Applied Mathematics Commons^™

The Inverse Elasto-Acoustic Problem, Patrick Grice

Dissertations

A stable and numerically efficient boundary integral method formulation of the elasto-acoustic problem is presented, based on Fourier analysis. The method generalizes well to multiple scattering. The Frechet derivative of the elasto-acoustic problem with respect to shape perturbations is derived, and geometric flow theory is used to design stable numerical methods for the simulation of moving boundaries. The shape derivative is used to define a regularized Gauss-Newton algorithm for shape fitting of elasto-acoustic scatterers.

Advances In Computational Methods For Sparsity-Promoting Linear Inverse Problems, Jonathan Lindbloom

Dartmouth College Ph.D Dissertations

Inverse problems arise throughout science and engineering, where indirect, incomplete, and noisy observations are used to recover unknown parameters of interest. In these applications, the corresponding forward or measurement models are often ill-conditioned or underdetermined, so direct inversion is unstable and regularization is required. This thesis develops computational methods for linear inverse problems in which the unknown is assumed to be approximately sparse in a transformed domain defined by a linear, possibly rank-deficient operator, such as a finite-difference matrix, with particular emphasis on large-scale problems.

The thesis makes three main contributions. First, it generalizes hierarchical Bayesian maximum a posteriori estimation …

Analytical And Numerical Approaches To Parameter Estimation In Damped Oscillatory Systems, Gracie Crooks, F. Ayça Çetinkaya

CODEE Journal

We investigate the inverse problem of identifying damping and stiffness parameters in one-dimensional damped oscillatory systems governed by second-order differential equations. Focusing on mass–spring–damper models, we analyze the qualitative behavior of solutions across underdamped, critically damped, and overdamped regimes, and derive explicit conditions for parameter recovery based on time-domain observations such as equilibrium crossings and turnaround points. Two numerical estimation methods are developed and compared: a finite-difference least-squares approach based on central difference approximations, and a finite element formulation derived from a variational framework using piecewise linear basis functions. Computational experiments using synthetic data assess the accuracy, stability, and noise …

Exploring Electrical Impedance Tomography Techniques For Breast Cancer Detection In Three Dimensions, Emily Jean Mader Corcoran

Dissertations (1934 -)

Electrical Impedance Tomography (EIT) is an imaging modality whose reconstruction problem is severely ill-posed; large changes in the interior conductivity can present as small changes in the data. Due to this ill-posedness, reconstructed images generally have low spatial resolution. However, EIT remains a promising area of study for many medical imaging applications, including breast imaging. In this dissertation, a proof of concept study is discussed in Project 1: "Machine Learning for Breast Cancer Detection" wherein machine learning techniques are used to classify breast tumors as malignant or benign from simulated EIT voltage data. Promising results in terms of accuracy and …

Comparison Of Methods For Creating Populations Of Models By Solving Stochastic Inverse Problems, Elizabeth Epstein

Theses

Given a parametric family of models and observational data, a researcher may be faced with an inverse problem: what distribution of parameters best creates a set of models that produce the observed data? Traditionally, Markov Chain Monte Carlo (MCMC) has commonly been used as a method to solve these stochastic inverse problems. In recent years, however, Generative Adversarial Networks (GANs) have been employed. The effectiveness of Markov Chain Monte Carlo methods as compared to a conditional generative adversarial network (cGAN) when applied to a family of models produced by a system of ordinary differential equations that model viral load over …

A Method Of Presenting Experimental Dependencies In Solving Inverse Problems, Anatoliy Fyodorovich Verlan, Miraziz Vorisovich Sagatov

Chemical Technology, Control and Management

Scientific interest in the creation, research and application of methods and means for solving inverse problems is determined by the need for the development of new methods of signal processing, as well as the increased complexity of inverse problems in relation to direct problems, since the latter are not correct from a mathematical point of view and have a number of peculiarities. Since the input information in inverse problems is experimental data determined with a certain error, the resulting solution can differ greatly from the exact solution. An arbitrary discretely given function of time can be approximated with a predetermined …

Inverse Spectral Problems For Spectral Data And Two Spectra Of N By N Tridiagonal Almost-Symmetric Matrices, Bayram Bala, Manaf D. Manafov, Abdullah Kablan

Applications and Applied Mathematics: An International Journal (AAM)

One way to study the spectral properties of Sturm-Liouville operators is difference equations. The coefficients of the second order difference equation which is equivalent Sturm-Liouville equation can be written as a tridiagonal matrix. One investigation area for tridiagonal matrix is finding eigenvalues, eigenvectors and normalized numbers. To determine these datas, we use the solutions of the second order difference equation and this investigation is called direct spectral problem. Furthermore, reconstruction of matrix according to some arguments is called inverse spectral problem. There are many methods to solve inverse spectral problems according to selecting the datas which are generalized spectral function, …

Joint Inversion Of Compact Operators, James Ford

Boise State University Theses and Dissertations

The first mention of joint inversion came in [22], where the authors used the singular value decomposition to determine the degree of ill-conditioning in inverse problems. The authors demonstrated in several examples that combining two models in a joint inversion, and effectively stacking discrete linear models, improved the conditioning of the problem. This thesis extends the notion of using the singular value decomposition to determine the conditioning of discrete joint inversion to using the singular value expansion to determine the well-posedness of joint linear operators. We focus on compact linear operators related to geophysical, electromagnetic subsurface imaging.

The operators are …

Optimization Schemes For The Inversion Of Bouguer Gravity Anomalies, Azucena Zamora

Open Access Theses & Dissertations

Data sets obtained from measurable physical properties of the Earth structure have helped advance the understanding of its tectonic and structural processes and constitute key elements for resource prospecting. 2-Dimensional (2-D) and 3-D models obtained from the inversion of geophysical data sets are widely used to represent the structural composition of the Earth based on physical properties such as density, seismic wave velocities, magnetic susceptibility, conductivity, and resistivity. The inversion of each one of these data sets provides structural models whose consistency depends on the data collection process, methodology, and overall assumptions made in their individual mathematical processes. Although sampling …

On Constrained Optimization Schemes For Joint Inversion Of Geophysical Datasets, Uram Anibal Sosa Aguirre

Open Access Theses & Dissertations

In the area of geological sciences, there exist several experimental techniques used to advance in the understanding of the Earth. We implement a joint inversion least-squares (LSQ) algorithm to characterize one dimensional Earth's structure by using seismic shear wave velocities as a model parameter. We use two geophysical datasets sensitive to shear velocities, namely Receiver Function and Surface Wave dispersion velocity observations, with a choice of an optimization method: Truncated Singular Value Decomposition (TSVD) or Primal-Dual Interior-Point (PDIP). The TSVD and the PDIP methods solve a regularized unconstrained and a constrained minimization problem, respectively. Both techniques include bounds into the …

A Sensitivity Matrix Methodology For Inverse Problem Formulation, Ariel Cintron-Arias, H. T. Banks, Alex Capaldi, Alun L. Lloyd

Mathematics and Computer Science Faculty Publications

We propose an algorithm to select parameter subset combinations that can be estimated using an ordinary least-squares (OLS) inverse problem formulation with a given data set. First, the algorithm selects the parameter combinations that correspond to sensitivity matrices with full rank. Second, the algorithm involves uncertainty quantification by using the inverse of the Fisher Information Matrix. Nominal values of parameters are used to construct synthetic data sets, and explore the effects of removing certain parameters from those to be estimated using OLS procedures. We quantify these effects in a score for a vector parameter defined using the norm of the …

A Sensitivity Matrix Methodology For Inverse Problem Formulation, Ariel Cintron-Arias, H. Banks, Alex Capaldi, Alun Lloyd

Mathematics and Statistics Faculty Publications

We propose an algorithm to select parameter subset combinations that can be estimated using an ordinary least-squares (OLS) inverse problem formulation with a given data set. First, the algorithm selects the parameter combinations that correspond to sensitivity matrices with full rank. Second, the algorithm involves uncertainty quantification by using the inverse of the Fisher Information Matrix. Nominal values of parameters are used to construct synthetic data sets, and explore the effects of removing certain parameters from those to be estimated using OLS procedures. We quantify these effects in a score for a vector parameter defined using the norm of the …

Singular Superposition/Boundary Element Method For Reconstruction Of Multi-Dimensional Heat Flux Distributions With Application To Film Cooling Holes, Mahmood Silieti, Eduardo Divo, Alain J. Kassab

Publications

A hybrid singularity superposition/boundary element-based inverse problem method for the reconstruction of multi-dimensional heat flux distributions is developed. Cauchy conditions are imposed at exposed surfaces that are readily reached for measurements while convective boundary conditions are unknown at surfaces that are not amenable to measurements such as the walls of the cooling holes. The purpose of the inverse analysis is to determine the heat flux distribution along cooling hole surfaces. This is accomplished in an iterative process by distributing a set of singularities (sinks) inside the physical boundaries of the cooling hole (usually along cooling hole centerline) with a given …

Determining The Shape Of A Resistor Grid, Esther Chiew, Vincent Selhorst-Jones

Mathematical Sciences Technical Reports (MSTR)

Impedance imaging has received a lot of attention in the past two decades, as a means for non-destructively imaging the interior of a conductive object. One injects a known electrical current pattern into an object at the exterior boundary, then measures the induced potential (voltage) on some portion of the boundary. The goal is to recover information about the interior conductivity of the object, which (we hope) influences the voltages we measure. Of course one can also use multiple input currents and measured voltages. A variation on this problem is that of "boundary identification," in which some portion of the …

Utilizing Thermal Testing For Recovering, James Preciado, Thomas Werne

Mathematical Sciences Technical Reports (MSTR)

Given a two-dimensional region that contains one or more circular voids, we develop mathematical methods to locate the center and radius of the voids based on thermal boundary data. These methods can be readily applied in the field of non-destructive evaluation.

Reconstruction Of An Unknown Boundary Portion From Cauchy Data In N- Dimensions, Kurt Bryan, Lester Caudill

Department of Math & Statistics Faculty Publications

We consider the inverse problem of determining the shape of some inaccessible portion of the boundary of a region in n dimensions from Cauchy data for the heat equation on an accessible portion of the boundary. The inverse problem is quite ill-posed, and nonlinear. We develop a Newton-like algorithm for solving the problem, with a simple and efficient means for computing the required derivatives, develop methods for regularizing the process, and provide computational examples.

Reconstruction Of Partially Conductive Cracks Using Boundary Data, David Mccune, Janine Haugh

Mathematical Sciences Technical Reports (MSTR)

This paper develops an algorithm for finding one or more non-insulated, pair-wise disjoint, linear cracks in a two dimensional region using boundary measurements.

Reconstruction Of An Unknown Boundary Portion From Cauchy Data In N-Dimensions, Kurt M. Bryan, Lester Caudill

Mathematical Sciences Technical Reports (MSTR)

We consider the inverse problem of determining the shape of some inacces­ sible portion of the boundary of a region in n dimensions from Cauchy data for the heat equation on an accessible portion of the boundary. The inverse problem is quite ill-posed, and nonlinear. We develop a Newton-like algorithm for solving the problem, with a simple and efficient means for computing the required derivatives, develop methods for regularizing the process, and provide computational examples

Determining The Length Of A One-Dimensional Bar, Natalya Yarlikina, Holly Walrath

Mathematical Sciences Technical Reports (MSTR)

In this paper we examine the inverse problem of determining the length of a one-dimensional bar from thermal measurements (temperature and heat flux) at one end of the bar (the "accessible" end); the other inaccessible end of the bar is assumed to be moving. We develop two different approaches to estimating the length of the bar, and show how one approach can also be adapted to find unknown boundary conditions at the inaccessible end of the bar.

A Review Of Selected Works On Crack Indentification, Kurt M. Bryan

Mathematical Sciences Technical Reports (MSTR)

We give a short survey of some of the results obtained within the last 10 years or so concerning crack identification using impedance imaging techniques. We touch upon uniqueness results, continuous dependence results, and computational algorithms.

Characterizing A Defect In A One-Dimensional Bar, Cynthia Gangi, Sameer Shah

Mathematical Sciences Technical Reports (MSTR)

We examine the inverse problem of locating and describing an internal point defect in a one­ dimensional rod W by controlling the heat inputs and measuring the subsequent temperatures at the boundary of W. We use a variation of the forward heat equation to model heat flow through W, then propose algorithms for locating an internal defect and quantifying the effect the defect has on the heat flow. We implement these algorithms, analyze the stability of the procedures, and provide several computational examples.

Fast Reconstruction Of Cracks Using Boundary Measurements, Nicholas A. Trainor, Rachel M. Krieger

Mathematical Sciences Technical Reports (MSTR)

This paper develops a fast algorithm for locating one or more perfectly insulating, pair-wise disjoint, linear cracks in a homogeneous two-dimensional electrical conductor, using boundary measurements.

Uniqueness For A Boundary Identification Problem In Thermal Imaging, Kurt Bryan, Lester Caudill

Department of Math & Statistics Faculty Publications

An inverse problem for an initial-boundary value problem is considered. The goal is to determine an unknown portion of the boundary of a region in ℝⁿ from measurements of Cauchy data on a known portion of the boundary. The dynamics in the interior of the region are governed by a differential operator of parabolic type. Utilizing a unique continuation result for evolution operators, along with the method of eigenfunction expansions, it is shown that uniqueness holds for a large and physically reasonable class of Cauchy data pairs.

Stability And Reconstruction For An Inverse Problem For The Heat Equations, Kurt M. Bryan, Lester Caudill

Mathematical Sciences Technical Reports (MSTR)

We examine the inverse problem of determining the shape of some unknown portion of the boundary of a region W from measurements of the Cauchy data for solutions to the heat equation on W. By suitably linearizing the inverse problem we obtain uniqueness and continuous dependence results. We propose an algorithm for recovering estimates of the unknown portion of the surface and use the insight gained from a detailed analysis of the inverse problem to regularize the inversion. Several computational examples are presented.

Uniqueness For A Boundary Identification Problem In Thermal Imaging, Kurt M. Bryan, Lester F. Caudill

Mathematical Sciences Technical Reports (MSTR)

An inverse problem for a parabolic initial-boundary value problem is considered. The goal is to determine an unknown portion of the boundary of a region in Rⁿ from measurements of Dirichlet data on a known portion of the boundary. It is shown that under reasonable hypotheses uniqueness results hold.

An Inverse Problem In Thermal Imaging, Kurt Bryan, Lester Caudill

Department of Math & Statistics Faculty Publications

This paper examines uniqueness and stability results for an inverse problem in thermal imaging. The goal is to identify an unknown boundary of an object by applying a heat flux and measuring the induced temperature on the boundary of the sample. The problem is studied in both the case in which one has data at every point on the boundary of the region and the case in which only finitely many measurements are available. An inversion procedure is developed and used to study the stability of the inverse problem for various experimental configurations.

Effective Behavior Of Clusters Of Microscopic Cracks Inside A Homogeneous Conductor, Kurt M. Bryan, Michael Vogelius

Mathematical Sciences Technical Reports (MSTR)

We study the effective behaviour of a periodic array of microscopic cracks inside a homoge­neous conductor. Special emphasis is placed on a rigorous study of the case in which the corresponding effective conductivity becomes nearly singular, due to the fact that adjacent cracks nearly touch. It is heuristically shown how thin clusters of such extremely close cracks may macroscopically appear as a single crack. The results have implications for our earlier work on impedance imaging.

An Inverse Problem In Thermal Language, Kurt M. Bryan, Lester Caudill

Mathematical Sciences Technical Reports (MSTR)

This paper examines uniqueness and stability results for an inverse problem in thermal imaging. The goal is to identify an unknown boundary of an object by applying a heat flux and measuring of the induced temperature on the boundary of the sample. The problem is studied both in the case in which one has of data at every point on the boundary of the region and the case in which only finitely many measurements are available. An inversion procedure is developed and used to study the stability of the inverse problem for various experimental configurations.

Reconstruction Of Semiconductor Doping Profile From Lbic Image, Weifu Fang, Kazufumi Ito

Mathematics and Statistics Faculty Publications

In this paper, the authors study the reconstruction of a semiconductor doping profile or, equivalently, the equilibrium potential, from its LBIC (laser-beam-induced current) image. For the one-dimensional case, the authors first characterize the attainable class of current measurements, and from this they show the nonuniqueness of the inverse problem. Then the reconstruction of the equilibrium potential is reduced to finding two constants subject to some constraints. A reconstruction algorithm is established based on a least squares formulation of the problem. The case of noise-collapsed data is also discussed. For a special case of two-dimensional domain, the authors apply the one-dimensional …

A Direct Method For The Inversion Of Physical Systems, Lester Caudill, Herschel Rabitz, Attila Askar

Department of Math & Statistics Faculty Publications

A general algorithm for the direct inversion of data to yield unknown functions entering physical systems is presented. Of particular interest are linear and non-linear dynamical systems. The potential broad applicability of this method is examined in the context of a number of coefficient-recovery problems for partial differential equations. Stability issues are addressed and a stabilization approach, based on inverse asymptotic tracking, is proposed. Numerical examples for a simple illustration are presented, demonstrating the effectiveness of the algorithm.