this provides strong evidence for two-gap sign-changing s-wave superconductivity.

Additionally spatial scanning tunneling spectroscopic studies are performed on mechanically exfoliated graphene and chemical vapor deposition grown graphene. In all cases lattice strain exerts a strong influence on the electronic properties of the sample. In particular topological defects give rise to pseudomagnetic fields (B ~ 50 Tesla) and charging effects resulting in quantized conductance peaks associated with the integer and fractional Quantum Hall States.

Finally, spectroscopic studies on the 3D-STI, Bi₂Se₃ found evidence of impurity resonance in the surface state. The impurities are in the unitary limit and the spectral resonances are localized spatially to within ~ 0.2 nm of the impurity. The spectral weight of the impurity resonance diverges as the Fermi energy approaches the Dirac point and the rapid recovery of the surface state suggests robust topological protection against perturbations that preserve time reversal symmetry.

The optoelectronic swept-frequency laser and its applications in ranging, three-dimensional imaging, and coherent beam combining of chirped-seed amplifiers

This thesis explores the design, construction, and applications of the optoelectronic swept-frequency laser (SFL). The optoelectronic SFL is a feedback loop designed around a swept-frequency (chirped) semiconductor laser (SCL) to control its instantaneous optical frequency, such that the chirp characteristics are determined solely by a reference electronic oscillator. The resultant system generates precisely controlled optical frequency sweeps. In particular, we focus on linear chirps because of their numerous applications. We demonstrate optoelectronic SFLs based on vertical-cavity surface-emitting lasers (VCSELs) and distributed-feedback lasers (DFBs) at wavelengths of 1550 nm and 1060 nm. We develop an iterative bias current predistortion procedure that enables SFL operation at very high chirp rates, up to 10^16 Hz/sec. We describe commercialization efforts and implementation of the predistortion algorithm in a stand-alone embedded environment, undertaken as part of our collaboration with Telaris, Inc. We demonstrate frequency-modulated continuous-wave (FMCW) ranging and three-dimensional (3-D) imaging using a 1550 nm optoelectronic SFL.

We develop the technique of multiple source FMCW (MS-FMCW) reflectometry, in which the frequency sweeps of multiple SFLs are "stitched" together in order to increase the optical bandwidth, and hence improve the axial resolution, of an FMCW ranging measurement. We demonstrate computer-aided stitching of DFB and VCSEL sweeps at 1550 nm. We also develop and demonstrate hardware stitching, which enables MS-FMCW ranging without additional signal processing. The culmination of this work is the hardware stitching of four VCSELs at 1550 nm for a total optical bandwidth of 2 THz, and a free-space axial resolution of 75 microns.

We describe our work on the tomographic imaging camera (TomICam), a 3-D imaging system based on FMCW ranging that features non-mechanical acquisition of transverse pixels. Our approach uses a combination of electronically tuned optical sources and low-cost full-field detector arrays, completely eliminating the need for moving parts traditionally employed in 3-D imaging. We describe the basic TomICam principle, and demonstrate single-pixel TomICam ranging in a proof-of-concept experiment. We also discuss the application of compressive sensing (CS) to the TomICam platform, and perform a series of numerical simulations. These simulations show that tenfold compression is feasible in CS TomICam, which effectively improves the volume acquisition speed by a factor ten.

We develop chirped-wave phase-locking techniques, and apply them to coherent beam combining (CBC) of chirped-seed amplifiers (CSAs) in a master oscillator power amplifier configuration. The precise chirp linearity of the optoelectronic SFL enables non-mechanical compensation of optical delays using acousto-optic frequency shifters, and its high chirp rate simultaneously increases the stimulated Brillouin scattering (SBS) threshold of the active fiber. We characterize a 1550 nm chirped-seed amplifier coherent-combining system. We use a chirp rate of 5*10^14 Hz/sec to increase the amplifier SBS threshold threefold, when compared to a single-frequency seed. We demonstrate efficient phase-locking and electronic beam steering of two 3 W erbium-doped fiber amplifier channels, achieving temporal phase noise levels corresponding to interferometric fringe visibilities exceeding 98%.

Innovations of wide-field optical-sectioning fluorescence microscopy : toward high-speed volumetric bio-imaging with simplicity

Optical microscopy has become an indispensable tool for biological researches since its invention, mostly owing to its sub-cellular spatial resolutions, non-invasiveness, instrumental simplicity, and the intuitive observations it provides. Nonetheless, obtaining reliable, quantitative spatial information from conventional wide-field optical microscopy is not always intuitive as it appears to be. This is because in the acquired images of optical microscopy the information about out-of-focus regions is spatially blurred and mixed with in-focus information. In other words, conventional wide-field optical microscopy transforms the three-dimensional spatial information, or volumetric information about the objects into a two-dimensional form in each acquired image, and therefore distorts the spatial information about the object. Several fluorescence holography-based methods have demonstrated the ability to obtain three-dimensional information about the objects, but these methods generally rely on decomposing stereoscopic visualizations to extract volumetric information and are unable to resolve complex 3-dimensional structures such as a multi-layer sphere.

The concept of optical-sectioning techniques, on the other hand, is to detect only two-dimensional information about an object at each acquisition. Specifically, each image obtained by optical-sectioning techniques contains mainly the information about an optically thin layer inside the object, as if only a thin histological section is being observed at a time. Using such a methodology, obtaining undistorted volumetric information about the object simply requires taking images of the object at sequential depths.

Among existing methods of obtaining volumetric information, the practicability of optical sectioning has made it the most commonly used and most powerful one in biological science. However, when applied to imaging living biological systems, conventional single-point-scanning optical-sectioning techniques often result in certain degrees of photo-damages because of the high focal intensity at the scanning point. In order to overcome such an issue, several wide-field optical-sectioning techniques have been proposed and demonstrated, although not without introducing new limitations and compromises such as low signal-to-background ratios and reduced axial resolutions. As a result, single-point-scanning optical-sectioning techniques remain the most widely used instrumentations for volumetric imaging of living biological systems to date.

In order to develop wide-field optical-sectioning techniques that has equivalent optical performance as single-point-scanning ones, this thesis first introduces the mechanisms and limitations of existing wide-field optical-sectioning techniques, and then brings in our innovations that aim to overcome these limitations. We demonstrate, theoretically and experimentally, that our proposed wide-field optical-sectioning techniques can achieve diffraction-limited optical sectioning, low out-of-focus excitation and high-frame-rate imaging in living biological systems. In addition to such imaging capabilities, our proposed techniques can be instrumentally simple and economic, and are straightforward for implementation on conventional wide-field microscopes. These advantages together show the potential of our innovations to be widely used for high-speed, volumetric fluorescence imaging of living biological systems.

General-domain compressible Navier-Stokes solvers exhibiting quasi-unconditional stability and high-order accuracy in space and time

This thesis presents a new class of solvers for the subsonic compressible Navier-Stokes equations in general two- and three-dimensional spatial domains. The proposed methodology incorporates: 1) A novel linear-cost implicit solver based on use of higher-order backward differentiation formulae (BDF) and the alternating direction implicit approach (ADI); 2) A fast explicit solver; 3) Dispersionless spectral spatial discretizations; and 4) A domain decomposition strategy that negotiates the interactions between the implicit and explicit domains. In particular, the implicit methodology is quasi-unconditionally stable (it does not suffer from CFL constraints for adequately resolved flows), and it can deliver orders of time accuracy between two and six in the presence of general boundary conditions. In fact this thesis presents, for the first time in the literature, high-order time-convergence curves for Navier-Stokes solvers based on the ADI strategy---previous ADI solvers for the Navier-Stokes equations have not demonstrated orders of temporal accuracy higher than one. An extended discussion is presented in this thesis which places on a solid theoretical basis the observed quasi-unconditional stability of the methods of orders two through six. The performance of the proposed solvers is favorable. For example, a two-dimensional rough-surface configuration including boundary layer effects at Reynolds number equal to one million and Mach number 0.85 (with a well-resolved boundary layer, run up to a sufficiently long time that single vortices travel the entire spatial extent of the domain, and with spatial mesh sizes near the wall of the order of one hundred-thousandth the length of the domain) was successfully tackled in a relatively short (approximately thirty-hour) single-core run; for such discretizations an explicit solver would require truly prohibitive computing times. As demonstrated via a variety of numerical experiments in two- and three-dimensions, further, the proposed multi-domain parallel implicit-explicit implementations exhibit high-order convergence in space and time, useful stability properties, limited dispersion, and high parallel efficiency.

An investigation of wind tunnel wall effects at high Mach numbers

This paper presents the results of an investigation of wind tunnel wall interference in a two-dimensional wind tunnel at high Mach numbers. The results are presented in the form of curves of lift coefficient versus the ratio of model chord to tunnel height, as functions of Mach number and angle of attack. The investigation was carried out by the authors at the Guggenheim Aeronautical Laboratory of the California Institute of Technology during the school year 1944-45.

Tests were carried out on the NACA low drag airfoil section 65,1-012 at Mach numbers from .60 to .80, and angles of attack of from 1 to 3 degrees. Models were 1", 2", 4" and 6" chord, giving values of the chord to tunnel height ration of .1 to .6. Schlieren photographs were made of shock waves where they occurred.

Thermopower in Two-Dimensional Electron Systems

The subject of this thesis is the measurement and interpretation of thermopower in high-mobility two-dimensional electron systems (2DESs). These 2DESs are realized within state-of-the-art GaAs/AlGaAs heterostructures that are cooled to temperatures as low as T = 20 mK. Much of this work takes place within strong magnetic fields where the single-particle density of states quantizes into discrete Landau levels (LLs), a regime best known for the quantum Hall effect (QHE). In addition, we review a novel hot-electron technique for measuring thermopower of 2DESs that dramatically reduces the influence of phonon drag.

Early chapters concentrate on experimental materials and methods. A brief overview of GaAs/AlGaAs heterostructures and device fabrication is followed by details of our cryogenic setup. Next, we provide a primer on thermopower that focuses on 2DESs at low temperatures. We then review our experimental devices, temperature calibration methods, as well as measurement circuits and protocols.

Latter chapters focus on the physics and thermopower results in the QHE regime. After reviewing the basic phenomena associated with the QHE, we discuss thermopower in this regime. Emphasis is given to the relationship between diffusion thermopower and entropy. Experimental results demonstrate this relationship persists well into the fractional quantum Hall (FQH) regime.

Several experimental results are reviewed. Unprecedented observations of the diffusion thermopower of a high-mobility 2DES at temperatures as high as T = 2 K are achieved using our hot-electron technique. The composite fermion (CF) effective mass is extracted from measurements of thermopower at LL filling factor ν = 3/2. The thermopower versus magnetic field in the FQH regime is shown to be qualitatively consistent with a simple entropic model of CFs. The thermopower at ν = 5/2 is shown to be quantitatively consistent with the presence of non-Abelian anyons. An abrupt collapse of thermopower is observed at the onset of the reentrant integer quantum Hall effect (RIQHE). And the thermopower at temperatures just above the RIQHE transition suggests the existence of an unconventional conducting phase.

Small scale turbulence in high Karlovitz number premixed flames

The purpose of this thesis is to characterize the behavior of the smallest turbulent scales in high Karlovitz number (Ka) premixed flames. These scales are particularly important in the two-way coupling between turbulence and chemistry and better understanding of these scales will support future modeling efforts using large eddy simulations (LES). The smallest turbulent scales are studied by considering the vorticity vector, ω, and its transport equation.

Due to the complexity of turbulent combustion introduced by the wide range of length and time scales, the two-dimensional vortex-flame interaction is first studied as a simplified test case. Numerical and analytical techniques are used to discern the dominate transport terms and their effects on vorticity based on the initial size and strength of the vortex. This description of the effects of the flame on a vortex provides a foundation for investigating vorticity in turbulent combustion.

Subsequently, enstrophy, ω² = ω • ω, and its transport equation are investigated in premixed turbulent combustion. For this purpose, a series of direct numerical simulations (DNS) of premixed n-heptane/air flames are performed, the conditions of which span a wide range of unburnt Karlovitz numbers and turbulent Reynolds numbers. Theoretical scaling analysis along with the DNS results support that, at high Karlovitz number, enstrophy transport is controlled by the viscous dissipation and vortex stretching/production terms. As a result, vorticity scales throughout the flame with the inverse of the Kolmogorov time scale, τ_η, just as in homogeneous isotropic turbulence. As τ_η is only a function of the viscosity and dissipation rate, this supports the validity of Kolmogorov’s first similarity hypothesis for sufficiently high Ka numbers (Ka ≳ 100). These conclusions are in contrast to low Karlovitz number behavior, where dilatation and baroclinic torque have a significant impact on vorticity within the flame. Results are unaffected by the transport model, chemical model, turbulent Reynolds number, and lastly the physical configuration.

Next, the isotropy of vorticity is assessed. It is found that given a sufficiently large value of the Karlovitz number (Ka ≳ 100) the vorticity is isotropic. At lower Karlovitz numbers, anisotropy develops due to the effects of the flame on the vortex stretching/production term. In this case, the local dynamics of vorticity in the strain-rate tensor, S, eigenframe are altered by the flame. At sufficiently high Karlovitz numbers, the dynamics of vorticity in this eigenframe resemble that of homogeneous isotropic turbulence.

Combined, the results of this thesis support that both the magnitude and orientation of vorticity resemble the behavior of homogeneous isotropic turbulence, given a sufficiently high Karlovitz number (Ka ≳ 100). This supports the validity of Kolmogorov’s first similarity hypothesis and the hypothesis of local isotropy under these condition. However, dramatically different behavior is found at lower Karlovitz numbers. These conclusions provides/suggests directions for modeling high Karlovitz number premixed flames using LES. With more accurate models, the design of aircraft combustors and other combustion based devices may better mitigate the detrimental effects of combustion, from reducing CO₂ and soot production to increasing engine efficiency.