ℓ1\ell^1-Analysis Minimization and Generalized (Co-)Sparsity: When Does Recovery Succeed?

This paper investigates the problem of signal estimation from undersampled noisy sub-Gaussian measurements under the assumption of a cosparse model. Based on generalized notions of sparsity, we derive novel recovery guarantees for the $\ell^{1}$-analysis basis pursuit, enabling highly accurate predictions of its sample complexity. The corresponding bounds on the number of required measurements do explicitly depend on the Gram matrix of the analysis operator and therefore particularly account for its mutual coherence structure. Our findings defy conventional wisdom which promotes the sparsity of analysis coefficients as the crucial quantity to study. In fact, this common paradigm breaks down completely in many situations of practical interest, for instance, when applying a redundant (multilevel) frame as analysis prior. By extensive numerical experiments, we demonstrate that, in contrast, our theoretical sampling-rate bounds reliably capture the recovery capability of various examples, such as redundant Haar wavelets systems, total variation, or random frames. The proofs of our main results build upon recent achievements in the convex geometry of data mining problems. More precisely, we establish a sophisticated upper bound on the conic Gaussian mean width that is associated with the underlying $\ell^{1}$-analysis polytope. Due to a novel localization argument, it turns out that the presented framework naturally extends to stable recovery, allowing us to incorporate compressible coefficient sequences as well

Spatial statistics and analysis of earth's ionosphere

Thesis (Ph.D.)--Boston UniversityThe ionosphere, a layer of Earths upper atmosphere characterized by energetic charged particles, serves as a natural plasma laboratory and supplies proxy diagnostics of space weather drivers in the magnetosphere and the solar wind. The ionosphere is a highly dynamic medium, and the spatial structure of observed features (such as auroral light emissions, charge density, temperature, etc.) is rich with information when analyzed in the context of fluid, electromagnetic, and chemical models. Obtaining measurements with higher spatial and temporal resolution is clearly advantageous. For instance, measurements obtained with a new electronically-steerable incoherent scatter radar (ISR) present a unique space-time perspective compared to those of a dish-based ISR. However, there are unique ambiguities for this modality which must be carefully considered. The ISR target is stochastic, and the fidelity of fitted parameters (ionospheric densities and temperatures) requires integrated sampling, creating a tradeoff between measurement uncertainty and spatio-temporal resolution. Spatial statistics formalizes the relationship between spatially dispersed observations and the underlying process(es) they represent. A spatial process is regarded as a random field with its distribution structured (e.g., through a correlation function) such that data, sampled over a spatial domain, support inference or prediction of the process. Quantification of uncertainty, an important component of scientific data analysis, is a core value of spatial statistics. This research applies the formalism of spatial statistics to the analysis of Earth's ionosphere using remote sensing diagnostics. In the first part, we consider the problem of volumetric imaging using phased-array ISR based on optimal spatial prediction ("kriging"). In the second part, we develop a technique for reconstructing two-dimensional ion flow fields from line-of-sight projections using Tikhonov regularization. In the third part, we adapt our spatial statistical approach to global ionospheric imaging using total electron content (TEC) measurements derived from navigation satellite signals

A Spectral Discontinuous Galerkin method for incompressible flow with Applications to turbulence

In this thesis we develop a numerical solution method for the instationary incompressible Navier-Stokes equations. The approach is based on projection methods for discretization in time and a higher order discontinuous Galerkin discretization in space. We propose an upwind scheme for the convective term that chooses the direction of flux across cell interfaces by the mean value of the velocity and has favorable properties in the context of DG. We present new variants of solenoidal projection operators in the Helmholtz decomposition which are indeed discrete projection operators. The discretization is accomplished on quadrilateral or hexahedral meshes where sum-factorization in tensor product finite elements can be exploited. Sum-factorization significantly reduces algorithmic complexity during assembling. In this thesis we thereby build efficient scalable matrix-free solvers and preconditioners to tackle the arising subproblems in the discretization. Conservation properties of the numerical method are demonstrated for both problems with exact solution and turbulent flows. Finally, the presented DG solver enables long time stable direct numerical simulations of the Navier-Stokes equations. As an application we perform computations on a model of the atmospheric boundary layer and demonstrate the existence of surface renewal

Diffeomorphic Transformations for Time Series Analysis: An Efficient Approach to Nonlinear Warping

The proliferation and ubiquity of temporal data across many disciplines has sparked interest for similarity, classification and clustering methods specifically designed to handle time series data. A core issue when dealing with time series is determining their pairwise similarity, i.e., the degree to which a given time series resembles another. Traditional distance measures such as the Euclidean are not well-suited due to the time-dependent nature of the data. Elastic metrics such as dynamic time warping (DTW) offer a promising approach, but are limited by their computational complexity, non-differentiability and sensitivity to noise and outliers. This thesis proposes novel elastic alignment methods that use parametric \& diffeomorphic warping transformations as a means of overcoming the shortcomings of DTW-based metrics. The proposed method is differentiable \& invertible, well-suited for deep learning architectures, robust to noise and outliers, computationally efficient, and is expressive and flexible enough to capture complex patterns. Furthermore, a closed-form solution was developed for the gradient of these diffeomorphic transformations, which allows an efficient search in the parameter space, leading to better solutions at convergence. Leveraging the benefits of these closed-form diffeomorphic transformations, this thesis proposes a suite of advancements that include: (a) an enhanced temporal transformer network for time series alignment and averaging, (b) a deep-learning based time series classification model to simultaneously align and classify signals with high accuracy, (c) an incremental time series clustering algorithm that is warping-invariant, scalable and can operate under limited computational and time resources, and finally, (d) a normalizing flow model that enhances the flexibility of affine transformations in coupling and autoregressive layers.Comment: PhD Thesis, defended at the University of Navarra on July 17, 2023. 277 pages, 8 chapters, 1 appendi

Achievable Rate and Modulation for Bandlimited Channels with Oversampling and 1-Bit Quantization at the Receiver

Sustainably realizing applications of the future with high performance demands requires that energy efficiency becomes a central design criterion for the entire system. For example, the power consumption of the analog-to-digital converter (ADC) can become a major factor when transmitting at large bandwidths and carrier frequencies, e.g., for ultra-short range high data rate communication. The consumed energy per conversion step increases with the sampling rate such that high resolution ADCs become unfeasible in the sub-THz regime at the very high sampling rates required. This makes signaling schemes adapted to 1-bit quantizers a promising alternative. We therefore quantify the performance of bandlimited 1-bit quantized wireless communication channels using techniques like oversampling and faster-than-Nyquist (FTN) signaling to compensate for the loss of achievable rate. As a limiting case, we provide bounds on the mutual information rate of the hard bandlimited 1-bit quantized continuous-time – i.e., infinitely oversampled – additive white Gaussian noise channel in the mid-to-high signal-to-noise ratio (SNR) regime. We derive analytic expressions using runlength encoded input signals. For real signals the maximum value of the lower bound on the spectral efficiency in the high-SNR limit was found to be approximately 1.63 bit/s/Hz. Since in practical scenarios the oversampling ratio remains finite, we derive bounds on the achievable rate of the bandlimited oversampled discrete-time channel. These bounds match the results of the continuous-time channel remarkably well. We observe spectral efficiencies up to 1.53 bit/s/Hz in the high-SNR limit given hard bandlimitation. When excess bandwidth is tolerable, spectral efficiencies above 2 bit/s/Hz per domain are achievable w.r.t. the 95 %-power containment bandwidth. Applying the obtained bounds to a bandlimited oversampled 1-bit quantized multiple-input multiple-output channel, we show the benefits when using appropriate power allocation schemes. As a constant envelope modulation scheme, continuous phase modulation is considered in order to relieve linearity requirements on the power amplifier. Noise-free performance limits are investigated for phase shift keying (PSK) and continuous phase frequency shift keying (CPFSK) using higher-order modulation alphabets and intermediate frequencies. Adapted waveforms are designed that can be described as FTN-CPFSK. With the same spectral efficiency in the high-SNR limit as PSK and CPFSK, these waveforms provide a significantly improved bit error rate (BER) performance. The gain in SNR required for achieving a certain BER can be up to 20 dB.Die nachhaltige Realisierung von zukünftigen Übertragungssystemen mit hohen Leistungsanforderungen erfordert, dass die Energieeffizienz zu einem zentralen Designkriterium für das gesamte System wird. Zum Beispiel kann die Leistungsaufnahme des Analog-Digital-Wandlers (ADC) zu einem wichtigen Faktor bei der Übertragung mit großen Bandbreiten und Trägerfrequenzen werden, z. B. für die Kommunikation mit hohen Datenraten über sehr kurze Entfernungen. Die verbrauchte Energie des ADCs steigt mit der Abtastrate, so dass hochauflösende ADCs im Sub-THz-Bereich bei den erforderlichen sehr hohen Abtastraten schwer einsetzbar sind. Dies macht Signalisierungsschemata, die an 1-Bit-Quantisierer angepasst sind, zu einer vielversprechenden Alternative. Wir quantifizieren daher die Leistungsfähigkeit von bandbegrenzten 1-Bit-quantisierten drahtlosen Kommunikationssystemen, wobei Techniken wie Oversampling und Faster-than-Nyquist (FTN) Signalisierung eingesetzt werden, um den durch Quantisierung verursachten Verlust der erreichbaren Rate auszugleichen. Wir geben Grenzen für die Transinformationsrate des Extremfalls eines strikt bandbegrenzten 1-Bit quantisierten zeitkontinuierlichen – d.h. unendlich überabgetasteten – Kanals mit additivem weißen Gauß’schen Rauschen bei mittlerem bis hohem Signal-Rausch-Verhältnis (SNR) an. Wir leiten analytische Ausdrücke basierend auf lauflängencodierten Eingangssignalen ab. Für reelle Signale ist der maximale Wert der unteren Grenze der spektralen Effizienz im Hoch-SNR-Bereich etwa 1,63 Bit/s/Hz. Da die Überabtastrate in praktischen Szenarien endlich bleibt, geben wir Grenzen für die erreichbare Rate eines bandbegrenzten, überabgetasteten zeitdiskreten Kanals an. Diese Grenzen stimmen mit den Ergebnissen des zeitkontinuierlichen Kanals bemerkenswert gut überein. Im Hoch-SNR-Bereich sind spektrale Effizienzen bis zu 1,53 Bit/s/Hz bei strikter Bandbegrenzung möglich. Wenn Energieanteile außerhalb des Frequenzbandes tolerierbar sind, können spektrale Effizienzen über 2 Bit/s/Hz pro Domäne – bezogen auf die Bandbreite, die 95 % der Energie enthält – erreichbar sein. Durch die Anwendung der erhaltenen Grenzen auf einen bandbegrenzten überabgetasteten 1-Bit quantisierten Multiple-Input Multiple-Output-Kanal zeigen wir Vorteile durch die Verwendung geeigneter Leistungsverteilungsschemata. Als Modulationsverfahren mit konstanter Hüllkurve betrachten wir kontinuierliche Phasenmodulation, um die Anforderungen an die Linearität des Leistungsverstärkers zu verringern. Beschränkungen für die erreichbare Datenrate bei rauschfreier Übertragung auf Zwischenfrequenzen mit Modulationsalphabeten höherer Ordnung werden für Phase-shift keying (PSK) and Continuous-phase frequency-shift keying (CPFSK) untersucht. Weiterhin werden angepasste Signalformen entworfen, die als FTN-CPFSK beschrieben werden können. Mit der gleichen spektralen Effizienz im Hoch-SNR-Bereich wie PSK und CPFSK bieten diese Signalformen eine deutlich verbesserte Bitfehlerrate (BER). Die Verringerung des erforderlichen SNRs zur Erreichung einer bestimmten BER kann bis zu 20 dB betragen

A Study of Synchronization Techniques for Optical Communication Systems

The study of synchronization techniques and related topics in the design of high data rate, deep space, optical communication systems was reported. Data cover: (1) effects of timing errors in narrow pulsed digital optical systems, (2) accuracy of microwave timing systems operating in low powered optical systems, (3) development of improved tracking systems for the optical channel and determination of their tracking performance, (4) development of usable photodetector mathematical models for application to analysis and performance design in communication receivers, and (5) study application of multi-level block encoding to optical transmission of digital data

Fracture, Fatigue, and Structural Integrity of Metallic Materials and Components Undergoing Random or Variable Amplitude Loadings

Most metallic components and structures are subjected, in service, to random or variable amplitude loadings. There are many examples: vehicles subjected to loadings and vibrations caused by road irregularity and engine, structures exposed to wind, off-shore platforms undergoing wave-loadings, and so on. Just like constant amplitude loadings, random and variable amplitude loadings can make fatigue cracks initiate and propagate, even up to catastrophic failures. Engineers faced with the problem of estimating the structural integrity and the fatigue strength of metallic structures, or their propensity to fracture, usually make use of theoretical, numerical, or experimental approaches. This reprint collects a series of recent scientific contributions aimed at providing an up-to-date overview of approaches and case studies—theoretical, numerical or experimental—on several topics in the field of fracture, fatigue strength, and the structural integrity of metallic components subjected to random or variable amplitude loadings