Optimal uncertainty quantification via convex optimization and relaxation

Many engineering applications face the problem of bounding the expected value of a quantity of interest (performance, risk, cost, etc.) that depends on stochastic uncertainties whose probability distribution is not known exactly. Optimal uncertainty quantification (OUQ) is a framework that aims at obtaining the best bound in these situations by explicitly incorporating available information about the distribution. Unfortunately, this often leads to non-convex optimization problems that are numerically expensive to solve.

This thesis emphasizes on efficient numerical algorithms for OUQ problems. It begins by investigating several classes of OUQ problems that can be reformulated as convex optimization problems. Conditions on the objective function and information constraints under which a convex formulation exists are presented. Since the size of the optimization problem can become quite large, solutions for scaling up are also discussed. Finally, the capability of analyzing a practical system through such convex formulations is demonstrated by a numerical example of energy storage placement in power grids.

When an equivalent convex formulation is unavailable, it is possible to find a convex problem that provides a meaningful bound for the original problem, also known as a convex relaxation. As an example, the thesis investigates the setting used in Hoeffding's inequality. The naive formulation requires solving a collection of non-convex polynomial optimization problems whose number grows doubly exponentially. After structures such as symmetry are exploited, it is shown that both the number and the size of the polynomial optimization problems can be reduced significantly. Each polynomial optimization problem is then bounded by its convex relaxation using sums-of-squares. These bounds are found to be tight in all the numerical examples tested in the thesis and are significantly better than Hoeffding's bounds.

I. Spectroscopic evidence for slow vibrational relaxation in excited electronic states of diatomics in rare gas solids. II. Static crystal field effects in the electronic spectra of isotopically mixed benzene crystals

Part I

Studies of vibrational relaxation in excited electronic states of simple diatomic molecules trapped in solid rare-gas matrices at low temperatures are reported. The relaxation is investigated by monitoring the emission intensity from vibrational levels of the excited electronic state to vibrational levels of the ground electronic state. The emission was in all cases excited by bombardment of the doped rare-gas solid with X-rays.

The diatomics studied and the band systems seen are: N₂, Vegard-Kaplan and Second Positive systems; O₂, Herzberg system; OH and OD, A ²Σ⁺ - X²II_i system. The latter has been investigated only in solid Ne, where both emission and absorption spectra were recorded; observed fine structure has been partly interpreted in terms of slightly perturbed rotational motion in the solid. For N₂, OH, and OD emission occurred from v' > 0, establishing a vibrational relaxation time in the excited electronic state of the order, of longer than, the electronic radiative lifetime. The relative emission intensity and decay times for different v' progressions in the Vegard-Kaplan system are found to depend on the rare-gas host and the N₂ concentration, but are independent of temperature in the range 1.7°K to 30°K.

Part II

Static crystal field effects on the absorption, fluorescence, and phosphorescence spectra of isotopically mixed benzene crystals were investigated. Evidence is presented which demonstrate that in the crystal the ground, lowest excited singlet, and lowest triplet states of the guest deviate from hexagonal symmetry. The deviation appears largest in the lowest triplet state and may be due to an intrinsic instability of the ³B_1u state. High resolution absorption and phospho- rescence spectra are reported and analyzed in terms of site-splitting of degenerate vibrations and orientational effects. The guest phosphorescence lifetime for various benzene isotopes in C₆D₆ and sym-C₆H₃D₃ hosts is presented and discussed.

Investigations of charge-carrier dynamics at semiconductor/liquid interfaces

Theoretical and experimental investigations of charge-carrier dynamics at semiconductor/liquid interfaces, specifically with respect to interfacial electron transfer and surface recombination, are presented.

Fermi's golden rule has been used to formulate rate expressions for charge transfer of delocalized carriers in a nondegenerately doped semiconducting electrode to localized, outer-sphere redox acceptors in an electrolyte phase. The treatment allows comparison between charge-transfer kinetic data at metallic, semimetallic, and semiconducting electrodes in terms of parameters such as the electronic coupling to the electrode, the attenuation of coupling with distance into the electrolyte, and the reorganization energy of the charge-transfer event. Within this framework, rate constant values expected at representative semiconducting electrodes have been determined from experimental data for charge transfer at metallic electrodes. The maximum rate constant (i.e., at optimal exoergicity) for outer-sphere processes at semiconducting electrodes is computed to be in the range 10^-17-10^-16 cm⁴ s^-1, which is in excellent agreement with prior theoretical models and experimental results for charge-transfer kinetics at semiconductor/liquid interfaces.

Double-layer corrections have been evaluated for semiconductor electrodes in both depletion and accumulation conditions. In conjuction with the Gouy-Chapman-Stern model, a finite difference approach has been used to calculate potential drops at a representative solid/liquid interface. Under all conditions that were simulated, the correction to the driving force used to evaluate the interfacial rate constant was determined to be less than 2% of the uncorrected interfacial rate constant.

Photoconductivity decay lifetimes have been obtained for Si(111) in contact with solutions of CH₃OH or tetrahydrofuran containing one-electron oxidants. Silicon surfaces in contact with electrolyte solutions having Nernstian redox potentials > 0 V vs. SCE exhibited low effective surface recombination velocities regardless of the different surface chemistries. The formation of an inversion layer, and not a reduced density of electrical trap sites on the surface, is shown to be responsible for the long charge-carrier lifetimes observed for these systems. In addition, a method for preparing an air-stable, low surface recombination velocity Si surface through a two-step, chlorination/alkylation reaction is described.

Part I. Influence of membrane morphology on ion-exchange and charge propagation in composite polyelectrolyte electrode coatings. Part II. Dynamic surface tension measurements of polarization relaxation at mercury pool electrodes by the method of Wilhelmy.

Part I. Novel composite polyelectrolyte materials were developed that exhibit desirable charge propagation and ion-retention properties. The morphology of electrode coatings cast from these materials was shown to be more important for its electrochemical behavior than its chemical composition.

Part II. The Wilhelmy plate technique for measuring dynamic surface tension was extended to electrified liquid-liquid interphases. The dynamical response of the aqueous NaF-mercury electrified interphase was examined by concomitant measurement of surface tension, current, and applied electrostatic potential. Observations of the surface tension response to linear sweep voltammetry and to step function perturbations in the applied electrostatic potential (e.g., chronotensiometry) provided strong evidence that relaxation processes proceed for time-periods that are at least an order of magnitude longer than the time periods necessary to establish diffusion equilibrium. The dynamical response of the surface tension is analyzed within the context of non-equilibrium thermodynamics and a kinetic model that requires three simultaneous first order processes.

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.

Light-matter Interactions in Semiconductors and Metals: From Nitride Optoelectronics to Quantum Plasmonics

This thesis puts forth a theory-directed approach coupled with spectroscopy aimed at the discovery and understanding of light-matter interactions in semiconductors and metals.

The first part of the thesis presents the discovery and development of Zn-IV nitride materials.The commercial prominence in the optoelectronics industry of tunable semiconductor alloy materials based on nitride semiconductor devices, specifically InGaN, motivates the search for earth-abundant alternatives for use in efficient, high-quality optoelectronic devices. II-IV-N2 compounds, which are closely related to the wurtzite-structured III-N semiconductors, have similar electronic and optical properties to InGaN namely direct band gaps, high quantum efficiencies and large optical absorption coefficients. The choice of different group II and group IV elements provides chemical diversity that can be exploited to tune the structural and electronic properties through the series of alloys. The first theoretical and experimental investigation of the ZnSnxGe1−xN2 series as a replacement for III-nitrides is discussed here.

The second half of the thesis shows ab−initio calculations for surface plasmons and plasmonic hot carrier dynamics. Surface plasmons, electromagnetic modes confined to the surface of a conductor-dielectric interface, have sparked renewed interest because of their quantum nature and their broad range of applications. The decay of surface plasmons is usually a detriment in the field of plasmonics, but the possibility to capture the energy normally lost to heat would open new opportunities in photon sensors, energy conversion devices and switching. A theoretical understanding of plasmon-driven hot carrier generation and relaxation dynamics in the ultrafast regime is presented here. Additionally calculations for plasmon-mediated upconversion as well as an energy-dependent transport model for these non-equilibrium carriers are shown.

Finally, this thesis gives an outlook on the potential of non-equilibrium phenomena in metals and semiconductors for future light-based technologies.

I. Electron spin relaxation studies of manganese(II) complexes in acetonitrile. II. Nuclear spin relaxation studies of bromine in tetrabromomanganese(II) in acetonitrile

Magnetic resonance techniques have given us a powerful means for investigating dynamical processes in gases, liquids and solids. Dynamical effects manifest themselves in both resonance line shifts and linewidths, and, accordingly, require detailed analyses to extract desired information. The success of a magnetic resonance experiment depends critically on relaxation mechanisms to maintain thermal equilibrium between spin states. Consequently, there must be an interaction between the excited spin states and their immediate molecular environment which promote changes in spin orientation while excess magnetic energy is coupled into other degrees of freedom by non-radiative processes. This is well known as spin-lattice relaxation. Certain dynamical processes cause fluctuations in the spin state energy levels leading to spin-spin relaxation and, here again, the environment at the molecular level plays a significant role in the magnitude of interaction. Relatively few electron spin relaxation studies of solutions have been conducted and the present work is addressed toward the extension of our knowledge in this area and the retrieval of dynamical information from line shape analyses on a time scale comparable to diffusion controlled phenomena.

Specifically, the electron spin relaxation of three Mn⁺²3d⁵ complexes, Mn(CH₃CN)₆⁺², MnCl₄^-2 in acetonitrile has been studied in considerable detail. The effective spin Hamiltonian constants were carefully evaluated under a wide range of experimental conditions. Resonance widths of these Mn⁺² complexes were studied in the presence of various excess ligand ions and as a function of concentration, viscosity, temperature and frequency (X-band, ~9.5 Ԍ Hz and K-band, ~35 Ԍ Hz).

A number of interesting conclusions were drawn from these studies. For the Et₄NCl-₄^-2 system several relaxation mechanisms leading to resonance broadening were observed. One source appears to arise through spin-orbit interactions caused by modulation of the ligand field resulting from transient distortions of the complex imparted by solvent fluctuations in the immediate surroundings of the paramagnetic ion. An additional spin relaxation was assigned to the formation of ion pairs [Et₄N⁺…MnCl₄^-2] and it was possible to estimate the dissociation constant for this specie in acetonitrile.

The Bu₄NBr-MnBr₄^-2 study was considerably more interesting. As in the former case, solvent fluctuations and ion-pairing of the paramagnetic complex [Bu₄N⁺…MnBr₄^-2] provide significant relaxation for the electronic spin system. Most interesting, without doubt, is the onset of a new relaxation mechanism leading to resonance broadening which is best interpreted as chemical exchange. Thus, assuming that resonance widths were simply governed by electron spin state lifetimes, we were able to extract dynamical information from an interaction in which the initial and final states are the same

MnBr₄^-2 + Br^- = MnBr₄^-2 + Br^-.

The bimolecular rate constants were obtained at six different temperatures and their magnitudes suggested that the exchange is probably diffusion controlled with essentially a zero energy of activation. The most important source of spin relaxation in this system stems directly from dipolar interactions between the manganese 3d⁵ electrons. Moreover, the dipolar broadening is strongly frequency dependent indicating a deviation between the transverse and longitudinal relaxation times. We are led to the conclusion that the 3d⁵ spin states of ion-paired MnBr₄^-2 are significantly correlated so that dynamical processes are also entering the picture. It was possible to estimate the correlation time, ^Td, characterizing this dynamical process.

In Part II we study nuclear magnetic relaxation of bromine ions in the MnBr4-2-Bu4NBr-acetonitrile system. Essentially we monitor the 79Br and 81Br linewidths in response to the [MnBr4-2]/[Br-] ratio with the express purpose of supporting our contention that exchange is occurring between "free" bromine ions in the solvent and bromine in the first coordination sphere of the paramagnetic anion. The complexity of the system elicited a two-part study: (1) the linewidth behavior of Bu4NBr in anhydrous CH3CN in the absence of MnBr4-2 and (2) in the presence of MnBr4-2. It was concluded in study (1) that dynamical association, Bu4NBr k1= Bu4N+ + Br-, was modulating field-gradient interactions at frequencies high enough to provide an estimation of the unimolecular rate constant, k1. A comparison of the two isotopic bromine linewidth-mole fraction results led to the conclusion that quadrupole interactions provided the dominant relaxation mechanism. In study (2) the "residual" bromine linewidths for both 79Br and 81Br are clearly controlled by quadrupole interactions which appear to be modulated by very rapid dynamical processes other than molecular reorientation. We conclude that the "residual" linewidth has its origin in chemical exchange and that bromine nuclei exchange rapidly between a "free" solvated ion and the paramagnetic complex, MnBr4-2.

Mössbauer effect investigations of the hyperfine interactions and relaxation phenomena in salts of thulium

Two new phenomena have been observed in Mössbauer spectra: a temperature-dependent shift of the center of gravity of the spectrum, and an asymmetric broadening of the spectrum peaks. Both phenomena were observed in thulium salts. In the temperature range 1˚K ≤ T ≤ 5˚K the observed shift has an approximate inverse temperature dependence. We explain this on the basis of a Van Vleck type of interaction between the magnetic moment of two nearly degenerate electronic levels and the magnetic moment of the nucleus. From the size of the shift we are able to deduce an “effective magnetic field” H = (6.0 ± 0.1) x 10⁶ Gauss, which is proportional to ‹r^-3›_M‹G|J|E› where ‹r^-3›_M is an effective magnetic radial integral for the 4f electrons and |G› and |E› are the lowest 4f electronic states in Tm Cl₃·6H₂O. From the temperature dependence of the shift we have derived a preliminary value of 1 cm^-1 for the splitting of these two states. The observed asymmetric line broadening is independent of temperature in the range 1˚K ≤ T ≤ 5˚K, but is dependent on the concentration of thulium ions in the crystal. We explain this broadening on the basis of spin-spin interactions between thulium ions. From size and concentration dependence of the broadening we are able to deduce a spin-spin relaxation time for Tm Cl₃·6H₂O of the order of 10^-11 sec.

I. Proton magnetic resonance of polynucleotides and transfer RNA. II. Electron spin relaxation studies of manganese (II) complexes in acetonitrile

Part I. Proton Magnetic Resonance of Polynucleotides and Transfer RNA.

Proton magnetic resonance was used to follow the temperature dependent intramolecular stacking of the bases in the polynucleotides of adenine and cytosine. Analysis of the results on the basis of a two state stacked-unstacked model yielded values of -4.5 kcal/mole and -9.5 kcal/mole for the enthalpies of stacking in polyadenylic and polycytidylic acid, respectively.

The interaction of purine with these molecules was also studied by pmr. Analysis of these results and the comparison of the thermal unstacking of polynucleotides and short chain nucleotides indicates that the bases contained in stacks within the long chain poly nucleotides are, on the average, closer together than the bases contained in stacks in the short chain nucleotides.

Temperature and purine studies were also carried out with an aqueous solution of formylmethionine transfer ribonucleic acid. Comparison of these results with the results of similar experiments with the homopolynucleotides of adenine, cytosine and uracil indicate that the purine is probably intercalating into loop regions of the molecule.

The solvent denaturation of phenylalanine transfer ribonucleic acid was followed by pmr. In a solvent mixture containing 83 volume per cent dimethylsulf oxide and 17 per cent deuterium oxide, the tRNA molecule is rendered quite flexible. It is possible to resolve resonances of protons on the common bases and on certain modified bases.

Part II. Electron Spin Relaxation Studies of Manganese (II) Complexes in Acetonitrile.

The electron paramagnetic resonance spectra of three Mn⁺² complexes, [Mn(CH₃CN)₆]⁺², [MnCl₄]^-2, and [MnBr₄]^-2, in acetonitrile were studied in detail. The objective of this study was to relate changes in the effective spin Hamiltonian parameters and the resonance line widths to the structure of these molecular complexes as well as to dynamical processes in solution.

Of the three systems studied, the results obtained from the [Mn(CH₃CN)₆]⁺² system were the most straight-forward to interpret. Resonance broadening attributable to manganese spin-spin dipolar interactions was observed as the manganese concentration was increased.

In the [MnCl₄]^-2 system, solvent fluctuations and dynamical ion-pairing appear to be significant in determining electron spin relaxation.

In the [MnBr₄]^-2 system, solvent fluctuations, ion-pairing, and Br- ligand exchange provide the principal means of electron spin relaxation. It was also found that the spin relaxation in this system is dependent upon the field strength and is directly related to the manganese concentration. A relaxation theory based on a two state collisional model was developed to account for the observed behavior.

Classical and Quantum Effects in Plasmonic Metals

The field of plasmonics exploits the unique optical properties of metallic nanostructures to concentrate and manipulate light at subwavelength length scales. Metallic nanostructures get their unique properties from their ability to support surface plasmons– coherent wave-like oscillations of the free electrons at the interface between a conductive and dielectric medium. Recent advancements in the ability to fabricate metallic nanostructures with subwavelength length scales have created new possibilities in technology and research in a broad range of applications.

In the first part of this thesis, we present two investigations of the relationship between the charge state and optical state of plasmonic metal nanoparticles. Using experimental bias-dependent extinction measurements, we derive a potential- dependent dielectric function for Au nanoparticles that accounts for changes in the physical properties due to an applied bias that contribute to the optical extinction. We also present theory and experiment for the reverse effect– the manipulation of the carrier density of Au nanoparticles via controlled optical excitation. This plasmoelectric effect takes advantage of the strong resonant properties of plasmonic materials and the relationship between charge state and optical properties to eluci- date a new avenue for conversion of optical power to electrical potential.

The second topic of this thesis is the non-radiative decay of plasmons to a hot-carrier distribution, and the distribution’s subsequent relaxation. We present first-principles calculations that capture all of the significant microscopic mechanisms underlying surface plasmon decay and predict the initial excited carrier distributions so generated. We also preform ab initio calculations of the electron-temperature dependent heat capacities and electron-phonon coupling coefficients of plasmonic metals. We extend these first-principle methods to calculate the electron-temperature dependent dielectric response of hot electrons in plasmonic metals, including direct interband and phonon-assisted intraband transitions. Finally, we combine these first-principles calculations of carrier dynamics and optical response to produce a complete theoretical description of ultrafast pump-probe measurements, free of any fitting parameters that are typical in previous analyses.