Theses and Dissertations from UMD
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Item Mean-field Approaches in Multi-agent Systems: Learning and Control(2023) Tirumalai, Amoolya; Baras, John S; Electrical Engineering; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)In many settings in physics, chemistry, biology, and sociology, when individuals (particles) interact in large collectives, they begin to behave in emergent ways. This is to say that their collective behavior is altogether different from their individual behavior. In physics and chemistry, particles interact through the various forces, and this results in the rich behavior of the phases of matter. A particularly interesting case arises in the dynamics of gaseous star formation. In models of star formation, the gases are subject to the attractive gravitational force, and perhaps viscosity, electromagnetism, or thermal fluctuations. Depending on initial conditions, and inclusion of additional forces in the models, a variety of interesting configurations can arise, from dense nodules of gas to swirling vortices. In biology and sociology, these interactions (forces) can be explicitly tied to chemical or physical phenomena, as in the case of microbial chemotaxis, or they can be more abstract or virtual, as in the case of bird flocking or human pedestrian traffic. We focus on the latter cases in this work. In collective animal or human traffic, we do not say that animals or humans are explicity subject to physical forces that causes them to move in alignment with each other, or whatever else. Rather, they behave as if there were such forces. In short, we use the language and notation of physics and forces as a convenient tool to build our understanding. We do so since natural phenomena are rich with sophisticated and adaptive behavior. Bird flocks rapidly adapt to avoid collisions, to fly around obstacles, and to confuse predators. Engineers today can only dream of building drone swarms with such plasticity. An important question to answer is how one takes a model of interacting individuals and builds a model of a collective. Once one answers this question, another immediately follows: how do we take these models of collectives and use them to discover representations of natural phenomena? Then, can we use these models to build methods to control such phenomena, assuming suitable actuation? Once these questions are answered, our understanding of collective dynamics will improve, broadening the applications we can tackle. In this thesis, we study collective dynamics via mean-field theory. In mean-field theory, an individual is totally anonymous, and so can be removed or permuted from a large collective without changing the collective dynamics significantly. More specifically, when any individual is excluded from the definition of the empirical measure of all the individuals, those empirical measures converge to the same measure, termed the mean-field measure. The mean-field measure is governed by the forward Kolmogorov equation. In certain scenarios where an analogy can be drawn to particle dynamics, these forward Kolmogorov equations can be converted to compressible Euler equations. When optimal control problems are posed on the particle dynamics, in the mean-field limit we obtain a forward Kolmogorov equation coupled to a backward Hamilton-Jacobi-Bellman (-Isaacs) equation (or a stationary analogue of these). This system of equations describes the solution to the mean-field game. The first two problems we explore in this thesis are focused on the system identification (inverse) problem: discover a model of collective dynamics from data. In these problems, we study a generalized hydrodynamic Cucker-Smale-type model of flocking in a bounded region of 3D space. We first prove existence of weak bounded energy solutions and a weak-strong uniqueness principle for our model. Then, we use the model to learn a representation of the dynamics of data associated to a synthetic bird flock. The second two problems we study focus on the control (forward) problem: learn an approximately optimal control for collective dynamics online. We study this first in a relatively simple state-and-control-constrained mean-field game on traffic. In this case, the mean-field term is contained only in the mean-field game's cost. We first numerically study a finite horizon version of this problem. The approach for the first problem is not online. Then, we take an infinite horizon version, and we form a system of approximate dynamic programming ODE-PDEs from the exact dynamic programming PDEs. This approach results in online learning and adapting of the control to the dynamics. We prove this ODE-PDE system has a unique weak solution via semigroup and successive approximation methods. We present a numerical example, and discuss the tradeoffs in this approach. We conclude the thesis by summarizing our results, and discussing future directions and applications in theoretical and practical settings.Item Part I: On the Stability threshold of Couette flow in a uniform magnetic field; Part II: Quantitative convergence to equilibrium for hypoelliptic stochastic differential equations with small noise(2021) Liss, Kyle; Bedrossian, Jacob; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)This dissertation contains two parts. In Part I, We study the stability of the Couette flow (y,0,0) in the presence of a uniform magnetic field a*(b, 0, 1) on TxRxT using the 3D incompressible magnetohydrodynamics (MHD) equations. We consider the inviscid, perfect conductor limit Re^(-1) = Rm^(-1) << 1 and prove that for strong and suitably oriented background fields the Couette flow is asymptotically stable to perturbations that are O(Re^(-1)) in the Sobolev space H^N. More precisely, we establish the decay estimates predicted by a linear stability analysis and show that the perturbations u(t,x+yt,y,z) and b(t,x+yt,y,z) remain O(Re^(-1)) in H^M for some 1 << M(b) < N. In the Navier-Stokes case, high regularity control on the perturbation in a coordinate system adapted to the mixing of the Couette flow is known only under the stronger assumption of O(Re^(-3/2)) data. The improvement in the MHD setting is possible because the magnetic field induces time oscillations that partially suppress the lift-up effect, which is the primary transient growth mechanism for the Navier-Stokes equations linearized around Couette flow. In Part II, we study the convergence rate to equilibrium for a family of Markov semigroups (parametrized by epsilon>0) generated by a class of hypoelliptic stochastic differential equations on R^d, including Galerkin truncations of the incompressible Navier-Stokes equations, Lorenz-96, and the shell model SABRA. In the regime of vanishing, balanced noise and dissipation, we obtain a sharp (in terms of scaling) quantitative estimate on the exponential convergence in terms of the small parameter epsilon. By scaling, this regime implies corresponding optimal results both for fixed dissipation and large noise limits or fixed noise and vanishing dissipation limits. As part of the proof, and of independent interest, we obtain uniform-in-epsilon upper and lower bounds on the density of the stationary measure. Upper bounds are obtained by a hypoelliptic Moser iteration, the lower bounds by a De Giorgi-type iteration (both uniform in epsilon). The spectral gap estimate on the semigroup is obtained by a weak Poincar\'e inequality argument combined with quantitative hypoelliptic regularization of the time-dependent problem.Item BATTERY STATE OF CHARGE ESTIMATION BASED ON DATA-DRIVEN MODELS WITH MOVING WINDOW FILTERS AND PHYSICS-BASED MODELS WITH EFFICIENT SOLID-PHASE DIFFUSION PDES SOLVED BY THE OPTIMIZED PROJECTION METHOD(2018) He, Wei; Pecht, Michael; Mechanical Engineering; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)State of charge (SOC) estimation is one of the most important functions of battery management systems (BMSs), which is defined as the percentage of the remaining charge inside the battery to its maximum capacity. SOC indicates when the battery needs to be recharged. It is necessary for many battery management applications, for example, charge/discharge control, remaining useful time/ driving range predictions, and battery power capability estimations. Inaccurate SOC estimations can lead to user dissatisfaction, mission failures, and premature battery failures. This thesis focuses on the development of advanced battery models and algorithms for SOC estimations. Two SOC estimation approaches are investigated, including electrochemical models and data-driven models. Electrochemical models have intrinsic advantages for SOC estimation since it can relate battery internal physical parameters, e.g. lithium concentrations, to SOC. However, the computational complexity of the electrochemical model is the major obstacle for its application in a real-time BMS. To address this problem, an efficient solution for the solid phase diffusion equations in the electrochemical model is developed based on projection with optimized basis functions. The developed method generates 20 times fewer equations compared with finite difference-based methods, without losing accuracy. The results also show that the developed method is three times more efficient compared with the conventional projection-based method. Then, a novel moving window filter (MWF) algorithm is developed to infer SOC based on the electrochemical model. MWF converges to true values nearly 15 times faster compared with unscented Kalman filter in experimental test cases. This work also develops a data-driven SOC estimation approach. Traditional data-driven approaches, e.g. neural network, have generalization problems. For example, the model over-fits to training data and generate erroneous results in the testing data. This thesis investigates algorithms to improve the generalization capability of the data-driven model. An algorithm is developed to select optimal neural network structure and training data inputs. Then, a hybrid approach is developed by combining the neural network and MWF to provide stable SOC estimations. The results show that the SOC estimation error can be reduced from 8% to less 4% compared with the original neural network approach.Item Asymptotic problems for stochastic partial differential equations(2015) Salins, Michael; Cerrai, Sandra; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Stochastic partial differential equations (SPDEs) can be used to model systems in a wide variety of fields including physics, chemistry, and engineering. The main SPDEs of interest in this dissertation are the semilinear stochastic wave equations which model the movement of a material with constant mass density that is exposed to both determinstic and random forcing. Cerrai and Freidlin have shown that on fixed time intervals, as the mass density of the material approaches zero, the solutions of the stochastic wave equation converge uniformly to the solutions of a stochastic heat equation, in probability. This is called the Smoluchowski-Kramers approximation. In Chapter 2, we investigate some of the multi-scale behaviors that these wave equations exhibit. In particular, we show that the Freidlin-Wentzell exit place and exit time asymptotics for the stochastic wave equation in the small noise regime can be approximated by the exit place and exit time asymptotics for the stochastic heat equation. We prove that the exit time and exit place asymptotics are characterized by quantities called quasipotentials and we prove that the quasipotentials converge. We then investigate the special case where the equation has a gradient structure and show that we can explicitly solve for the quasipotentials, and that the quasipotentials for the heat equation and wave equation are equal. In Chapter 3, we study the Smoluchowski-Kramers approximation in the case where the material is electrically charged and exposed to a magnetic field. Interestingly, if the system is frictionless, then the Smoluchowski-Kramers approximation does not hold. We prove that the Smoluchowski-Kramers approximation is valid for systems exposed to both a magnetic field and friction. Notably, we prove that the solutions to the second-order equations converge to the solutions of the first-order equation in an $L^p$ sense. This strengthens previous results where convergence was proved in probability.