The research presented in this dissertation is on the development of advanced reduced rank adaptive signal processing for airborne radar space-time adaptive processing (STAP) and steering vector mismatch robustness. This is an important area of research in the field of airborne radar signal processing since practical STAP algorithms should be robust against various kinds of mismatch errors. The clutter return in an airborne radar has widely spread Doppler frequencies; therefore STAP, a two-dimensional adaptive filtering algorithm is required for effective clutter and jamming cancellation. Real-world effects in nonhomogeneous environments increase the number of adaptive degrees of freedom required to adequately suppress interference. The increasing computational complexity and the need to estimate the interference from a limited sample support make full rank STAP impractical. The research presented here shows that the reduced rank multistage Wiener filter (MWF) provides significant subspace compression better than any previous techniques in a nonhomogeneous environment. In addition, the impact of steering vector mismatch will also be examined on the MWF. In an airborne radar environment, it is well known that calibration errors and steering vector mismatch can seriously degrade adaptive array performance and result in signal cancellation. These errors can be caused by many non-ideal factors such as beam steering angle errors, multipath propagation, and phase errors due to array imperfections. Since the MWF centrally features the steering vector on its formulation, it is important to assess the impact of steering vector mismatch. In this dissertation, several novel techniques for increasing robustness are examined and applied to the MWF. These include derivative constraints, quiescent pattern control (QPC) techniques, and covariance matrix tapers (CMT). This research illustrates that a combination of CMT and QPC, denoted CMTQ, is very effective at mitigating the impact of steering vector mismatch. Use of CMTQ augmentation provides the steering vector mismatch robustness that we desire while improving the reduced-rank and reduced sample characteristics of the MWF. Results using Monte Carlo simulations and experimental Multichannel Airborne Radar Measurements (MCARM) data confirm that the use of CMTQ gives superior performance to steering vector errors at a much reduced rank and sample support as compared to conventional techniques. / Ph. D.

Numerical models of PDE systems can involve very large matrix equations, but feedback controllers for these systems must be computable in real time to be implemented on physical systems. Classical control design methods produce controllers of the same order as the numerical models. Therefore, reduced order control design is vital for practical controllers. The main contribution of this research is a method of control order reduction that uses a newly developed low order basis. The low order basis is obtained by applying Proper Orthogonal Decomposition (POD) to a set of functional gains, and is referred to as the functional gain POD basis. Low order controllers resulting from the functional gain POD basis are compared with low order controllers resulting from more commonly used time snapshot POD bases, with the two dimensional heat equation as a test problem. The functional gain POD basis avoids subjective criteria associated with the time snapshot POD basis and provides an equally effective low order controller with larger stability radii. An efficient and effective methodology is introduced for using a low order basis in reduced order compensator design. This method combines "design-then-reduce" and "reduce-then-design" philosophies. The desirable qualities of the resulting reduced order compensator are verified by application to Burgers' equation in numerical experiments. / Ph. D.

Despite the fact that higher-order tensors (HOTs) plague electronic structure methods and severely limits the modeling of interesting chemistry problems, introduction and application of higher-order tensor (HOT) decompositions, specifically the canonical polyadic (CP) decomposition, is fairly limited. The CP decomposition is an incredibly useful sparse tensor factorization that has the ability to disentangle all correlated modes of a tensor. However the complexities associated with CP decomposition have made its application in electronic structure methods difficult. Some of the major issues related to CP decomposition are a product of the mathematics of computing the decomposition: determining the exact CP rank is a non-polynomially hard problem, finding stationary points for rank-R approximations require non-linear optimization techniques, and inexact CP approximations can introduce a large degree of error into tensor networks. While other issues are a result of the construction of computer architectures. For example, computer processing units (CPUs) are organized in a way to maximize the efficiency of dense linear algebra and, thus, the performance of routine tensor algebra kernels, like the Khatri-Rao product, is limited. In this work, we seek to reduce the complexities associated with the CP decomposition and create a route for others to develop reduced-scaling electronic structure theory methods using the CP decomposition. In Chapter 2, we introduce the robust tensor network approximation. This approximation is a way to, in general, eliminate the leading-order error associated with approximated tensors in a network. We utilize the robust network approximation to significantly increase the accuracy of approximating density fitting (DF) integral tensors using rank-deficient CP decompositions in the particle-particle ladder (PPL) diagram of the coupled cluster method with single and double substitutions (CCSD). We show that one can produce results with negligible error in chemically relevant energy differences using a CP rank roughly the same size as the DF fitting basis; which is a significantly smaller rank requirement than found using either a nonrobust approximation or similar grid initialized CP approximations (the pseudospectral (PS) and tensor hypercontraction (THC) approximations). Introduction of the CP approximation, formally, reduces the complexity of the PPL diagram from 𝓞(N⁶) to 𝓞(N⁵) and, using the robust approximation, we are able to observe a cost reduction in CCSD calculations for systems as small as a single water molecule. In Chapter 3, we further demonstrate the utility of the robust network approximation and, in addition, we construct a scheme to optimize a grid-free CP decomposition of the order-four Coulomb integral tensor in 𝓞(N⁴) time. Using these ideas, we reduce the complexity of ten bottleneck contractions from 𝓞(N⁶) to 𝓞(N⁵) in the Laplace transform (LT) formulation of the perturbative triple, (T), correction to CCSD. We show that introducing CP into the LT (T) method with a CP rank roughly the size of the DF fitting basis reduces the cost of computing medium size molecules by a factor of about 2.5 and introduces negligible error into chemically relevant energy differences. Furthermore, we implement these low-cost algorithms using newly developed, optimized tensor algebra kernels in the massively-parallel, block-sparse TiledArray [Calvin, et. al Chemical Reviews 2021 121 (3), 1203-1231] tensor framework. / Doctor of Philosophy / Electronic structure methods and accurate modeling of quantum chemistry have developed alongside the advancements in computer infrastructures. Increasingly large and efficient computers have allowed researchers to model remarkably large chemical systems. Sadly, for as fast as computer infrastructures grow (Moores law predicts that the number of transistors in a computer will double every 18 months) the cost of electronic structure methods grows more quickly. One of the least expensive electronic structure methods, Hartree Fock (HF), grows quartically with molecular size; this means that doubling the size of a molecule increase the number of computer operations by a factor of 16. However, it is known that when chemical systems become sufficiently large, the amount of physical information added to the system grows linearly with system size.[Goedecker, et. al. Comput. Sci. Eng., 2003, 5, (4), 14-21] Unfortunately, standard implementations of electronic structure methods will never achieve linear scaling; the disparity between actual cost and physical scaling of molecules is a result of storing and manipulating data using dense tensors and is known as the curse of dimensionality.[Bellman, Adaptive Control Processes, 1961, 2045, 276] Electronic structure theorists, in their desire to apply accurate methods to increasingly large systems, have known for some time that the cost of conventional algorithms is unreasonably high. These theorists have found that one can reveal sparsity and develop reduced-complexity algorithms using matrix decomposition techniques. However, higher-order tensors (HOTs), tensors with more than two modes, are routinely necessary in algorithm formulations. Matrix decompositions applied to HOTs are not necessarily straight-forward and can have no effect on the limiting behavior of an algorithm. For example, because of the positive definiteness of the Coulomb integral tensor, it is possible to perform a Cholesky decomposition (CD) to reduce the complexity of tensor from an order-4 tensor to a product of order-3 tensors.[Beebe, et. al. Int. J. Quantum Chem., 1977, 12, 683-705] However, using the CD approximated Coulomb integral tensors it is not possible to reduce the complexity of popular methods such as Hartree-Fock or coupled cluster theory. We believe that the next step to reducing the complexity of electronic structure methods is through the accurate application of HOT decompositions. In this work, we only consider a single HOT decomposition: the canonical polyadic (CP) decomposition which represents a tensor as a polyadic sum of products. The CP decomposition disentangles all modes of a tensor by representing an order-N tensor as N order-2 tensors. In this work, we construct the CP decomposition of tensors using algebraic optimization. Our goal, here, is to tackle one of the biggest issues associated with the CP decomposition: accurately approximating tensors and tensor networks. In Chapter 2, we develop a robust formulation to approximate tensor networks, a formulation which removes the leading-order error associated with tensor approximations in a network.[Pierce, et. al. J. Chem. Theory Comput., 2021 17 (4), 2217- 2230] We apply a robust CP approximation to the coupled cluster method with single and double substitutions (CCSD) to reduce the overall cost of the approach. Using this robust CP approximation we can compute CCSD, on average, 2.5-3 times faster and introduce negligibly small error in chemically relevant energy values. Furthermore in Chapter 3, we again use the robust CP network approximation in conjunction with a novel, low cost approach to compute order-four CP decompositions, to reduce the cost of 10 high cost computations in the the perturbative triple, (T), correction to CCSD. By removing these computations, we are able to reduce the cost of (T) by a factor of about 2.5 while introducing significantly small error.

In the effort to reduce emission and fuel consumption in industrial gas turbines, lean premixed combustion is utilized but is susceptible to thermoacoustic instabilities. These instabilities occur due to an in-phase relationship between acoustic pressure and unsteady heat release in a combustor. Thermoacoustic instabilities have been shown to cause structural damage and limit operability of combustors. To mitigate these instabilities, a variety of active and passive methods can be employed. The addition of porous inert media (PIM) is a passive mitigation technique that has been shown to be effective at mitigating an instability. Practical industrial application of a mitigation strategy requires early-stage design considerations such as reduced order modeling, which is often used to study a systems' stability response to geometric changes and mitigation approaches. These reduced order models rely on flame transfer functions (FTF) which numerically represent the relationship between heat release and acoustic perturbations. The accurate quantification of heat release is critical in the study of these instabilities and is a necessary component to improve the reduced order model's predictive capability. Heat release quantification presents numerous challenges. Previous work resolving heat release has used optical diagnostics. For perfectly premixed, laminar flames, it has been shown there are proportional relationships between OH* or CH* chemiluminescence to heat release. This is an ideal case; in reality, practical burners produce turbulent and partially premixed flames. Due to the additional straining of the flame caused by turbulence, the heat release is no longer proportional to chemiluminescence quantities. Also, partially premixed systems have spatially varying equivalence ratios and heat release rates, meaning analysis reliant on perfectly premixed assumptions cannot be used and techniques that can handle spatial variations is needed. The objective of this thesis is to incorporate PIM effects into a reduced order model and resolve quantities vital to understand how PIM is mitigating thermoacoustic instabilities in a partially premixed, turbulent combustion environment. The initial work presented in this thesis is the development of a reduced order model for predicting mode shapes and system stability with and without PIM. This was the first time that a reduced order model was developed to study PIM effects on the thermoacoustic response. Model development used a linear FTF and can predict the system frequency and stability response. Through the frequency response, mode shapes can be constructed which show the axial variation in acoustic values, along with node and anti-node locations. Stability trends can be predicted, such as the independent effects of system parameter variation, to determine its stability response. The model was compared to canonical case studies as well as experimental data with reasonable agreement. With PIM addition, it was shown that a combustor would be under stable operation at more flow conditions than without PIM. The work also shows the stability sensitivity to different porous parameters and PIM locations within the combustor. The model has been used to aid in the design of other combustion systems developed at Virginia Tech's Advanced Propulsion and Power Laboratory. To better understand how PIM is affecting the system stability and demonstrate measurements for the improvement of a numerical FTF, experimental work to resolve the spatially varying equivalence ratio fluctuations was conducted in an atmospheric, swirl-stabilized combustor. The experimental studies worked to improve the fundamental understanding of PIM and its mitigation effects through spatially and temporally resolved equivalence ratios during a self-excited instability. The experimental combustor has an optically accessible flame region which allowed for high speed chemiluminescence to be captured during the instability. Equivalence ratio values were calculated from a relation involving OH*/CH* chemiluminescence ratio. The acoustic perturbations were studied to show how the equivalence ratio fluctuations were being generated and coupling with the acoustic waves. The fluctuation in equivalence ratio showed about 65% variation around its mean value during the period of an instability cycle. When porous media was added to the system, the fluctuation in equivalence ratio was mitigated and a reduction in peak frequency (sound pressure level) SPL of 38 dB was observed. Changes in the spatial distribution of equivalence ratio with PIM addition were shown to produce a more stable operation. Effects such as locally richer burning and changes to recirculation zones promoted more stable operation with PIM addition. Testing with forced acoustic input was also conducted to quantify the flame response. The results demonstrated that a flame in a system with PIM responds differently than without PIM, highlighting the need to develop FTF for systems with PIM. This experimental study was the first to study equivalence ratio in a turbulent, partially premixed combustor using PIM as a mitigation technique. A final experimental investigation was conducted to resolve the spatially defined heat release and its fluctuation during a thermoacoustic instability period. This was the first time that heat release had been investigated in a partially premixed, thermoacoustically unstable system, using PIM as a migration method. Heat release was quantified using equivalence ratio, strain rate, OH* intensity, and a correction factor determined from a chemical kinetic solver. The heat release analysis built upon the equivalence ratio study with additional flow field analysis using PIV. The velocity vectors showed prominent corner and central recirculation zones in the no PIM case which have been shown to be feedback mechanisms that support instability formation. With PIM addition, these flow features were reduced and decoupled from the combustor inlet reactants. The velocity results were decomposed using a spectral proper orthogonal decomposition (SPOD) method. The energy breakdown showed how PIM reduced and distributed the energy in the flow structures, creating a more stable flow field. Heat release results with velocity information demonstrated the significant coupling mechanisms in the flow field that were mitigated with the PIM addition. The no PIM case showed high heat release areas being directly influenced by the incoming flow fluctuations. The feedback mechanisms, both mean flow and acoustic, have a direct path to the incoming flow to the combustor. In the PIM case, there is significant mixing and burning taking place in locations where it is less likely that feedback can reach the incoming flow to couple to form an instability. The methodology to quantify heat release provides a framework for quantifying a non-linear FTF with PIM. The development and testing to determine a non-linear FTF with PIM are reserved for future work and discussed in the final chapter. The methodologies and modeling conducted here provided insight and understanding to answer why PIM is effective at mitigating a thermoacoustic instability and how it can be studied using a reduced order numerical tool. / Doctor of Philosophy / In the effort to reduce emission and fuel consumption in industrial gas turbines, lean premixed combustion is utilized but is susceptible to thermoacoustic instabilities. These instabilities occur due to a relationship between acoustic pressure and unsteady heat release in a combustor. Thermoacoustic instabilities have been shown to cause structural damage and limit operability of combustors. To mitigate these instabilities, a variety of active and passive methods can be employed. The addition of porous inert media (PIM) is a passive mitigation technique that has been shown to be effective at mitigating an instability. Implementation of these mitigation strategies require early-stage design considerations such as reduced order modeling, which is often used to study a systems' stability response to geometric changes and mitigation approaches. These reduced order models rely on flame transfer functions (FTF) which numerically model the flame response. The accurate quantification of heat release is critical in the study of these instabilities and is a necessary component to improve the reduced order model's predicative capability. Heat release quantification presents numerous challenges. Previous work resolving heat release has used optical diagnostics with varying levels of success. For perfectly premixed, laminar flames, it has been shown there are proportional relationships between flame light emission and heat release. This is an ideal case; in reality, practical burners produce complex turbulent flames. Due to complex turbulent flame, the heat release is no longer proportional to the flame light emission quantities. Also, partially premixed systems have spatially variant flame quantities, meaning analyses reliant on perfectly premixed assumptions cannot be used and techniques that can handle spatial variations are required. The objective of this thesis is to incorporate PIM effects into a reduced order model and resolve quantities vital to understand how PIM is mitigating thermoacoustic instabilities in a partially premixed, turbulent combustion environment. The initial work presented in this thesis is the development of a reduced order model for predicting mode shapes and system stability with and without PIM. The model uses a simple relationship to model the flame response in an acoustic framework. To improve the model and understanding of PIM mitigation, experimental data such as the local heat release rates and equivalence ratios need to be quantified. An experimental technique was utilized on an optically accessible atmospheric, swirl-stabilized combustor, to resolve the spatially variant equivalence ratio and heat release rates. From these results, better understanding of how PIM is improving the stability in a combustion environment is shown. Quantities such as velocity, acoustic pressure, equivalence ratio, and heat release are all studied and used to explain the improved stability with PIM addition. The methodologies and modeling conducted here provided insight and understanding to answer why PIM is effective at mitigating a thermoacoustic instability and how it can be studied using a reduced order numerical tool. Furthermore, the present work provides a framework for quantifying spatially varying heat release measurements, which can be used to develop FTF for use with thermoacoustic modeling approaches.

An important role of Pin1 is to catalyze the cis-trans isomerization of pSer/Thr-Pro bonds; as such, it plays an important role in many cellular events through the effects of conformational change on the function of its biological substrates, including Cdc25, c-Jun, and p53. The expression of Pin1 correlates with cyclin D1 levels, which contributes to cancer cell transformation. Overexpression of Pin1 promotes tumor growth, while its inhibition causes tumor cell apoptosis. Because Pin1 is overexpressed in many human cancer tissues, including breast, prostate, and lung cancer tissues, it plays an important role in oncogenesis, making its study vital for the development of anti-cancer agents. Many inhibitors have been discovered for Pin1, including 1) several classes of designed inhibitors such as alkene isosteres, non-peptidic, small molecular Pin1 inhibitors, and indanyl ketones, and 2) several natural products such as juglone, pepticinnamin E analogues, PiB and its derivatives obtained from a library screen. These Pin1 inhibitors show promise in the development of novel diagnostic and therapeutic anticancer drugs due to their ability to block cell cycle progression. In order to develop potent Pin1 inhibitors, the concept of transition-state analogues was used for the design of three classes of compounds: ketoamide, ketone, and reduced amide analogues. Specifically, a convergent synthesis of α-ketoamide inhibitors of Pin1 was developed. An α-hydroxyorthothioester derivative of Ser was reacted directly with an aminyl synthon. The reaction was catalyzed by HgO and HgCl2 to form an α-hydroxyamide. Hydrolysis and coupling were combined in one step in 80% yield. Two diastereomers of a phospho-Ser-Pro α-ketoamide analogue were synthesized. The resulting IC50 values of 100 µM and 200 µM were surprisingly weak for the Pin1 peptidyl-prolyl isomerase. Diastereomeric ketones were synthesized by coupling cyclohexenyl lithium to the serine Weinreb amide, via the Michael addition of a carboxylate synthon. The IC50 values of the two ketone diastereomers were determined to be 260 μM and 61 μM, respectively. Five reduced amide inhibitors for Pin1 were synthesized through a selective reduction using borane. The most potent inhibitor was found to be Fmocâ pSerâ Ψ[CH2N]-Proâ tryptamine, which had an IC50 value of 6.3 µM. This represents a 4.5-fold better inhibition for Pin1 than a comparable cis-amide alkene isostere. The co-crystal structure of Acâ pSerâ Ψ[CH2N]-Proâ tryptamine bound to Pin1 was determined to 1.76 Ã resolution. Towards understanding the two proposed mechanisms of Pin1 catalysis, nucleophilic-additition mechanism and twisted-amide mechanism, three classes of Pin1 inhibitors (ketoamide, ketone, and reduced amide analogues) involving a total of nine compounds were synthesized and evaluated. The weak inhibitory activities of ketoamide and ketone analogues do not support the nucleophilic-addition mechanism, while the twisted-amide mechanism of Pin1 catalysis is promising based on the reduced amide inhibitors with good potencies. / Ph. D.

Heat transfer systems contain many sources of nonlinearity including temperature dependent material properties, radiation boundary conditions, and internal source terms. Despite progress in numerical simulations, producing accurate models that can predict these complex behaviors are still encumbered by lengthy processing times. Accurate models can be produced quickly by utilizing projection Reduced Order Modeling (ROM) techniques. For discretized systems, the Singular Value Decomposition technique is the preferred approach but has had limited success on treating nonlinearities. In this research, the treatment of nonlinear temperature dependent material properties was incorporated into a ROM. Additional sources of nonlinearities such as radiation boundary conditions, temperature dependent source heating terms, and complex geometry were also integrated. From the results, low conductivity, highly nonlinear material properties were predicted by the ROM within 1% of full order models, and additional nonlinearities were predicted within 8%. A study was then done to identify initial snapshots for use in developing a ROM that can accurately predict results across a wide range of inputs. From this, a step function was identified as being the most accurate and computationally efficient. The ROM was further investigated by a discretization study to assess computational gains in both 1D and 3D models as a function of mesh density. The lower mesh densities in the 1D and 3D ROMs resulted in moderate computational times (up to 40 times faster). However, highly discretized systems such as 5000 nodes in 1D and 125000 nodes in 3D resulted in computational gains on the order of 2000 to 3000 times faster than the full order model. / Ph. D.

A numerical study of turbulent combustion systems was pursued to examine different computational modeling techniques, namely computational fluid dynamics (CFD) and chemical reactor network (CRN) methods. Both methods have been studied and analyzed as individual techniques as well as a coupled approach to pursue better understandings of the mechanisms and interactions between turbulent flow and mixing, ignition behavior and pollutant formation. A thorough analysis and comparison of both turbulence models and chemistry representation methods was executed and simulations were compared and validated with experimental works. An extensive study of turbulence modeling methods, and the optimization of modeling techniques including turbulence intensity and computational domain size have been conducted. The final CFD model has demonstrated good predictive performance for different turbulent bluff-body flames. The NOx formation and the effects of fuel mixtures indicated that the addition of hydrogen to the fuel and non-flammable diluents like CO2 and H2O contribute to the reduction of NOx. The second part of the study focused on developing chemical models and methods that include the detailed gaseous reaction mechanism of GRI-Mech 3.0 but cost less computational time. A new chemical reactor network has been created based on the CFD results of combustion characteristics and flow fields. The proposed CRN has been validated with the temperature and species emission for different bluff-body flames and has shown the capability of being applied to general bluff-body systems. Specifically, the rate of production of NOx and the sensitivity analysis based on the CRN results helped to summarize the reduced reaction mechanism, which not only provided a promising method to generate representative reactions from hundreds of species and reactions in gaseous mechanism but also presented valuable information of the combustion mechanisms and NOx formation. Finally, the proposed reduced reaction mechanism from the sensitivity analysis was applied to the CFD simulations, which created a fully coupled process between CFD and CRN, and the results from the reduced reaction mechanism have shown good predictions compared with the probability density function method. / Ph. D.

Permanent magnet brushless dc (PMBDC) and synchronous machines (PMSM) drives are favored in variable speed applications for their high efficiency operation. Energy efficiency improvement in such motor drives is of interest in recent times because of rising cost of energy. Accordingly, two current control options for improving efficiency of these drives are taken for study and they are; (i) injecting sinusoidal and non-sinusoidal currents in PMBDC machines and (ii) lowering switching frequency of inverter driving the PMSM but without having significant low ordered sidebands of currents. Both these methods are applicable to existing types of permanent magnet motors and hence do not upset their existing optimized designs. / Ph. D.

This thesis proposes a large eddy simulation reduced order model (LES-ROM) framework for the numerical simulation of realistic flows. In this LES-ROM framework, the proper orthogonal decomposition (POD) is used to define the ROM basis and a POD differential filter is used to define the large ROM structures. An approximate deconvolution (AD) approach is used to solve the ROM closure problem and develop a new AD-ROM. This AD-ROM is tested in the numerical simulation of the one-dimensional Burgers equation with a small diffusion coefficient ( ν= 10⁻³). / Master of Science

This thesis investigates pool boiling heat transfer from bare and graphene-coated NiCr wires in a saturated liquid of FC 770, a fluorocarbon fluid. Of particular interest was the effect of graphene-oxide platelets, dip-coated onto the heater surface, in enhancing the nucleate boiling heat transfer (BHT) rates and the critical heat flux (CHF) value. In the course of the pool boiling experiment, the primary focus was on the reduction mechanism of graphene oxide. The transition from hydrophilic to hydrophobic behavior of the graphene oxide-coated surface was captured, and the attendant effects on surface wettability, porosity and thermal activity were observed. A parametric sensitivity analysis of these surface factors was performed to understand the CHF and BHT enhancement mechanisms. In the presence of graphene-oxide coating, the data indicated an increase of 50% in CHF. As the experiment continued, a partial reduction of graphene oxide occurred, accompanied by (a) further enhancement in the CHF to 77% larger compared to the bare wire. It was shown that the reduction of graphene oxide progressively altered the porosity and thermal conductivity of the coating layer without changing the wettability of FC 770. Further enhancement in CHF was explained in terms of improved porosity and thermal activity that resulted from the partial reduction of graphene-oxide. An implication of these results is that a graphene-oxide coating is potentially a viable option for thermal management of high-power electronics by immersion cooling technology. / Master of Science