Dynamical systems are mathematical models characterized by a set of differential or difference equations. Due to the increasing demand for more accuracy, the number of equations involved may reach the order of thousands and even millions. With so many equations, it often becomes computationally cumbersome to work with these large-scale dynamical systems. Model reduction aims to replace the original system with a reduced system of significantly smaller dimension which will still describe the important dynamics of the large-scale model. Interpolation is one method used to obtain the reduced order model. This requires that the reduced order model interpolates the full order model at selected interpolation points. Reduced order models are obtained through the Krylov reduction process, which involves solving a sequence of linear systems. The Iterative Rational Krylov Algorithm (IRKA) iterates this Krylov reduction process to obtain an optimal Η₂ reduced model. Especially in the large-scale setting, these linear systems often require employing inexact solves. The aim of this thesis is to investigate the impact of inexact solves on interpolatory model reduction. We considered preconditioning the linear systems, varying the stopping tolerances, employing GMRES and BiCG as the inexact solvers, and using different initial shift selections. For just one step of Krylov reduction, we verified theoretical properties of the interpolation error. Also, we found a linear improvement in the subspace angles between the inexact and exact subspaces provided that a good shift selection was used. For a poor shift selection, these angles often remained of the same order regardless of how accurately the linear systems were solved. These patterns were reflected in Η₂ and Η∞ errors between the inexact and exact subspaces, since these errors improved linearly with a good shift selection and were typically of the same order with a poor shift. We found that the shift selection also influenced the overall model reduction error between the full model and inexact model as these error norms were often several orders larger when a poor shift selection was used. For a given shift selection, the overall model reduction error typically remained of the same order for tolerances smaller than 1 x 10^-3, which suggests that larger tolerances for the inexact solver may be used without necessarily augmenting the model reduction error. With preconditioned linear systems as well as BiCG, we found smaller errors between the inexact and exact models while the order of the overall model reduction error remained the same. With IRKA, we observed similar patterns as with just one step of Krylov reduction. However, we also found additional benefits associated with using an initial guess in the inexact solve and by varying the tolerance of the inexact solve. / Master of Science

Reduced-order models have long been used to understand the behavior of nonlinear partial differential equations. Naturally, reduced-order modeling techniques come at the price of either computational accuracy or computation time. Optimization techniques are studied to improve either or both of these objectives and decrease the total computational cost of the problem. This thesis focuses on the dynamic mode decomposition (DMD) applied to nonlinear PDEs with periodic boundary conditions. It provides one study of an existing optimization framework for the DMD method known as the Optimized DMD and provides another study of a newly proposed optimization framework for the DMD method called the Split DMD. / Master of Science / The Navier-Stokes (NS) equations are the primary mathematical model for understanding the behavior of fluids. The existence and smoothness of the NS equations is considered to be one of the most important open problems in mathematics, and challenges in their numerical simulation is a barrier to understanding the physical phenomenon of turbulence. Due to the difficulty of studying this problem directly, simpler problems in the form of nonlinear partial differential equations (PDEs) that exhibit similar properties to the NS equations are studied as preliminary steps towards building a wider understanding of the field. Reduced-order models have long been used to understand the behavior of nonlinear partial differential equations. Naturally, reduced-order modeling techniques come at the price of either computational accuracy or computation time. Optimization techniques are studied to improve either or both of these objectives and decrease the total computational cost of the problem. This thesis focuses on the dynamic mode decomposition (DMD) applied to nonlinear PDEs with periodic boundary conditions. It provides one study of an existing optimization framework for the DMD method known as the Optimized DMD and provides another study of a newly proposed optimization framework for the DMD method called the Split DMD.

This thesis describes the challenges involved in detecting sources of acoustic noise using an autonomous underwater vehicle (AUV) in real world environments. The initial part of this thesis describes the developments made for redesigning an acoustic sensing system that can be used to estimate the relative bearing between a source of acoustic noise and an AUV. With an estimate of the relative bearing, the AUV can maneuver toward the source of noise. The class of algorithms that are used to estimate bearing angle are known as beamforming algorithms. A comparison of the performance of a variety of beamforming algorithms is presented. When estimating the bearing to a source of noise from a small AUV, the noise of the AUV, especially its propulsor, pose significant challenges. Toward the goal of active cancellation of AUV self-noise, we propose placing an additional hydrophone inside the AUV in order to estimate the AUV self-noise that appears on the exterior hydrophones that are used for bearing estimation. / Master of Science / A real world application using an autonomous underwater vehicle (AUV) is presented in this thesis. The application deals with detecting and estimating the relative location (bearing angle) between sources of acoustic noise and the AUV. The thesis starts by describing design changes made to target data sensing system inside the AUV for collecting and estimating the bearing angle. The estimation of bearing angle is done with a class of algorithms called beamforming algorithms whose performance comparison is presented on real world data. Operating the AUV propulsor yields inaccurate bearing angle estimations and thus presents a huge challenge for bearing estimation. We propose measuring AUV self-noise using additional sensors to move towards the goal of cancelling AUV self-noise and recovering target signal for accurate bearing estimation.

Dynamical systems are widely used to model physical phenomena and, in many cases, these physical phenomena are parameter dependent. In this thesis we investigate three prominent problems related to the simulation of parametric dynamical systems and develop the analysis and computational framework to solve each of them. In many cases we have access to data resulting from simulations of a parametric dynamical system for which an explicit description may not be available. We introduce the parametric AAA (p-AAA) algorithm that builds a rational approximation of the underlying parametric dynamical system from its input/output measurements, in the form of transfer function evaluations. Our algorithm generalizes the AAA algorithm, a popular method for the rational approximation of nonparametric systems, to the parametric case. We develop p-AAA for both scalar and matrix-valued data and study the impact of parameter scaling. Even though we present p-AAA with parametric dynamical systems in mind, the ideas can be applied to parametric stationary problems as well, and we include such examples. The solution of a dynamical system can often be expressed in terms of an eigenvalue problem (EVP). In many cases, the resulting EVP is nonlinear and depends on a parameter. A common approach to solving (nonparametric) nonlinear EVPs is to approximate them with a rational EVP and then to linearize this approximation. An existing algorithm can then be applied to find the eigenvalues of this linearization. The AAA algorithm has been successfully applied to this scheme for the nonparametric case. We generalize this approach by using our p-AAA algorithm to find a rational approximation of parametric nonlinear EVPs. We define a corresponding linearization that fits the format of the compact rational Krylov (CORK) algorithm for the approximation of eigenvalues. The simulation of dynamical systems may be costly, since the need for accuracy may yield a system of very large dimension. This cost is magnified in the case of parametric dynamical systems, since one may be interested in simulations for many parameter values. Interpolatory model order reduction (MOR) tackles this problem by creating a surrogate model that interpolates the original, is of much smaller dimension, and captures the dynamics of the quantities of interest well. We generalize interpolatory projection MOR methods from parametric linear to parametric bilinear systems. We provide necessary subspace conditions to guarantee interpolation of the subsystems and their first and second derivatives, including the parameter gradients and Hessians. Throughout the dissertation, the analysis is illustrated via various benchmark numerical examples. / Doctor of Philosophy / Simulation of mathematical models plays an important role in the development of science. There is a wide range of models and approaches that depend on the information available and the goal of the problem. In this dissertation we focus on three problems whose solution depends on parameters and for which we have either data resulting from simulations of the model or a explicit structure that describes the model. First, for the case when only data are available, we develop an algorithm that builds a data-driven approximation that is then easy to reevaluate. Second, we embed our algorithm in an already developed framework for the solution of a specific kind of model structure: nonlinear eigenvalue problems. Third, given a model with a specific nonlinear structure, we develop a method to build a model with the same structure, smaller dimension (for faster computation), and that provides an accurate approximation of the original model.

Instability of humanoid robots is a common problem, especially given external disturbances or difficult terrain. Even with the robustness of most whole body controllers, instability is inevitable given the right conditions. When these unstable events occur they can result in costly damage to the robot potentially causing a cease of normal functionality. Therefore, it is important to study and develop methods to control a humanoid robot during a fall to reduce the chance of critical damage. This thesis proposes joint angular velocity strategies to reduce the impact velocity resulting from a lateral, backward, or forward fall. These strategies were used on two and three link reduced order models to simulate a fall from standing height of a humanoid robot. The results of these simulations were then used on a full degree of freedom robot, Viginia Tech's humanoid robot ESCHER, to validate the efficacy of these strategies. By using angular velocity strategies for the knee and waist joint, the reduced order models resulted in a decrease in impact velocity of the center of mass by 58%, 87%, and 74% for a lateral, backward, and forward fall respectively in comparison to a rigid fall using the same initial conditions. Best case angular velocity strategies were then developed for various initial conditions for each falling direction. Finally, these parameters were implemented on the full degree of freedom robot which showed results similar to those of the reduced order models. / Master of Science / Instability of humanoid robots is a common problem, especially given external disturbances or difficult terrain. Even with the robustness of most whole body controllers, instability is inevitable given the right conditions. When these unstable events occur they can result in costly damage to the robot potentially causing a cease of normal functionality. Therefore, it is important to study and develop methods to control a humanoid robot during a fall to reduce the chance of critical damage. This thesis proposes strategies that rotate the joints at a constant rate to reduce damage resulting from a lateral, backward, or forward fall. These strategies were used on two and three link simplistic models to simulate a fall from standing height of a humanoid robot. The results of these simulations were then used on a full robot, Viginia Tech’s humanoid robot ESCHER, to validate the efficacy of these strategies. By constant joint rotation strategies for the knee and waist joint, the simplistic models resulted in a decrease in impact velocity of the center of mass by 58%, 87%, and 74% for a lateral, backward, and forward fall respectively in comparison to a rigid fall using the same initial conditions. Best case joint rotation strategies were then developed for various initial conditions for each falling direction. Finally, these parameters were implemented on the full robot which showed results similar to those of the reduced order models.

While computational fluid dynamics (CFD) simulations are an indispensable tool in modern aerospace engineering design, they bear a severe computational burden in applications where simulation results must be found quickly or repeatedly. Therefore, creating computationally inexpensive models that can capture complex fluid behaviors is a long-sought-after goal. As a result, methods to construct these reduced order models (ROMs) have seen increasing research interest. Still, parameter dependent high-speed flows that contain shock waves are a particularly challenging class of problems that introduces many complications in ROM frameworks. To make approximations in a linear space, ROM techniques for these problems require that basis functions are transformed such that discontinuities are aligned into a consistent reference frame. Techniques to construct these transformations, however, fail when the topology of shocks is not consistent between data snapshots. In this work, we first identify key features of these topology changes, and how that constrains transformations of this kind. We then construct a new modeling framework that can effectively deal with shockwave interactions that are known to cause failures. The capabilities of the resulting model were evaluated by analyzing supersonic flows over a wedge and a forward-facing step. In the case of the forward-facing step, when shock topology changes with Mach number, our method exhibits significant accuracy improvements. Suggestions for further developments and improvements to our methodology are also identified and discussed / Master of Science / While computational fluid dynamics (CFD) simulations are an indispensable tool in modern aerospace engineering design, they bear a severe computational burden in applications where simulation results must be found quickly or repeatedly. Therefore, creating computationally inexpensive models that can capture complex fluid behaviors is a long-sought-after goal. As a result, methods to construct these reduced order models (ROMs) have seen increasing research interest. Still, high-speed flows that contain shock waves are a particularly challenging class of problems that introduces many complications in ROM frameworks. First, we identify some of the common failure modes in previous ROM methodologies. We then construct a new modeling framework that can effectively deal with shockwave interactions that are known to cause these failures. The capabilities of the resulting model were evaluated by analyzing supersonic flows over a wedge and a forward-facing step. In cases where previous modeling frameworks are known to fail, our method exhibits significant accuracy improvements. Suggestions for further developments and improvements to our methodology are also identified and discussed.

Thesis advisor: John J. Piderit / This paper examines state and county-exclusive incentives on battery electric vehicle (BEV) registration in the United States. Using two main methods, a differences-in-differences method and a sigmoidal growth rate equation, I examine the impact of non-federal incentives on the total amount of electric vehicles between 2017 and 2022, as well as estimate the years that each state will reach its net-zero goals for carbon emissions in the transportation sector. I hope to provide a deeper understanding of the effectiveness of incentive policy, based on differing levels of incentive policy between regions, in order to best increase electric vehicle adoption in a cost-effective method. In addition, I hope that my estimates of net-zero projections will serve as a beneficial comparison to track states’ respective progress towards sustainable energy in vehicles. These findings can be used to assist policymakers in determining appropriate BEV adoption policies based on regional consumer demographics and needs, as well as visualize a timeline for the next century of rapid electric vehicle growth. / Thesis (BA) — Boston College, 2024. / Submitted to: Boston College. Morrissey School of Arts and Sciences. / Discipline: Economics. / Discipline: Departmental Honors.

No / Propargylic sulfones can be cis-hydrogenated using commercial zinc powder and ammonium chloride in THF¿water at room temperature, the major products being the corresponding (Z)-allylic sulfones. Other reducible groups (alkene, benzyloxy) are not affected. Allenylsulfones are implicated in one of the possible reaction pathways.

Dynamical systems and their numerical simulation are very important for investigating physical and technical problems. The more accuracy is desired, the more equations are needed to reach the desired level of accuracy. This leads to large-scale dynamical systems. The problem is that computations become infeasible due to the limitations on time and/or memory in large-scale settings. Another important issue is numerical ill-conditioning. These are the main reasons for the need of model reduction, i.e. replacing the original system by a reduced system of much smaller dimension. Then one uses the reduced models in order to simulate or control processes. The main goal of this thesis is to investigate an interpolation-based approach to the weighted-H2 model reduction problem. Nonetheless, first we will discuss the regular (unweighted) H2 model reduction problem. We will re-visit the interpolation conditions for H2-optimality, also known as Meier-Luenberger conditions, and discuss how to obtain an optimal reduced order system via projection. After having introduced the H2-norm and the unweighted-H2 model reduction problem, we will introduce the weighted-H2 model reduction problem. We will first derive a new error expression for the weighted-H2 model reduction problem. This error expression illustrates the significance of interpolation at the mirror images of the reduced system poles and the original system poles, as in the unweighted case. However, in the weighted case this expression yields that interpolation at the mirror images of the poles of the weighting system is also significant. Finally, based on the new weighted-H2 error expression, we will propose an iteratively corrected interpolation-based algorithm for the weighted-H2 model reduction problem. Moreover we will present new optimality conditions for the weighted-H2 approximation. These conditions occur as structured orthogonality conditions similar to those for the unweighted case which were derived by Antoulas, Beattie and Gugercin. We present several numerical examples to illustrate the effectiveness of the proposed approach and compare it with the frequency-weighted balanced truncation method. We observe that, for virtually all of our numerical examples, the proposed method outperforms the frequency-weighted balanced truncation method. / Master of Science

By shearing activated sludge using a high shear rotor stator device, bioavailable proteinaceous material can be produced. Operation at elevated temperatures, serves to increase the amount of material that is rendered soluble (<0.45 um) and biodegradable. The storage of sludge under anoxic condition prior to shearing does not appear to enhance solublization of solids, though deflocculation and deterioration of dewaterablility was observed. Anaerobic digestibility appears to be enhanced by the addition of a high shear as shown by increases in gas production and volatile solids destruction. The dewatering properties of activated sludge, measured by capillary suction time, deteriorated with the addition of sheared solids, but reaeration resulted in near complete recovery. The role of iron and iron chemistry plays a critical role in the activated sludge. Iron apparently selectively removes protein, in particular material ranging in the 1.5 um to 30K size range. The addition of ferric iron was found to increase SVI and decrease zone-settling velocity, when added to reactors with mechanically disintegrated sludges. Similar trends were not observed in reactors dosed with ferrous iron. Preliminary results suggest that the ferric/ferrous redox chemistry may serve to enhance floc structure, as observed by increased settling velocity and shear resistance for sludges dosed with ferrous sulfate. / Master of Science