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Can One Hear...? An Exploration Into Inverse Eigenvalue Problems Related to Musical Instruments
Title
Can One Hear...? An Exploration Into Inverse Eigenvalue Problems Related to Musical Instruments.
Creator
Adams, Christine, Nashed, M, Mohapatra, Ram, Kaup, David, University of Central Florida
Abstract / Description
The central theme of this thesis deals with problems related to the question, (")Can one hear the shape of a drum?(") first posed formally by Mark Kac in 1966. More precisely, can one determine the shape of a membrane with fixed boundary from the spectrum of the associated differential operator? For this paper, Kac received both the Lester Ford Award and the Chauvant Prize of the Mathematical Association of America. This problem has received a great deal of attention in the past forty years...
Show moreThe central theme of this thesis deals with problems related to the question, (")Can one hear the shape of a drum?(") first posed formally by Mark Kac in 1966. More precisely, can one determine the shape of a membrane with fixed boundary from the spectrum of the associated differential operator? For this paper, Kac received both the Lester Ford Award and the Chauvant Prize of the Mathematical Association of America. This problem has received a great deal of attention in the past forty years and has led to similar questions in completely different contexts such as (")Can one hear the shape of a graph associated with the Schr(&)#246;dinger operator?("), (")Can you hear the shape of your throat?("), (")Can you feel the shape of a manifold with Brownian motion?("), (")Can one hear the crack in a beam?("), (")Can one hear into the sun?("), etc. Each of these topics deals with inverse eigenvalue problems or related inverse problems. For inverse problems in general, the problem may or may not have a solution, the solution may not be unique, and the solution does not necessarily depend continuously on perturbation of the data. For example, in the case of the drum, it has been shown that the answer to Kac's question in general is (")no.(") However, if we restrict the class of drums, then the answer can be yes. This is typical of inverse problems when a priori information and restriction of the class of admissible solutions and/or data are used to make the problem well-posed. This thesis provides an analysis of shapes for which the answer to Kac's question is positive and a variety of interesting questions on this problem and its variants, including cases that remain open. This thesis also provides a synopsis and perspectives of other types of (")can one hear(") problems mentioned above. Another part of this thesis deals with aspects of direct problems related to musical instruments.
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Date Issued
2013
Identifier
CFE0004643, ucf:49886
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0004643
Solution of linear ill-posed problems using overcomplete dictionaries
Title
Solution of linear ill-posed problems using overcomplete dictionaries.
Creator
Gupta, Pawan, Pensky, Marianna, Swanson, Jason, Zhang, Teng, Foroosh, Hassan, University of Central Florida
Abstract / Description
In this dissertation, we consider an application of overcomplete dictionaries to the solution of general ill-posed linear inverse problems. In the context of regression problems, there has been an enormous amount of effort to recover an unknown function using such dictionaries. While some research on the subject has been already carried out, there are still many gaps to address. In particular, one of the most popular methods, lasso, and its variants, is based on minimizing the empirical...
Show moreIn this dissertation, we consider an application of overcomplete dictionaries to the solution of general ill-posed linear inverse problems. In the context of regression problems, there has been an enormous amount of effort to recover an unknown function using such dictionaries. While some research on the subject has been already carried out, there are still many gaps to address. In particular, one of the most popular methods, lasso, and its variants, is based on minimizing the empirical likelihood and unfortunately, requires stringent assumptions on the dictionary, the so-called, compatibility conditions. Though compatibility conditions are hard to satisfy, it is well known that this can be accomplished by using random dictionaries. In the first part of the dissertation, we show how one can apply random dictionaries to the solution of ill-posed linear inverse problems with Gaussian noise. We put a theoretical foundation under the suggested methodology and study its performance via simulations and real-data example. In the second part of the dissertation, we investigate the application of lasso to the linear ill-posed problems with non-Gaussian noise. We have developed a theoretical background for the application of lasso to such problems and studied its performance via simulations.
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Date Issued
2019
Identifier
CFE0007811, ucf:52345
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0007811
APPLICATION OF TRAINED POD-RBF TO INTERPOLATION IN HEAT TRANSFER AND FLUID MECHANICS
Title
APPLICATION OF TRAINED POD-RBF TO INTERPOLATION IN HEAT TRANSFER AND FLUID MECHANICS.
Creator
Ashley, Rebecca A, Kassab, Alain, University of Central Florida
Abstract / Description
To accurately model or predict future operating conditions of a system in engineering or applied mechanics, it is necessary to understand its fundamental principles. These may be the material parameters, defining dimensional characteristics, or the boundary conditions. However, there are instances when there is little to no prior knowledge of the system properties or conditions, and consequently, the problem cannot be modeled accurately. It is therefore critical to define a method that can...
Show moreTo accurately model or predict future operating conditions of a system in engineering or applied mechanics, it is necessary to understand its fundamental principles. These may be the material parameters, defining dimensional characteristics, or the boundary conditions. However, there are instances when there is little to no prior knowledge of the system properties or conditions, and consequently, the problem cannot be modeled accurately. It is therefore critical to define a method that can identify the desired characteristics of the current system without accumulating extensive computation time. This thesis formulates an inverse approach using proper orthogonal decomposition (POD) with an accompanying radial basis function (RBF) interpolation network. This method is capable of predicting the desired characteristics of a specimen even with little prior knowledge of the system. This thesis first develops a conductive heat transfer problem, and by using the truncated POD - RBF interpolation network, temperature values are predicted given a varying Biot number. Then, a simple bifurcation problem is modeled and solved for velocity profiles while changing the mass flow rate. This bifurcation problem provides the data and foundation for future research into the left ventricular assist device (LVAD) and implementation of POD - RBF. The trained POD - RBF inverse approach defined in this thesis can be implemented in several applications of engineering and mechanics. It provides model reduction, error filtration, regularization and an improvement over previous analysis utilizing computational fluid dynamics (CFD).
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Date Issued
2018
Identifier
CFH2000279, ucf:45782
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFH2000279
INVERSE BOUNDARY ELEMENT/GENETIC ALGORITHM METHOD FOR RECONSTRUCTION OF MULTI-DIMENSIONAL HEAT FLUX DISTRIBUTIONS WITH FILM COOLING APPLICATIONS
Title
INVERSE BOUNDARY ELEMENT/GENETIC ALGORITHM METHOD FOR RECONSTRUCTION OF MULTI-DIMENSIONAL HEAT FLUX DISTRIBUTIONS WITH FILM COOLING APPLICATIONS.
Creator
Silieti, Mahmood, Kassab, Alain, University of Central Florida
Abstract / Description
A methodology is formulated for the solution of the inverse problem concerned with the reconstruction of multi-dimensional heat fluxes for film cooling applications. The motivation for this study is the characterization of complex thermal conditions in industrial applications such as those encountered in film cooled turbomachinery components. The heat conduction problem in the metal endwall/shroud is solved using the boundary element method (bem), and the inverse problem is solved using a...
Show moreA methodology is formulated for the solution of the inverse problem concerned with the reconstruction of multi-dimensional heat fluxes for film cooling applications. The motivation for this study is the characterization of complex thermal conditions in industrial applications such as those encountered in film cooled turbomachinery components. The heat conduction problem in the metal endwall/shroud is solved using the boundary element method (bem), and the inverse problem is solved using a genetic algorithm (ga). Thermal conditions are overspecified at exposed surfaces amenable to measurement, while the temperature and surface heat flux distributions are unknown at the film cooling hole/slot walls. The latter are determined in an iterative process by developing two approaches. The first approach, developed for 2d applications, solves an inverse problem whose objective is to adjust the film cooling hole/slot wall temperatures and heat fluxes until the temperature and heat flux at the measurement surfaces are matched in an overall heat conduction solution. The second approach, developed for 2d and 3d applications, is to distribute a set of singularities (sinks) at the vicinity of the cooling slots/holes surface inside a fictitious extension of the physical domain or along cooling hole centerline with a given initial strength distribution. The inverse problem iteratively alters the strength distribution of the singularities (sinks) until the measuring surfaces heat fluxes are matched. The heat flux distributions are determined in a post-processing stage after the inverse problem is solved. The second approach provides a tremendous advantage in solving the inverse problem, particularly in 3d applications, and it is recommended as the method of choice for this class of problems. It can be noted that the ga reconstructed heat flux distributions are robust, yielding accurate results to both exact and error-laden inputs. In all cases in this study, results from experiments are simulated using a full conjugate heat transfer (cht) finite volume models which incorporate the interactions of the external convection in the hot turbulent gas, internal convection within the cooling plena, and the heat conduction in the metal endwall/shroud region. Extensive numerical investigations are undertaken to demonstrate the significant importance of conjugate heat transfer in film cooling applications and to identify the implications of various turbulence models in the prediction of accurate and more realistic surface temperatures and heat fluxes in the cht simulations. These, in turn, are used to provide numerical inputs to the inverse problem. Single and multiple cooling slots, cylindrical cooling holes, and fan-shaped cooling holes are considered in this study. The turbulence closure is modeled using several two-equation approach, the four-equation turbulence model, as well as five and seven moment reynolds stress models. The predicted results, by the different turbulence models, for the cases of adiabatic and conjugate models, are compared to experimental data reported in the open literature. Results show the significant effects of conjugate heat transfer on the temperature field in the film cooling hole region, and the additional heating up of the cooling jet itself. Moreover, results from the detailed numerical studies presented in this study validate the inverse problem approaches and reveal good agreement between the bem/ga reconstructed heat fluxes and the cht simulated heat fluxes along the inaccessible cooling slot/hole walls
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Date Issued
2004
Identifier
CFE0000166, ucf:52896
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0000166
PARAMETER ESTIMATION IN HEAT TRANSFER AND ELASTICITY USING TRAINED POD-RBF NETWORK INVERSE METHODS
Title
PARAMETER ESTIMATION IN HEAT TRANSFER AND ELASTICITY USING TRAINED POD-RBF NETWORK INVERSE METHODS.
Creator
Rogers, Craig, Kassab, Alain, University of Central Florida
Abstract / Description
In applied mechanics it is always necessary to understand the fundamental properties of a system in order to generate an accurate numerical model or to predict future operating conditions. These fundamental properties include, but are not limited to, the material parameters of a specimen, the boundary conditions inside of a system, or essential dimensional characteristics that define the system or body. However in certain instances there may be little to no knowledge about the systems...
Show moreIn applied mechanics it is always necessary to understand the fundamental properties of a system in order to generate an accurate numerical model or to predict future operating conditions. These fundamental properties include, but are not limited to, the material parameters of a specimen, the boundary conditions inside of a system, or essential dimensional characteristics that define the system or body. However in certain instances there may be little to no knowledge about the systems conditions or properties; as a result the problem cannot be modeled accurately using standard numerical methods. Consequently, it is critical to define an approach that is capable of identifying such characteristics of the problem at hand. In this thesis, an inverse approach is formulated using proper orthogonal decomposition (POD) with an accompanying radial basis function (RBF) network to estimate the current material parameters of a specimen with little prior knowledge of the system. Specifically conductive heat transfer and linear elasticity problems are developed in this thesis and modeled with a corresponding finite element (FEM) or boundary element (BEM) method. In order to create the truncated POD-RBF network to be utilized in the inverse approach, a series of direct FEM or BEM solutions are used to generate a statistical data set of temperatures or deformations in the system or body, each having a set of various material parameters. The data set is then transformed via POD to generate an orthonormal basis to accurately solve for the desired material characteristics using the Levenberg-Marquardt (LM) algorithm. For now, the LM algorithm can be simply defined as a direct relation to the minimization of the Euclidean norm of the objective Least Squares function(s). The trained POD-RBF inverse technique outlined in this thesis provides a flexible by which this inverse approach can be implemented into various fields of engineering and mechanics. More importantly this approach is designed to offer an inexpensive way to accurately estimate material characteristics or properties using nondestructive techniques. While the POD-RBF inverse approach outlined in this thesis focuses primarily in application to conduction heat transfer, elasticity, and fracture mechanics, this technique is designed to be directly applicable to other realistic conditions and/or industries.
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Date Issued
2010
Identifier
CFE0003267, ucf:48517
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0003267
Electrical Conductivity Imaging via Boundary Value Problems for the 1-Laplacian
Title
Electrical Conductivity Imaging via Boundary Value Problems for the 1-Laplacian.
Creator
Veras, Johann, Tamasan, Alexandru, Mohapatra, Ram, Nashed, M, Dogariu, Aristide, University of Central Florida
Abstract / Description
We study an inverse problem which seeks to image the internal conductivity map of a body by one measurement of boundary and interior data. In our study the interior data is the magnitude of the current density induced by electrodes. Access to interior measurements has been made possible since the work of M. Joy et al. in early 1990s and couples two physical principles: electromagnetics and magnetic resonance. In 2007 Nachman et al. has shown that it is possible to recover the conductivity...
Show moreWe study an inverse problem which seeks to image the internal conductivity map of a body by one measurement of boundary and interior data. In our study the interior data is the magnitude of the current density induced by electrodes. Access to interior measurements has been made possible since the work of M. Joy et al. in early 1990s and couples two physical principles: electromagnetics and magnetic resonance. In 2007 Nachman et al. has shown that it is possible to recover the conductivity from the magnitude of one current density field inside. The method now known as Current Density Impedance Imaging is based on solving boundary value problems for the 1-Laplacian in an appropriate Riemann metric space. We consider two types of methods: the ones based on level sets and a variational approach, which aim to solve specific boundary value problem associated with the 1-Laplacian. We will address the Cauchy and Dirichlet problems with full and partial data, and also the Complete Electrode Model (CEM). The latter model is known to describe most accurately the voltage potential distribution in a conductive body, while taking into account the transition of current from the electrode to the body. For the CEM the problem is non-unique. We characterize the non-uniqueness, and explain which additional measurements fix the solution. Multiple numerical schemes for each of the methods are implemented to demonstrate the computational feasibility.
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Date Issued
2014
Identifier
CFE0005437, ucf:50388
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0005437
ELECTRICAL CAPACITANCE VOLUME TOMOGRAPHY OF HIGH CONTRAST DIELECTRICS USING A CUBOID GEOMETRY
Title
ELECTRICAL CAPACITANCE VOLUME TOMOGRAPHY OF HIGH CONTRAST DIELECTRICS USING A CUBOID GEOMETRY.
Creator
Nurge, Mark, Schelling, Patrick, University of Central Florida
Abstract / Description
An Electrical Capacitance Volume Tomography system has been created for use with a new image reconstruction algorithm capable of imaging high contrast dielectric distributions. The electrode geometry consists of two 4 x 4 parallel planes of copper conductors connected through custom built switch electronics to a commercially available capacitance to digital converter. Typical electrical capacitance tomography (ECT) systems rely solely on mutual capacitance readings to reconstruct images of...
Show moreAn Electrical Capacitance Volume Tomography system has been created for use with a new image reconstruction algorithm capable of imaging high contrast dielectric distributions. The electrode geometry consists of two 4 x 4 parallel planes of copper conductors connected through custom built switch electronics to a commercially available capacitance to digital converter. Typical electrical capacitance tomography (ECT) systems rely solely on mutual capacitance readings to reconstruct images of dielectric distributions. This dissertation presents a method of reconstructing images of high contrast dielectric materials using only the self capacitance measurements. By constraining the unknown dielectric material to one of two values, the inverse problem is no longer ill-determined. Resolution becomes limited only by the accuracy and resolution of the measurement circuitry. Images were reconstructed using this method with both synthetic and real data acquired using an aluminum structure inserted at different positions within the sensing region. Comparisons with standard two dimensional ECT systems highlight the capabilities and limitations of the electronics and reconstruction algorithm.
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Date Issued
2007
Identifier
CFE0001591, ucf:47119
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0001591
Estimation and clustering in statistical ill-posed linear inverse problems
Title
Estimation and clustering in statistical ill-posed linear inverse problems.
Creator
Rajapakshage, Rasika, Pensky, Marianna, Swanson, Jason, Zhang, Teng, Bagci, Ulas, Foroosh, Hassan, University of Central Florida
Abstract / Description
The main focus of the dissertation is estimation and clustering in statistical ill-posed linear inverse problems. The dissertation deals with a problem of simultaneously estimating a collection of solutions of ill-posed linear inverse problems from their noisy images under an operator that does not have a bounded inverse, when the solutions are related in a certain way. The dissertation defense consists of three parts. In the first part, the collection consists of measurements of temporal...
Show moreThe main focus of the dissertation is estimation and clustering in statistical ill-posed linear inverse problems. The dissertation deals with a problem of simultaneously estimating a collection of solutions of ill-posed linear inverse problems from their noisy images under an operator that does not have a bounded inverse, when the solutions are related in a certain way. The dissertation defense consists of three parts. In the first part, the collection consists of measurements of temporal functions at various spatial locations. In particular, we studythe problem of estimating a three-dimensional function based on observations of its noisy Laplace convolution. In the second part, we recover classes of similar curves when the class memberships are unknown. Problems of this kind appear in many areas of application where clustering is carried out at the pre-processing step and then the inverse problem is solved for each of the cluster averages separately. As a result, the errors of the procedures are usually examined for the estimation step only. In both parts, we construct the estimators, study their minimax optimality and evaluate their performance via a limited simulation study. In the third part, we propose a new computational platform to better understand the patterns of R-fMRI by taking into account the challenge of inevitable signal fluctuations and interpretthe success of dynamic functional connectivity approaches. Towards this, we revisit an auto-regressive and vector auto-regressive signal modeling approach for estimating temporal changes of the signal in brain regions. We then generate inverse covariance matrices fromthe generated windows and use a non-parametric statistical approach to select significant features. Finally, we use Lasso to perform classification of the data. The effectiveness of theproposed method is evidenced in the classification of R-fMRI scans
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Date Issued
2019
Identifier
CFE0007710, ucf:52450
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0007710
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