We develop a systematic approach to the problem of finding the perturbative expansion for the topological pressure for an analytic expanding dynamics (/, M) on a Riemannian manifold M. The method is based on the spectral analysis of the transfer operator C. We show that in typical cases, when / depends real-analytically on a set of perturbing parameters ,", the related operators C~ form an analytic family. This gives rise to the rigorous construction of the power series expansion for the pressure via the analytic perturbation theory for eigenvalues, [Kato]. Consequently, the pressure and related dynamical indices, such as dimension spectra, Lyapunov exponents, escape rates and Renyi entropies inherit the real-analyticity in ~ from (I,M). / Ph. D.

The transport of material in geophysical fluid flows is a problem with important implications for fields as diverse as: agriculture, aviation, human health, disaster response, and weather forecasting. Due to the unsteady nature of geophysical flows, predicting how material will be transported in these systems can often be challenging. Tools from dynamical systems theory can help to improve the prediction of material transport by revealing important transport structures. These transport structures reveal areas of the flow where fluid parcels, and thus material transported by those parcels, are likely to converge or diverge. Typically, these transport structures have been uncovered by the use of Lagrangian diagnostics. Unfortunately, calculating Lagrangian diagnostics can often be time consuming and computationally expensive. Recently new Eulerian diagnostics have been developed. These diagnostics are faster and less expensive to compute, while still revealing important transport structures in fluid flows. Because Eulerian diagnostics are so new, there is still much about them and their connection to Lagrangian diagnostics that is unknown. This dissertation will fill in some of this gap and provide a mathematical bridge between Lagrangian and Eulerian diagnostics. This dissertation is composed of three projects. These projects represent theoretical, numerical, and experimental advances in the understanding of Eulerian diagnostics and their relationship to Lagrangian diagnostics. The first project rigorously explores the deep mathematical relationship that exists between Eulerian and Lagrangian diagnostics. It proves that some of the new Eulerian diagnostics are the limit of Lagrangian diagnostics as integration time of the velocity field goes to zero. Using this discovery, a new Eulerian diagnostic, infinitesimal-time Lagrangian coherent structures is developed. The second project develops a methodology for estimating local Eulerian diagnostics from wind velocity data measured by a fixed-wing unmanned aircraft system (UAS) flying in circular arcs. Using a simulation environment, it is shown that the Eulerian diagnostic estimates from UAS measurements approximate the true local Eulerian diagnostics and can predict the passage of Lagrangian diagnostics. The third project applies Eulerian diagnostics to experimental data of atmospheric wind measurements. These are then compared to Eulerian diagnostics as calculated from a numerical weather simulation to look for indications of Lagrangian diagnostics. / Doctor of Philosophy / How particles are moved by fluid flows, such as the oceanic currents and the atmospheric winds, is a problem with important implications for fields as diverse as: agriculture, aviation, human health, disaster response, and weather forecasting. Because these fluid flows tend to change over time, predicting how particles will be moved by these flows can often be challenging. Fortunately, mathematical tools exist which can reveal important geometric features in these flows. These geometric features can help us to visualize regions where particles are likely to come together or spread apart, as they are moved by the flow. In the past, these geometric features have been uncovered by using methods which look at the trajectories of particles in the flow. These methods are referred to as Lagrangian, in honor of the Italian mathematician Joseph-Louis Lagrange. Unfortunately, calculating the trajectories of particles can be a time consuming and computationally expensive process. Recently, new methods have been developed which look at how the speed of the flow changes in space. These new methods are referred to as Eulerian, in honor of the Swiss mathematician Leonhard Euler. These new Eulerian methods are faster and less expensive to calculate, while still revealing important geometric features within the flow. Because these Eulerian methods are so new, there is still much that we do not know about them and their connection to the older Lagrangian methods. This dissertation will fill in some of this gap and provide a mathematical bridge between these two methodologies. This dissertation is composed of three projects. These projects represent theoretical, numerical, and experimental advances in the understanding of these new Eulerian methods and their relationship to the older Lagrangian methods. The first project explores the deep mathematical relationship that exists between Eulerian and Lagrangian diagnostic tools. It mathematically proves that some of the new Eulerian diagnostics are the limit of Lagrangian diagnostics as the trajectory’s integration times is decreased to zero. Taking advantage of this discovery, a new Eulerian diagnostic is developed, called infinitesimal-time Lagrangian coherent structures. The second project develops a technique for estimating local Eulerian diagnostics using wind speed measures from a single fixed-wing unmanned aircraft system (UAS) flying in a circular path. Using computer simulations, we show that the Eulerian diagnostics as calculated from UAS measurements provide a reasonable estimate of the true local Eulerian diagnostics. Furthermore, we show that these Eulerian diagnostics can be used to estimate the local Lagrangian diagnostics. The third project applies these Eulerian diagnostics to real-world wind speed measurements. These results are then compared to Eulerian diagnostics that were calculated from a computer simulation to look for indications of Lagrangian diagnostics.

Dynamics of social processes in populations, such as the spread of emotions, influence, language, mass movements, and warfare (often referred to individually and collectively as contagions), are increasingly studied because of their social, political, and economic impacts. Discrete dynamical systems (discrete in time and discrete in agent states) are often used to quantify contagion propagation in populations that are cast as graphs, where vertices represent agents and edges represent agent interactions. We refer to such formulations as graph dynamical systems. For social applications, threshold models are used extensively for agent state transition rules (i.e., for vertex functions). In its simplest form, each agent can be in one of two states (state 0 (1) means that an agent does not (does) possess a contagion), and an agent contracts a contagion if at least a threshold number of its distance-1 neighbors already possess it. The transition to state 0 is not permitted. In this study, we extend threshold models in three ways. First, we allow transitions to states 0 and 1, and we study the long-term dynamics of these bithreshold systems, wherein there are two distinct thresholds for each vertex; one governing each of the transitions to states 0 and 1. Second, we extend the model from a binary vertex state set to an arbitrary number r of states, and allow transitions between every pair of states. Third, we analyze a recent hierarchical model from the literature where inputs to vertex functions take into account subgraphs induced on the distance-1 neighbors of a vertex. We state, prove, and analyze conditions characterizing long-term dynamics of all of these models. / Master of Science

A graph dynamical system (GDS) is a theoretical construct that can be used to simulate and analyze the dynamics of a wide spectrum of real world processes which can be modeled as networked systems. One of our goals is to compute the phase space of a system, and for this, even 30-vertex graphs present a computational challenge. This is because the number of state transitions needed to compute the phase space is exponential in the number of graph vertices. These problems thus produce memory and execution speed challenges. To address this, we devise various MapReduce programming paradigms that can be used to characterize system state transitions, compute phase spaces, functional equivalence classes, dynamic equivalence classes and cycle equivalence classes of dynamical systems. We also evaluate these paradigms and analyze their suitability for modeling different GDSs. / Master of Science

Generally, large-scale dynamical systems pose tremendous computational difficulties when applied in numerical simulations. In order to overcome these challenges we use several model reduction techniques. For stable linear models these techniques work very well and provide good approximations for the full model. However, large-scale unstable systems arise in many applications. Many of the known model reduction methods are not very robust, or in some cases, may not even work if we are dealing with unstable systems. When approximating an unstable system by a reduced order model, accuracy is not the only concern. We also need to consider the structure of the reduced order model. Often, it is important that the number of unstable poles in the reduced system is the same as the number of unstable poles in the original system. The Iterative Rational Krylov Algorithm (IRKA) is a robust model reduction technique which is used to locally reduce stable linear dynamical systems optimally in the ℋ₂-norm. While we cannot guarantee that IRKA reduces an unstable model optimally, there are no numerical obstacles to the reduction of an unstable model via IRKA. In this thesis, we investigate IRKA's behavior when it is used to reduce unstable models. We also consider systems for which we cannot obtain a first order realization of the transfer function. We can use Realization-independent IRKA to obtain a reduced order model which does not preserve the structure of the original model. In this paper, we implement a structure preserving algorithm for systems with nonlinear frequency dependency. / Master of Science

This thesis investigates the numerical treatment of nonlinear eigenvalue problems. These problems are defined by the condition $T(lambda) v = boldsymbol{0}$, with $T: C to C^{n times n}$, where we seek to compute the scalar-vector pairs, $lambda in C$ and nonzero $ v in C^{n}$. The first contribution of this work connects recent contour integration methods to the theory and practice of system identification. This observation leads us to explore rational interpolation for system realization, producing a Loewner matrix contour integration technique. The second development of this work studies the application of rational interpolation to the function $T(z)^{-1}$, where we use the poles of this interpolant to approximate the eigenvalues of $T$. We then expand this idea to several iterative methods, where at each step the approximate eigenvalues are taken as new interpolation points. We show that the case where one interpolation point is used is theoretically equivalent to Newton's method for a particular scalar function. / Master of Science / This thesis investigates the numerical treatment of nonlinear eigenvalue problems. The solutions to these problems often reveal characteristics of an underlying physical system. One popular methodology for handling these problems uses contour integrals to compute a set of the solutions. The first contribution of this work connects these contour integration methods to the theory and practice of system identification. This leads us to explore other techniques for system identification, resulting in a new method. Another common methodology approximates the nonlinear problem directly. The second development of this work studies the application of rational interpolation for this purpose. We then use this idea to form several iterative methods, where at each step the approximate solutions are taken to be new interpolation points. We show that the case where one interpolation point is used is theoretically equivalent to Newton’s method for a particular scalar function.

The problem of establishing local existence and uniqueness of solutions to systems of differential equations is well understood and has a long history. However, the problem of proving global existence and uniqueness is more difficult and fails even for some very simple ordinary differential equations. It is still not known if the 3D Navier-Stokes equation have global unique solutions and this open problem is one of the Millennium Prize Problems. However, many of these mathematical models are extremely useful in the understanding of complex physical systems. For years people have considered methods for modifying these equations in order to obtain models that still capture the observed fundamental physics, but for which one can rigorously establish global results. In this thesis we focus on a taming method to achieve this goal and apply taming to modeling and numerical problems. The method is also applied to a class of nonlinear differential equations with conservative nonlinearities and to Burgers’ Equation with Neumann boundary conditions. Numerical results are presented to illustrate the ideas. / Master of Science

Dynamical systems are useful tools for modeling many complex physical phenomena. In many situations, we do not have access to the governing equations to create these models. Instead, we have access to data in the form of input-output measurements. Data-driven approaches use these measurements to construct reduced order models (ROMs), a small scale model that well approximates the true system, directly from input/output data. Frequency domain data-driven methods, which require access to values (and in some cases to derivatives) of the transfer function, have been very successful in constructing high-fidelity ROMs from data. However, at times this frequency domain data can be difficult to obtain or one might have only access to time-domain data. Recently, Burohman et al. [2020] introduced a framework to approximate transfer function values using only time-domain data. We first discuss improvements to this method to allow a more efficient and more robust numerical implementation. Then, we develop an algorithm that performs optimal-H2 approximation using purely time-domain data; thus significantly extending the applicability of H2-optimal approximation without a need for frequency domain sampling. We also investigate how well other established frequency-based ROM techniques (such as the Loewner Framework, Adaptive Anderson-Antoulas Algorithm, and Vector Fitting) perform on this identified data, and compare them to the optimal-H2 model. / Master of Science / Dynamical systems are useful tools for modeling many phenomena found in physics, chemistry, biology, and other fields of science. A dynamical system is a system of ordinary differential equations (ODEs), together with a state to output mapping. These typically result from a spatial discretization of a partial differential equation (PDE). For every dynamical system, there is a corresponding transfer function in the frequency domain that directly links an input to the system with its corresponding output. For some phenomena where the underlying system does not have a known governing PDE, we are forced to use observations of system input-output behavior to construct models of the system. Such models are called data-driven models. If in addition, we seek a model that can well approximate the true system while keeping the number of degrees of freedom low (e.g., for fast simulation of the system or lightweight memory requirements), we refer to the resulting model as a reduced order model (ROM). There are well established ROM methods that assume access to transfer function input-output data, but such data may be costly or impossible to obtain. This thesis expands upon a method introduced by Burohman et al. [2020] to infer values and derivatives of the transfer function using time domain input-output data. The first contribution of this thesis is to provide a robust and efficient implementation for the data informativity framework. We then provide an algorithm for constructing a ROM that is optimal in a frequency domain sense from time domain data. Finally, we investigate how other established frequency domain ROM techniques perform on the learned frequency domain data.

Hypothesis-free learning is increasingly popular given the large amounts of data becoming available. Structure learning, a hypothesis-free approach, of graphical models is a field of growing interest due to the power of such models and lack of domain knowledge when applied on complex real-world data. State-of-the-art techniques improve on scalability of structure learning, which is often characterized by a large problem space. Nonetheless, these techniques still suffer computational bottlenecks that are yet to be approached. In this work, we focus on two popular models: dynamical linear systems and Markov random fields. For each case, we investigate major computational bottlenecks of baseline learning techniques. Next, we propose two frameworks that provide higher scalability using appropriate problem reformulation and efficient structure based heuristics. We perform experiments on synthetic and real data to validate our theoretical analysis. Current results show that we obtain a quality similar to expensive baseline techniques but with higher scalability. / Master of Science / Structure learning of graphical models is the process of understanding the interactions and influence between the variables of a given system. A few examples of such systems are road traffic systems, stock markets, and social networks. Learning the structure uncovers the invisible inter-variables relationships that govern their evolution. This process is key to qualitative analysis and forecasting. A classic approach to obtain the structure is through domain experts. For example, a financial expert could draw a graphical structure that encodes the relationships between different software companies based on his knowledge in the field. However, the absence of domain experts in the case of complex and heterogeneous systems has been a great motivation for the field of data driven, hypothesis free structure learning. Current techniques produce good results but unfortunately require a high computational cost and are often slow to execute. In this work, we focus on two popular graphical models that require computationally expensive structure learning methods. We first propose theoretical analysis of the high computational cost of current techniques. Next, we propose a novel approach for each model. Our proposed methods perform structure learning faster than baseline methods and provide a higher scalability to systems of large number of variables and large datasets as shown in our theoretical analysis and experimental results.

There are still many open questions regarding spatiotemporal chaos although many well developed theories exist for chaos in time. Rayleigh-B'enard convection is a paradigmatic example of spatiotemporal chaos that is also experimentally accessible. Discoveries uncovered using numerics can often be compared with experiments which can provide new physical insights. Lyapunov diagnostics can provide important information about the dynamics of small perturbations for chaotic systems. Covariant Lyapunov vectors reveal the true direction of perturbation growth and decay. The degree of hyperbolicity can also be quantified by the covariant Lyapunov vectors. To know whether a dynamical system is hyperbolic is important for the development of a theoretical understanding. In this thesis, the degree of hyperbolicity is calculated for chaotic Rayleigh-B'enard convection. For the values of the Rayleigh number explored, it is shown that the dynamics are non-hyperbolic. The spatial distribution of the covariant Lyapunov vectors is different for the different Lyapunov vectors. Localization is used to quantify this variation. The spatial localization of the covariant Lyapunov vectors has a decreasing trend as the order of the Lyapunov vector increases. The spatial localization of the covariant Lyapunov vectors are found to be related to the instantaneous Lyapunov exponents. The correlation is stronger as the order of the Lyapunov vector decreases. The covariant Lyapunov vectors are also computed using a spectral element approach. This allows an exploration of the covariant Lyapunov vectors in larger domains and for experimental conditions. The finite conductivity and finite thickness of the lateral boundaries of an experimental convection domain is also studied. Results are presented for the variation of the Nusselt number and fractal dimension for different boundary conditions. The fractal dimension changes dramatically with the variation of the finite conductivity. / Ph. D. / There are still many open questions regarding chaos. Rayleigh-Bènard convection is a type of natural convection which occurs when a fluid is placed between a hot bottom plate and a cold top plate. Rayleigh-Bènard convection is a classical model to explore chaos in space and time. The major application of Rayleigh-Bènard convection is weather prediction which is an extremely difficult problem of intense interest. The governing equations can only be solved using supercomputing resources. The main reason for this difficulty is the presence of a very large number of degrees of freedom that may influence the weather. To reduce the number of degrees of freedom by only including ones that contribute significantly is a difficult problem. In this thesis, vectors describing the growth of disturbances have been calculated for Rayleigh-Bènard convection. These vectors give us information about which regions in space are more important than others. For weather example, the knowledge of these vectors would tell us which regions are important. With this information, scientists and engineers can focus on the important regions and possibly improve their long term predictions. These vectors also yield the number of degrees of freedom to characterize a chaotic system, on average. In this thesis, this number is also explored for Rayleigh-Bènard convection. This thesis extends the calculation of these vectors to a realistic fluid model which gives us new insights into fundamental questions about chaos in space and time.