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Variational inclusions with general over-relaxed proximal point and variational-like inequalities with densely pseudomonotonicity
Title
Variational inclusions with general over-relaxed proximal point and variational-like inequalities with densely pseudomonotonicity.
Creator
Nguyen, George, Mohapatra, Ram, Han, Deguang, Shuai, Zhisheng, Xu, Mengyu, University of Central Florida
Abstract / Description
This dissertation focuses on the existence and uniqueness of the solutions of variational inclusion and variational inequality problems and then attempts to develop efficient algorithms to estimate numerical solutions for the problems. The dissertation consists a total of five chapters. Chapter 1 is an introduction to variational inequality problems, variational inclusion problems, monotone operators, and some basic definitions and preliminaries from convex analysis. Chapter 2 is a studyof a...
Show moreThis dissertation focuses on the existence and uniqueness of the solutions of variational inclusion and variational inequality problems and then attempts to develop efficient algorithms to estimate numerical solutions for the problems. The dissertation consists a total of five chapters. Chapter 1 is an introduction to variational inequality problems, variational inclusion problems, monotone operators, and some basic definitions and preliminaries from convex analysis. Chapter 2 is a studyof a general class of nonlinear implicit inclusion problems. The objective of this study is to explore how to omit the Lipschitz continuity condition by using an alternating approach to the proximal point algorithm to estimate the numerical solution of the implicit inclusion problems. In chapter 3 we introduce generalized densely relaxed ? ? ? pseudomonotone operators and generalized relaxed ? ? ? proper quasimonotone operators as well as relaxed ? ? ? quasimonotone operators. Using these generalized monotonicity notions, we establish the existence results for the generalized variational-like inequality in the general setting of Banach spaces. In chapter 4, we use the auxiliary principle technique to introduce a general algorithm for solutions of the densely relaxed pseudomonotone variational-like inequalities. Chapter 5 is the chapter concluding remarks and scope for future work.
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Date Issued
2019
Identifier
CFE0007693, ucf:52410
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0007693
In Quest of Bernstein Inequalities Rational Functions, Askey-Wilson Operator, and Summation Identities for Entire Functions
Title
In Quest of Bernstein Inequalities Rational Functions, Askey-Wilson Operator, and Summation Identities for Entire Functions.
Creator
Puwakgolle Gedara, Rajitha, Li, Xin, Mohapatra, Ram, Ismail, Mourad, Xu, Mengyu, University of Central Florida
Abstract / Description
The title of the dissertation gives an indication of the material involved with the connecting thread throughout being the classical Bernstein inequality (and its variants), which provides an estimate to the size of the derivative of a given polynomial on a prescribed set in the complex plane, relative to the size of the polynomial itself on the same set. Chapters 1 and 2 lay the foundation for the dissertation. In Chapter 1, we introduce the notations and terminology that will be used...
Show moreThe title of the dissertation gives an indication of the material involved with the connecting thread throughout being the classical Bernstein inequality (and its variants), which provides an estimate to the size of the derivative of a given polynomial on a prescribed set in the complex plane, relative to the size of the polynomial itself on the same set. Chapters 1 and 2 lay the foundation for the dissertation. In Chapter 1, we introduce the notations and terminology that will be used throughout. Also a brief historical recount is given on the origin of the Bernstein inequality, which dated back to the days of the discovery of the Periodic table by the Russian Chemist Dmitri Mendeleev. In Chapter 2, we narrow down the contents stated in Chapter 1 to the problems we were interested in working during the course of this dissertation. Henceforth, we present a problem formulation mainly for those results for which solutions or partial solutions are provided in the subsequent chapters.Over the years Bernstein inequality has been generalized and extended in several directions. In Chapter \ref{Bern-ineq}, we establish rational analogues to some Bernstein-type inequalities for restricted zeros and prescribed poles. Our inequalities extend the results for polynomials, especially which are themselves improved versions of the classical Erd\"{o}s-Lax and Tur\'{a}n inequalities. In working towards proving our results, we establish some auxiliary results, which may be of interest on their own. Chapters \ref{AW-on-polynomials} and \ref{AW-on-entire} focus on the research carried out with the Askey-Wilson operator applied on polynomials and entire functions (of exponential type) respectively.In Chapter 4, we first establish a Riesz-type interpolation formula on the interval $[-1,1]$ for the Askey-Wilson operator. In consequence, a sharp Bernstein inequality and a Markov inequality are obtained when differentiation is replaced by the Askey-Wilson operator. Moreover, an inverse approximation theorem is proved using a Bernstein-type inequality in $L^2-$space. We conclude this chapter with an overconvergence result which is applied to characterize all $q$-differentiable functions of Brown and Ismail. Chapter \ref{AW-on-entire} is devoted to an intriguing application of the Askey-Wilson operator. By applying it on the Sampling Theorem on entire functions of exponential type, we obtain a series representation formula, which is what we called an extended Boas' formula. Its power in discovering interesting summation formulas, some known and some new will be demonstrated. As another application, we are able to obtain a couple of Bernstein-type inequalities.In the concluding chapter, we state some avenues where this research can progress.
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Date Issued
2018
Identifier
CFE0007237, ucf:52220
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0007237
Viscous Dissipation Effects On Acoustic Instabilities In Combustion Chambers
Title
Viscous Dissipation Effects On Acoustic Instabilities In Combustion Chambers.
Creator
Flores, Wilmer, Ahmed, Kareem, Kapat, Jayanta, Bhattacharya, Samik, Xu, Mengyu, University of Central Florida
Abstract / Description
Combustion chambers are naturally prone to acoustic instabilities that originate from flame propagation. Passive devices such as combustor chamber baffles, resonators, and injection liners have proven to attenuate acoustic instabilities degradate the integrity of engine components. Acoustic energy viscous dissipation effects are measured and quantified for new designs and arrangements implemented in tested suppression devices. Two passive suppression devices are introduced which exhibit new...
Show moreCombustion chambers are naturally prone to acoustic instabilities that originate from flame propagation. Passive devices such as combustor chamber baffles, resonators, and injection liners have proven to attenuate acoustic instabilities degradate the integrity of engine components. Acoustic energy viscous dissipation effects are measured and quantified for new designs and arrangements implemented in tested suppression devices. Two passive suppression devices are introduced which exhibit new baffle arrangement and combustion liner design. Audio acoustic equipment excites chamber acoustic instabilities and microphones receive acoustic pressure wave amplitudes. Using this technique viscous damping effects from acoustic sound waves are measured in un-reacting static and flow conditions. An extensive study on damping enhancements to tangential acoustic mode instabilities was explored. A baffle insert was designed with staggered offset injector baffle blades to evaluate viscous damping effects on tangential acoustic instabilities. Tangential acoustic wave energy dissipation is characterized through decay rates measurements. It was concluded that a staggered offset baffle blades with a constant outer versus inner varying injector exhibits the highest attenuation rate. Changes to baffle blades shows a 2T mode experiences the greatest damping enhancement. An empirical expression is derived from curve fitting decay rates for tangential modes and demonstrates acoustic behavior to follow a non-linear correlation. A new auxetic s-shape structure is incorporated into a combustion liner that was coupled with a Helmholtz resonator. The investigation focuses on viscous damping acoustic effects comparing circles to auxetic designs within grazing and bias flow conditions. A series of experiments were conducted that characterized flow discharge behavior, acoustic impedance, acoustic rig that couples bias and grazing flow. Auxetic designs display enhanced absorption qualities at high frequency bandwidths compared to traditional circles. S-shapes with a 60(&)deg; injection angle demonstrates superior viscous damping absorption characteristics. A higher differential pressure highlights a reduction in absorption coefficient measurements.
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Date Issued
2019
Identifier
CFE0007630, ucf:52514
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0007630
Automated Synthesis of Memristor Crossbar Networks
Title
Automated Synthesis of Memristor Crossbar Networks.
Creator
Chakraborty, Dwaipayan, Jha, Sumit Kumar, Leavens, Gary, Ewetz, Rickard, Valliyil Thankachan, Sharma, Xu, Mengyu, University of Central Florida
Abstract / Description
The advancement of semiconductor device technology over the past decades has enabled the design of increasingly complex electrical and computational machines. Electronic design automation (EDA) has played a significant role in the design and implementation of transistor-based machines. However, as transistors move closer toward their physical limits, the speed-up provided by Moore's law will grind to a halt. Once again, we find ourselves on the verge of a paradigm shift in the computational...
Show moreThe advancement of semiconductor device technology over the past decades has enabled the design of increasingly complex electrical and computational machines. Electronic design automation (EDA) has played a significant role in the design and implementation of transistor-based machines. However, as transistors move closer toward their physical limits, the speed-up provided by Moore's law will grind to a halt. Once again, we find ourselves on the verge of a paradigm shift in the computational sciences as newer devices pave the way for novel approaches to computing. One of such devices is the memristor -- a resistor with non-volatile memory.Memristors can be used as junctional switches in crossbar circuits, which comprise of intersecting sets of vertical and horizontal nanowires. The major contribution of this dissertation lies in automating the design of such crossbar circuits -- doing a new kind of EDA for a new kind of computational machinery. In general, this dissertation attempts to answer the following questions:a. How can we synthesize crossbars for computing large Boolean formulas, up to 128-bit?b. How can we synthesize more compact crossbars for small Boolean formulas, up to 8-bit?c. For a given loop-free C program doing integer arithmetic, is it possible to synthesize an equivalent crossbar circuit?We have presented novel solutions to each of the above problems. Our new, proposed solutions resolve a number of significant bottlenecks in existing research, via the usage of innovative logic representation and artificial intelligence techniques. For large Boolean formulas (up to 128-bit), we have utilized Reduced Ordered Binary Decision Diagrams (ROBDDs) to automatically synthesize linearly growing crossbar circuits that compute them. This cutting edge approach towards flow-based computing has yielded state-of-the-art results. It is worth noting that this approach is scalable to n-bit Boolean formulas. We have made significant original contributions by leveraging artificial intelligence for automatic synthesis of compact crossbar circuits. This inventive method has been expanded to encompass crossbar networks with 1D1M (1-diode-1-memristor) switches, as well. The resultant circuits satisfy the tight constraints of the Feynman Grand Prize challenge and are able to perform 8-bit binary addition. A leading edge development for end-to-end computation with flow-based crossbars has been implemented, which involves methodical translation of loop-free C programs into crossbar circuits via automated synthesis. The original contributions described in this dissertation reflect the substantial progress we have made in the area of electronic design automation for synthesis of memristor crossbar networks.
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Date Issued
2019
Identifier
CFE0007609, ucf:52528
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0007609
Sampling and Reconstruction of Spatial Signals
Title
Sampling and Reconstruction of Spatial Signals.
Creator
Cheng, Cheng, Li, Xin, Sun, Qiyu, Yong, Jiongmin, Liu, Zhe, Xu, Mengyu, University of Central Florida
Abstract / Description
Digital processing of signals f may start from sampling on a discrete set ?, f ?? f(?_n), ?_n ??.The sampling theory is one of the most basic and fascinating topics in applied mathematics and in engineering sciences. The most well known form is the uniform sampling theorem for band-limited/wavelet signals, that gives a framework for converting analog signals into sequences of numbers. Over the past decade, the sampling theory has undergone a strong revival and the standard sampling paradigm...
Show moreDigital processing of signals f may start from sampling on a discrete set ?, f ?? f(?_n), ?_n ??.The sampling theory is one of the most basic and fascinating topics in applied mathematics and in engineering sciences. The most well known form is the uniform sampling theorem for band-limited/wavelet signals, that gives a framework for converting analog signals into sequences of numbers. Over the past decade, the sampling theory has undergone a strong revival and the standard sampling paradigm is extended to non-bandlimited signals including signals in reproducing kernel spaces (RKSs), signals with finite rate of innovation (FRI) and sparse signals, and to nontraditional sampling methods, such as phaseless sampling.In this dissertation, we first consider the sampling and Galerkin reconstruction in a reproducing kernel space. The fidelity measure of perceptual signals, such as acoustic and visual signals, might not be well measured by least squares. In the first part of this dissertation, we introduce a fidelity measure depending on a given sampling scheme and propose a Galerkin method in Banach space setting for signal reconstruction. We show that the proposed Galerkin method provides a quasi-optimal approximation, and the corresponding Galerkin equations could be solved by an iterative approximation-projection algorithm in a reproducing kernel subspace of Lp.A spatially distributed network contains a large amount of agents with limited sensing, data processing, and communication capabilities. Recent technological advances have opened up possibilities to deploy spatially distributed networks for signal sampling and reconstruction. We introduce a graph structure for a distributed sampling and reconstruction system by coupling agents in a spatially distributed network with innovative positions of signals. We split a distributed sampling and reconstruction system into a family of overlapping smaller subsystems, and we show that the stability of the sensing matrix holds if and only if its quasi-restrictions to those subsystems have l_2 uniform stability. This new stability criterion could be pivotal for the design of a robust distributed sampling and reconstruction system against supplement, replacement and impairment of agents, as we only need to check the uniform stability of affected subsystems. We also propose an exponentially convergent distributed algorithm for signal reconstruction, that provides a suboptimal approximation to the original signal in the presence of bounded sampling noises.Phase retrieval (Phaseless Sampling and Reconstruction) arises in various fields of science and engineering. It consists of reconstructing a signal of interest from its magnitude measurements. Sampling in shift-invariant spaces is a realistic model for signals with smooth spectrum. We consider phaseless sampling and reconstruction of real-valued signals in a shift-invariant space from their magnitude measurements on the whole Euclidean space and from their phaseless samples taken on a discrete set with finite sampling density. We find an equivalence between nonseparability of signals in a shift-invariant space and their phase retrievability with phaseless samples taken on the whole Euclidean space. We also introduce an undirected graph to a signal and use connectivity of the graph to characterize the nonseparability of high-dimensional signals. Under the local complement property assumption on a shift-invariant space, we find a discrete set with finite sampling density such that signals in shift-invariant spaces, that are determined by their magnitude measurements on the whole Euclidean space, can be reconstructed in a stable way from their phaseless samples taken on that discrete set. We also propose a reconstruction algorithm which provides a suboptimal approximation to the original signal when its noisy phaseless samples are available only.
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Date Issued
2017
Identifier
CFE0006726, ucf:51889
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0006726