Numerical Analysis and Computation Commons^™

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 …

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 …

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.

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.

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.