Current Search: Singularities (x)
View All Items
(1 - 7 of 7)
- Title
- PROBING RANDOM MEDIA WITH SINGULAR WAVES.
- Creator
-
Schwartz, Chaim, Dogariu, Aristide, University of Central Florida
- Abstract / Description
-
In recent years a resurgence of interest in wave singularities (of which optical vortices are a prominent example), light angular momentum and the relations between them has occurred. Many applications in various areas of linear and non-linear optics have been based on studying effects related to angular momentum and optical vortices. This dissertation examines the use of such wave singularities for studying the light propagation in highly inhomogeneous media and the relationship to angular...
Show moreIn recent years a resurgence of interest in wave singularities (of which optical vortices are a prominent example), light angular momentum and the relations between them has occurred. Many applications in various areas of linear and non-linear optics have been based on studying effects related to angular momentum and optical vortices. This dissertation examines the use of such wave singularities for studying the light propagation in highly inhomogeneous media and the relationship to angular momentum transfer. Angular momentum carried by light can be, in many cases, divided in two terms. The… first one relates to the polarization of light and can be associated, in the quantum description, to the spin of a photon. The second is determined by the electromagnetic …field distribution and, in analogy to atomic physics, is associated with the orbital angular momentum (OAM) of a photon. Under the paraxial approximation appropriate for the case of beam propagation, the two terms do not couple. However, each of them can be modified by the interaction with different media in which the light propagates through processes which involve angular momentum exchange. The decoupling of spin and orbital parts of light angular momentum can not, in general, be assumed for non paraxial propagation in turbid media, especially when backscattering is concerned. In Chapter 3 of this dissertation, scattering effects on angular momentum of light are discussed both for the single and multiple scattering processes. It is demonstrated for the first time that scattering from a spherically symmetric scattering potential, couples the spin and the OAM such that the total angular momentum ‡flux density in conserved in every direction. Remarkably, the conservation of angular momentum occurs also for some classes of multiple scattering trajectories and this phenomenon manifests itself in ubiquitous polarization patterns observed in back-scattering from turbid media. It is newly shown in this dissertation that the polarization patterns a result of OAM carrying optical vortices which have a geometrical origin. These geometrical phase vortices are analyzed using the helicity space approach for optical geometrical phase (Berry phase). This approach, introduced in the con- text of random media, elucidates several aspects specific to propagation in helicity preserving and non-preserving scattering trajectories. Another aspect of singular waves interaction with turbid media relates to singularities embedded in the incident waves. Chapter 4 of the dissertation discusses how the phase distribution associated with an optical vortex leads to changes in the spatial correlations of the electromagnetic field. This change can be used to control the properties of the effect of enhanced backscattering in a way which allows inferring the optical properties of the medium. A detailed theoretical and experimental study of this effect is presented here for the first time for both double-pass geometries and diffusive media. It is also demonstrated that this novel experimental technique can be used to determine the optical properties of turbid media and, moreover, it permits to sense the depth of reflective inclusions in opaque media. When considering a regime of weakly inhomogeneous media, the paraxial approximation is still valid and therefore the spin and OAM do not couple. If, In addition, the medium is optically isotropic then the polarization is not affected. However, when the medium is non-axially symmetric for any specific realization, the OAM does change as a result of interaction with the medium. This effect can be studied using a newly developed method of coherent modes coupling which is presented in Chapter 5. This approach allows studying the power spread across propagating modes which carry different orbital angular momentum. The powerful concept of coherent modes coupling can be applied to fully coherent, fully polarized sources as well to partially coherent, partially polarized ones. An example of this scattering regime is atmospheric turbulence and the propagation through turbulence is thoroughly examined in Chapter 5. The results included in this dissertation are of fundamental relevance for a variety of applications which involves probing different types of random media. Such applications include remote sensing in atmospheric and maritime environments, optical techniques for biomedical diagnostics, optical characterization procedures in material sciences and others.
Show less - Date Issued
- 2006
- Identifier
- CFE0001174, ucf:46852
- Format
- Document (PDF)
- PURL
- http://purl.flvc.org/ucf/fd/CFE0001174
- Title
- Weighted Low-Rank Approximation of Matrices:Some Analytical and Numerical Aspects.
- Creator
-
Dutta, Aritra, Li, Xin, Sun, Qiyu, Mohapatra, Ram, Nashed, M, Shah, Mubarak, University of Central Florida
- Abstract / Description
-
This dissertation addresses some analytical and numerical aspects of a problem of weighted low-rank approximation of matrices. We propose and solve two different versions of weighted low-rank approximation problems. We demonstrate, in addition, how these formulations can be efficiently used to solve some classic problems in computer vision. We also present the superior performance of our algorithms over the existing state-of-the-art unweighted and weighted low-rank approximation algorithms...
Show moreThis dissertation addresses some analytical and numerical aspects of a problem of weighted low-rank approximation of matrices. We propose and solve two different versions of weighted low-rank approximation problems. We demonstrate, in addition, how these formulations can be efficiently used to solve some classic problems in computer vision. We also present the superior performance of our algorithms over the existing state-of-the-art unweighted and weighted low-rank approximation algorithms.Classical principal component analysis (PCA) is constrained to have equal weighting on the elements of the matrix, which might lead to a degraded design in some problems. To address this fundamental flaw in PCA, Golub, Hoffman, and Stewart proposed and solved a problem of constrained low-rank approximation of matrices: For a given matrix $A = (A_1\;A_2)$, find a low rank matrix $X = (A_1\;X_2)$ such that ${\rm rank}(X)$ is less than $r$, a prescribed bound, and $\|A-X\|$ is small.~Motivated by the above formulation, we propose a weighted low-rank approximation problem that generalizes the constrained low-rank approximation problem of Golub, Hoffman and Stewart.~We study a general framework obtained by pointwise multiplication with the weight matrix and consider the following problem:~For a given matrix $A\in\mathbb{R}^{m\times n}$ solve:\begin{eqnarray*}\label{weighted problem}\min_{\substack{X}}\|\left(A-X\right)\odot W\|_F^2~{\rm subject~to~}{\rm rank}(X)\le r,\end{eqnarray*}where $\odot$ denotes the pointwise multiplication and $\|\cdot\|_F$ is the Frobenius norm of matrices.In the first part, we study a special version of the above general weighted low-rank approximation problem.~Instead of using pointwise multiplication with the weight matrix, we use the regular matrix multiplication and replace the rank constraint by its convex surrogate, the nuclear norm, and consider the following problem:\begin{eqnarray*}\label{weighted problem 1}\hat{X} (&)=(&) \arg \min_X \{\frac{1}{2}\|(A-X)W\|_F^2 +\tau\|X\|_\ast\},\end{eqnarray*}where $\|\cdot\|_*$ denotes the nuclear norm of $X$.~Considering its resemblance with the classic singular value thresholding problem we call it the weighted singular value thresholding~(WSVT)~problem.~As expected,~the WSVT problem has no closed form analytical solution in general,~and a numerical procedure is needed to solve it.~We introduce auxiliary variables and apply simple and fast alternating direction method to solve WSVT numerically.~Moreover, we present a convergence analysis of the algorithm and propose a mechanism for estimating the weight from the data.~We demonstrate the performance of WSVT on two computer vision applications:~background estimation from video sequences~and facial shadow removal.~In both cases,~WSVT shows superior performance to all other models traditionally used. In the second part, we study the general framework of the proposed problem.~For the special case of weight, we study the limiting behavior of the solution to our problem,~both analytically and numerically.~In the limiting case of weights,~as $(W_1)_{ij}\to\infty, W_2=\mathbbm{1}$, a matrix of 1,~we show the solutions to our weighted problem converge, and the limit is the solution to the constrained low-rank approximation problem of Golub et. al. Additionally, by asymptotic analysis of the solution to our problem,~we propose a rate of convergence.~By doing this, we make explicit connections between a vast genre of weighted and unweighted low-rank approximation problems.~In addition to these, we devise a novel and efficient numerical algorithm based on the alternating direction method for the special case of weight and present a detailed convergence analysis.~Our approach improves substantially over the existing weighted low-rank approximation algorithms proposed in the literature.~Finally, we explore the use of our algorithm to real-world problems in a variety of domains, such as computer vision and machine learning. Finally, for a special family of weights, we demonstrate an interesting property of the solution to the general weighted low-rank approximation problem. Additionally, we devise two accelerated algorithms by using this property and present their effectiveness compared to the algorithm proposed in Chapter 4.
Show less - Date Issued
- 2016
- Identifier
- CFE0006833, ucf:51789
- Format
- Document (PDF)
- PURL
- http://purl.flvc.org/ucf/fd/CFE0006833
- Title
- INFORMATION RETRIEVAL PERFORMANCE ENHANCEMENT USING THE AVERAGE STANDARD ESTIMATOR AND THE MULTI-CRITERIA DECISION WEIGHTED SET OF PERFORMANCE MEASURES.
- Creator
-
AHRAM, TAREQ, McCauley-Bush, Pamela, University of Central Florida
- Abstract / Description
-
Information retrieval is much more challenging than traditional small document collection retrieval. The main difference is the importance of correlations between related concepts in complex data structures. These structures have been studied by several information retrieval systems. This research began by performing a comprehensive review and comparison of several techniques of matrix dimensionality estimation and their respective effects on enhancing retrieval performance using singular...
Show moreInformation retrieval is much more challenging than traditional small document collection retrieval. The main difference is the importance of correlations between related concepts in complex data structures. These structures have been studied by several information retrieval systems. This research began by performing a comprehensive review and comparison of several techniques of matrix dimensionality estimation and their respective effects on enhancing retrieval performance using singular value decomposition and latent semantic analysis. Two novel techniques have been introduced in this research to enhance intrinsic dimensionality estimation, the Multi-criteria Decision Weighted model to estimate matrix intrinsic dimensionality for large document collections and the Average Standard Estimator (ASE) for estimating data intrinsic dimensionality based on the singular value decomposition (SVD). ASE estimates the level of significance for singular values resulting from the singular value decomposition. ASE assumes that those variables with deep relations have sufficient correlation and that only those relationships with high singular values are significant and should be maintained. Experimental results over all possible dimensions indicated that ASE improved matrix intrinsic dimensionality estimation by including the effect of both singular values magnitude of decrease and random noise distracters. Analysis based on selected performance measures indicates that for each document collection there is a region of lower dimensionalities associated with improved retrieval performance. However, there was clear disagreement between the various performance measures on the model associated with best performance. The introduction of the multi-weighted model and Analytical Hierarchy Processing (AHP) analysis helped in ranking dimensionality estimation techniques and facilitates satisfying overall model goals by leveraging contradicting constrains and satisfying information retrieval priorities. ASE provided the best estimate for MEDLINE intrinsic dimensionality among all other dimensionality estimation techniques, and further, ASE improved precision and relative relevance by 10.2% and 7.4% respectively. AHP analysis indicates that ASE and the weighted model ranked the best among other methods with 30.3% and 20.3% in satisfying overall model goals in MEDLINE and 22.6% and 25.1% for CRANFIELD. The weighted model improved MEDLINE relative relevance by 4.4%, while the scree plot, weighted model, and ASE provided better estimation of data intrinsic dimensionality for CRANFIELD collection than Kaiser-Guttman and Percentage of variance. ASE dimensionality estimation technique provided a better estimation of CISI intrinsic dimensionality than all other tested methods since all methods except ASE tend to underestimate CISI document collection intrinsic dimensionality. ASE improved CISI average relative relevance and average search length by 28.4% and 22.0% respectively. This research provided evidence supporting a system using a weighted multi-criteria performance evaluation technique resulting in better overall performance than a single criteria ranking model. Thus, the weighted multi-criteria model with dimensionality reduction provides a more efficient implementation for information retrieval than using a full rank model.
Show less - Date Issued
- 2008
- Identifier
- CFE0002426, ucf:47747
- Format
- Document (PDF)
- PURL
- http://purl.flvc.org/ucf/fd/CFE0002426
- Title
- INITIAL-VALUE TECHNIQUE FOR SINGULARLY PERTURBED TWO POINT BOUNDARY VALUE PROBLEMS VIA CUBIC SPLINE.
- Creator
-
Negron, Luis, Mohapatra, Ram, University of Central Florida
- Abstract / Description
-
A recent method for solving singular perturbation problems is examined. It is designed for the applied mathematician or engineer who needs a convenient, useful tool that requires little preparation and can be readily implemented using little more than an industry-standard software package for spreadsheets. In this paper, we shall examine singularly perturbed two point boundary value problems with the boundary layer at one end point. An initial-value technique is used for its solution by...
Show moreA recent method for solving singular perturbation problems is examined. It is designed for the applied mathematician or engineer who needs a convenient, useful tool that requires little preparation and can be readily implemented using little more than an industry-standard software package for spreadsheets. In this paper, we shall examine singularly perturbed two point boundary value problems with the boundary layer at one end point. An initial-value technique is used for its solution by replacing the problem with an asymptotically equivalent first order problem, which is, in turn, solved as an initial value problem by using cubic splines. Numerical examples are provided to show that the method presented provides a fine approximation of the exact solution. The first chapter provides some background material to the cubic spline and boundary value problems. The works of several authors and a comparison of different solution methods are also discussed. Finally, some background into the specific singularly perturbed boundary value problems is introduced. The second chapter contains calculations and derivations necessary for the cubic spline and the initial value technique which are used in the solutions to the boundary value problems. The third chapter contains some worked numerical examples and the numerical data obtained along with most of the tables and figures that describe the solutions. The thesis concludes with some reflections on the results obtained and some discussion of the error bounds on the calculated approximations to the exact solutions for the numeric examples discussed.
Show less - Date Issued
- 2010
- Identifier
- CFE0003460, ucf:48398
- Format
- Document (PDF)
- PURL
- http://purl.flvc.org/ucf/fd/CFE0003460
- Title
- Buidling Lax Integrable Variable-Coefficient Generalizations to Integrable PDEs and Exact Solutions to Nonlinear PDEs.
- Creator
-
Russo, Matthew, Choudhury, Sudipto, Moore, Brian, Schober, Constance, Christodoulides, Demetrios, University of Central Florida
- Abstract / Description
-
This dissertation is composed of two parts. In Part I a technique based on extended Lax Pairs isfirst considered to derive variable-coefficient generalizations of various Lax-integrable NLPDE hierarchies recently introduced in the literature. It is demonstrated that the technique yields Lax- or S-integrable nonlinear partial differential equations (PDEs) with both time- and space-dependent coefficients which are thus more general than almost all cases considered earlier via other methods such...
Show moreThis dissertation is composed of two parts. In Part I a technique based on extended Lax Pairs isfirst considered to derive variable-coefficient generalizations of various Lax-integrable NLPDE hierarchies recently introduced in the literature. It is demonstrated that the technique yields Lax- or S-integrable nonlinear partial differential equations (PDEs) with both time- and space-dependent coefficients which are thus more general than almost all cases considered earlier via other methods such as the Painleve Test, Bell Polynomials, and various similarity methods. However, this technique, although operationally effective, has the significant disadvantage that, for any integrable system with spatiotemporally varying coefficients, one must 'guess' a generalization of the structure of the known Lax Pair for the corresponding system with constant coefficients. Motivated by the somewhat arbitrary nature of the above procedure, we present a generalization to the well known Estabrook-Wahlquist prolongation technique which provides a systematic procedure for the derivation of the Lax representation. In order to obtain a nontrivial Lax representation we must impose differential constraints on the variable coefficients present in the nlpde. The resulting constraints determine a class of equations which represent generalizations to a previously known integrable constant coefficient nlpde. We demonstrate the effectiveness of this technique by deriving variable-coefficient generalizations to the nonlinear Schrodinger (NLS) equation, derivative NLS equation, PT-symmetric NLS, fifth-order KdV, and three equations in the MKdV hierarchy. In Part II of this dissertation, we introduce three types of singular manifold methods which have been successfully used in the literature to derive exact solutions to many nonlinear PDEs extending over a wide range of applications. The singular manifold methods considered are: truncated Painleve analysis, Invariant Painleve analysis, and a generalized Hirota expansion method. We then consider the KdV and KP-II equations as instructive examples before using each method to derive nontrivial solutions to a microstructure PDE and two generalized Pochhammer-Chree equations.
Show less - Date Issued
- 2016
- Identifier
- CFE0006173, ucf:51144
- Format
- Document (PDF)
- PURL
- http://purl.flvc.org/ucf/fd/CFE0006173
- Title
- Implementation And Performance Comparisons For The Crisfield And Stiff Arc Length Methods In FEA.
- Creator
-
Silvers, Thomas, Gordon, Ali, Nicholson, David, Kassab, Alain, Ilie, Marcel, University of Central Florida
- Abstract / Description
-
In Nonlinear Finite Element Analysis (FEA) applied to structures, displacements at which the tangent stiffness matrix KT becomes singular are called critical points, and correspond to instabilities such as buckling or elastoplastic softening (e.g., necking). Prior to the introduction of Arc Length Methods (ALMs), critical points posed severe computational challenges, which was unfortunate since behavior at instabilities is of great interest as a precursor to structural failure. The original...
Show moreIn Nonlinear Finite Element Analysis (FEA) applied to structures, displacements at which the tangent stiffness matrix KT becomes singular are called critical points, and correspond to instabilities such as buckling or elastoplastic softening (e.g., necking). Prior to the introduction of Arc Length Methods (ALMs), critical points posed severe computational challenges, which was unfortunate since behavior at instabilities is of great interest as a precursor to structural failure. The original ALM was shown to be capable in some circumstances of continued computation at critical points, but limited success and unattractive features of the formulation were noted and addressed in extensive subsequent research. The widely used Crisfield Cylindrical and Spherical ALMs may be viewed as representing the 'state-of-the-art'. The more recent Stiff Arc Length method, which is attractive on fundamental grounds, was introduced in 2004, but without implementation, benchmarking or performance assessment. The present thesis addresses (a) implementation and (b) performance comparisons for the Crisfield and Stiff methods, using simple benchmarks formulated to incorporate elastoplastic softening. It is seen that, in contrast to the Crisfield methods, the Stiff ALM consistently continues accurate computation at, near and beyond critical points.
Show less - Date Issued
- 2012
- Identifier
- CFE0004277, ucf:49544
- Format
- Document (PDF)
- PURL
- http://purl.flvc.org/ucf/fd/CFE0004277
- Title
- Automated Hybrid Singularity Superposition and Anchored Grid Pattern BEM Algorithm for the Solution of the Inverse Geometric Problem.
- Creator
-
Ni, Marcus, Kassab, Alain, Divo, Eduardo, Chopra, Manoj, University of Central Florida
- Abstract / Description
-
A method for solving the inverse geometrical problem is presented by reconstructing the unknown subsurface cavity geometry using boundary element methods, a genetic algorithm, and Nelder-Mead non-linear simplex optimization. The heat conduction problem is solved utilizing the boundary element method, which calculates the difference between the measured temperature at the exposed surface and the computed temperature under the current update of the unknown subsurface flaws and cavities. In a...
Show moreA method for solving the inverse geometrical problem is presented by reconstructing the unknown subsurface cavity geometry using boundary element methods, a genetic algorithm, and Nelder-Mead non-linear simplex optimization. The heat conduction problem is solved utilizing the boundary element method, which calculates the difference between the measured temperature at the exposed surface and the computed temperature under the current update of the unknown subsurface flaws and cavities. In a first step, clusters of singularities are utilized to solve the inverse problem and to identify the location of the centroid(s) of the subsurface cavity(ies)/flaw(s). In a second step, the reconstruction of the estimated cavity(ies)/flaw(s) geometry(ies) is accomplished by utilizing an anchored grid pattern upon which cubic spline knots are restricted to move in the search for unknown geometry. Solution of the inverse problem is achieved using a genetic algorithm accelerated with the Nelder-Mead non-linear simplex. To optimize the cubic spline interpolated geometry, the flux (Neumann) boundary conditions are minimized using a least squares functional. The automated algorithm successfully reconstructs single and multiple subsurface cavities within two dimensional mediums. The solver is also shown to accurately predict cavity geometries with random noise in the boundary condition measurements. Subsurface cavities can be difficult to detect based on their location. By applying different boundary conditions to the same geometry, more information is supplied at the boundary, and the subsurface cavity is easily detected despite its low heat signature effect at the boundaries. Extensions to three-dimensional applications are outlined.
Show less - Date Issued
- 2013
- Identifier
- CFE0004900, ucf:49644
- Format
- Document (PDF)
- PURL
- http://purl.flvc.org/ucf/fd/CFE0004900
