Sparsity plays an essential role in a number of modern algorithms. This thesis examines how we can incorporate additional structural information within the sparsity profile and formulate a richer class of learning approaches. The focus is on Bayesian techniques for promoting sparsity and developing novel priors and inference schemes.

The thesis begins by showing how structured sparsity can be used to recover simultaneously block sparse signals in the presence of outliers. The approach is validated with empirical results on synthetic data experiments as well as the multiple measurement face recognition problem.

In the next portion of the thesis, the focus is on how structured sparsity can be used to extend approaches for dictionary learning. Dictionary learning refers to decomposing a data matrix into the product of a dictionary and coefficient matrix, subject to a sparsity constraint on the coefficient matrix.

Chapter 3 studies structure in the form of non-negativity constraints on the unknowns, which is referred to as the sparse non-negative least squares (S-NNLS) problem. It presents a unified framework for S-NNLS based on a novel prior on the sparse codes and provides an

efficient multiplicative inference procedure. It then extends the framework to sparse non-negative matrix factorization (S-NMF) and proves that the proposed approach is guaranteed to converge to a set of stationary points for both the S-NNLS and a subclass of the S-NMF problems.

Finally, Chapter 4 addresses the problem of learning dictionaries for multimodal datasets. It presents the multimodal sparse Bayesian dictionary learning (MSBDL) algorithm. The MSBDL algorithm is able to leverage information from all available data modalities through a joint sparsity constraint on each modality’s sparse codes without restricting the coefficients themselves to be equal. The proposed framework offers a considerable amount of flexibility to practitioners and addresses many of the shortcomings of existing multimodal dictionary learning approaches. Unlike existing approaches, MSBDL allows the dictionaries for each data modality to have

different cardinality. In addition, MSBDL can be used in numerous scenarios, from small datasets to extensive datasets with large dimensionality. MSBDL can also be used in supervised settings.

The objective of this paper is to present an engineering model based on fundamentally sound but simplified treatment of mass diffusion phenomena for practical predictions of the effective diffusion coefficient of gases through closed-cell foams. A special attention was paid to stating all assumptions and simplifications that define the range of applicability of the proposed model. The model developed is based on the electrical circuit analogy, and on the first principles. The analysis suggests that the effective diffusion coefficient through the foam can be expressed as a product of the geometric factor and the gas diffusion coefficient through the foam membrane. Validation against experimental data available in the literature gives satisfactory results. Discrepancies between the model predictions and experimental data have been observed for gases with high solubility in the condensed phase for which Henry's law does not apply. Finally, further experimental data concerning both the foam morphology and the diffusion coefficient in the membrane are needed to fully validate the model.

This paper presents an approach for predicting the thickness of isothermal foams produced by blowing gas in a liquid solution under steady-state conditions. The governing equation for the transient foam thickness has been non-dimensionalized, and two dimensionless numbers have been identified to describe the formation and stability of this type of foam: \Pi_1 = {Re}/{Fr} and \Pi_2 = Ca H_{\infty}/r_0. Physical interpretation of the dimensionless numbers has been proposed, a power-law type relation has been assumed between \Pi_1 and \Pi_2 (i.e., \Pi_2=K \Pi^n_1). Experimental data available in the literature have been used to determine the empirical parameters of the correlation K and n. The experimental conditions cover a wide range of viscosity, density, surface tension, gas superficial velocity, and average bubble radius. The model is valid for foams formed from high viscosity liquids bubbled with nitrogen, air, helium, hydrogen, and argon injected through single, multi-orifice nozzles or porous medium. Comparison between the correlation developed and the experimental data yields reasonable agreements (within 35% error), given the broadness of the bubble radius distribution around the mean value, the uncertainty of the experimental data and of the thermophysical properties. Predictions have been found to be very sensitive to the average bubble radius. A more refined model is still needed which should be supported by a careful experimental studies. Finally, suggestions are given to extend the present work to foams generated from low viscosity solutions.

This paper presents a simple, experimentally validated approach to analyze the transient formation of a foam layer produced by injecting gas bubbles in a foaming solution. Based on experimental observations, three different regimes in the transient growth of the foam have been identified as a function of the superficial gas velocity. A model based on the mass conservation equation for the gas phase in the foam combined with three different models for the average porosity is proposed. It is shown that for practical calculations a constant average porosity equal to 0.82 can be used. The model predictions show very good agreement with experimental data for low superficial gas velocity and provide an upper limit of the foam thickness for intermediate and large superficial gas velocities. The paper discusses the physical mechanisms that may occur during the foam formation and the effects of the superficial gas velocity on the foam dynamics. The present analysis speculates several mechanisms for the bursting of the bubbles at the top of the foams and proposes the framework for more fundamental and detailed studies.

This paper presents a complete set of coupled equations that govern the bubble transport in three-dimensional gravity driven flow. The model accounts for bubbles growth or shrinkage due to pressure and temperature changes as well as for multiple gas diffusion in and out of the bubbles but neglects bubble coalescence, break-up, and nucleation. The model applies to glass melting furnaces but it could be extended to other two-phase flow applications such as metal and polymer processings, passive cooling systems, and two-phase flow around naval surface ship. Governing equations are given for the key variables which are, in the present case, (1) the refining agent concentration, (2) the gas species dissolved in the liquid phase, and (3) the bubble radius, gas molar fraction, and density function. The method of solution is discussed and based on the modified method of characteristics.