Browsing by Subject "Convex optimization"
Now showing 1 - 5 of 5
- Results Per Page
- Sort Options
Item Convergence analysis of the approximate proximal splitting method for non-smooth convex optimization(2014-05) Kadkhodaie Elyaderani, MojtabaConsider a class of convex minimization problems for which the objective function is the sum of a smooth convex function and a non-smooth convex regularity term. This class of problems includes several popular applications such as compressive sensing and sparse group LASSO. In this thesis, we introduce a general class of approximate proximal splitting (APS) methods for solving such minimization problems. Methods in the APS class include many well-known algorithms such as the proximal splitting method (PSM), the block coordinate descent method (BCD) and the approximate gradient projection methods for smooth convex optimization. We establish the linear convergence of APS methods under a local error bound assumption. Since the latter is known to hold for compressive sensing and sparse group LASSO problems, our analysis implies the linear convergence of the BCD method for these problems without strong convexity assumption.Item Low-complexity stochastic modeling of wall-bounded shear flows(2016-12) Zare, ArminTurbulent flows are ubiquitous in nature and they appear in many engineering applications. Transition to turbulence, in general, increases skin-friction drag in air/water vehicles compromising their fuel-efficiency and reduces the efficiency and longevity of wind turbines. While traditional flow control techniques combine physical intuition with costly experiments, their effectiveness can be significantly enhanced by control design based on low-complexity models and optimization. In this dissertation, we develop a theoretical and computational framework for the low-complexity stochastic modeling of wall-bounded shear flows. Part I of the dissertation is devoted to the development of a modeling framework which incorporates data-driven techniques to refine physics-based models. We consider the problem of completing partially known sample statistics in a way that is consistent with underlying stochastically driven linear dynamics. Neither the statistics nor the dynamics are precisely known. Thus, our objective is to reconcile the two in a parsimonious manner. To this end, we formulate optimization problems to identify the dynamics and directionality of input excitation in order to explain and complete available covariance data. For problem sizes that general-purpose solvers cannot handle, we develop customized optimization algorithms based on alternating direction methods. The solution to the optimization problem provides information about critical directions that have maximal effect in bringing model and statistics in agreement. In Part II, we employ our modeling framework to account for statistical signatures of turbulent channel flow using low-complexity stochastic dynamical models. We demonstrate that white-in-time stochastic forcing is not sufficient to explain turbulent flow statistics and develop models for colored-in-time forcing of the linearized Navier-Stokes equations. We also examine the efficacy of stochastically forced linearized NS equations and their parabolized equivalents in the receptivity analysis of velocity fluctuations to external sources of excitation as well as capturing the effect of the slowly-varying base flow on streamwise streaks and Tollmien-Schlichting waves. In Part III, we develop a model-based approach to design surface actuation of turbulent channel flow in the form of streamwise traveling waves. This approach is capable of identifying the drag reducing trends of traveling waves in a simulation-free manner. We also use the stochastically forced linearized NS equations to examine the Reynolds number independent effects of spanwise wall oscillations on drag reduction in turbulent channel flows. This allows us to extend the predictive capability of our simulation-free approach to high Reynolds numbers.Item An optimization based empirical mode decomposition scheme(University of Minnesota. Institute for Mathematics and Its Applications, 2012-02) Huang, Boqiang; Kunoth, AngelaItem Resource Allocation for Green Cloud Networks under Uncertainty: Stochastic, Robust and Big Data-driven Approaches(2016-09) CHEN, TIANYIMajor improvements have propelled the development of worldwide Internet systems during the past decade. To meet the growing demand in massive data processing, a large number of geographically-distributed data centers begin to surge in the era of data deluge and information explosion. Along with their remarkable expansion, contemporary cloud networks are being challenged by the growing concerns about global warming, due to their substantial energy consumption. Hence, the infrastructure of future data centers must be energy-efficient and sustainable. Fortunately, supporting technologies of smart grids, big data analytics and machine learning, are also developing rapidly. These considerations motivate well the present thesis, which mainly focuses on developing interdisciplinary approaches to offer sustainable resource allocation for future cloud networks, by leveraging three intertwining research subjects. The modern smart grid has many new features and advanced capabilities including e.g., high penetration of renewable energy sources, and dynamic pricing based demand-side management. Clearly, by integrating these features into the cloud network infrastructure, it becomes feasible to realize its desiderata of reliability, energy-efficiency and sustainability. Yet, full benefits of the renewable energy (e.g., wind and solar) can only be harnessed by properly mitigating its intrinsically stochastic nature, which is still a challenging task. This prompts leveraging the huge volume of historical data to reduce the stochasticity of online decision making. Specifically, valuable insights from big data analytics can enable a markedly improved resource allocation policy by learning historical user and environmental patterns. Relevant machine learning approaches can further uncover “hidden insights” from historical relationships and trends in massive datasets. Targeting this goal, the present thesis systematically studies resource allocation tasks for future sustainable cloud networks under uncertainty. With an eye towards realistic scenarios, the thesis progressively adapts elegant mathematical models, optimization frameworks, and develops low complexity algorithms from three different aspects: stochastic (Chapters 2 and 3), robust (Chapter 4), and big data-driven approaches (Chapter 5). The resultant algorithms are all numerically efficient with optimality guarantees, and most of them are also amenable to a distributed implementation.