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ANALYSIS OF KOLMOGOROV'S SUPERPOSITION THEOREM AND ITS IMPLEMENTATION IN APPLICATIONS WITH LOW AND HIGH DIMENSIONAL DATA.
Title
ANALYSIS OF KOLMOGOROV'S SUPERPOSITION THEOREM AND ITS IMPLEMENTATION IN APPLICATIONS WITH LOW AND HIGH DIMENSIONAL DATA.
Creator
Bryant, Donald, Li, Xin, University of Central Florida
Abstract / Description
In this dissertation, we analyze Kolmogorov's superposition theorem for high dimensions. Our main goal is to explore and demonstrate the feasibility of an accurate implementation of Kolmogorov's theorem. First, based on Lorentz's ideas, we provide a thorough discussion on the proof and its numerical implementation of the theorem in dimension two. We present computational experiments which prove the feasibility of the theorem in applications of low dimensions (namely, dimensions...
Show moreIn this dissertation, we analyze Kolmogorov's superposition theorem for high dimensions. Our main goal is to explore and demonstrate the feasibility of an accurate implementation of Kolmogorov's theorem. First, based on Lorentz's ideas, we provide a thorough discussion on the proof and its numerical implementation of the theorem in dimension two. We present computational experiments which prove the feasibility of the theorem in applications of low dimensions (namely, dimensions two and three). Next, we present high dimensional extensions with complete and detailed proofs and provide the implementation that aims at applications with high dimensionality. The amalgamation of these ideas is evidenced by applications in image (two dimensional) and video (three dimensional) representations, the content based image retrieval, video retrieval, de-noising and in-painting, and Bayesian prior estimation of high dimensional data from the fields of computer vision and image processing.
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Date Issued
2008
Identifier
CFE0002236, ucf:47909
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0002236
Automated Hybrid Singularity Superposition and Anchored Grid Pattern BEM Algorithm for the Solution of the Inverse Geometric Problem
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.
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Date Issued
2013
Identifier
CFE0004900, ucf:49644
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0004900