Kurt Bryan

The

Googles success derives in large part from its PageRank algorithm, which ranks the importance of web pages according to an eigenvector of a weighted link matrix. Analysis of the PageRank formula provides a wonderful applied topic for a linear algebra course. Instructors may assign this article as a project to more advanced students or spend one or two lectures presenting the material with assigned homework from the exercises. This material also complements the discussion of Markov chains in matrix algebra. Maple and Mathematica files supporting this material can be found at www.rose-hulman.edu/~bryan.

25,000,000,000 Eigenvector: The Linear Algebra behind Google

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 both in 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.

AN INVERSE PROBLEM IN THERMAL IMAGING

The problem of identification of a collection of finitely many cracks inside a planar domain is considered. The data used for the identification consist of measurements of the electrostatic boundary potentials induced by prescribed current fluxes. It is shown that a collection of n or fewer cracks is uniquely identified by boundary measurements corresponding to

A uniqueness result concerning the identification of a collection of cracks from finitely many electrostatic boundary measurements

n + 1

Stability and Reconstruction for an Inverse Problem for the Heat Equation

specific current fluxes, consisting entirely of electrode pairs.

Reconstruction of an unknown boundary portion from Cauchy data in n dimensions

We examine the inverse problem of determining the shape of some unknown portion of the boundary of a region from measurements of the Cauchy data for solutions to the heat equation on . 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.

A computational algorithm to determine crack locations from electrostatic boundary measurements. The case of multiple cracks

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.

A REVIEW OF SELECTED WORKS ON CRACK IDENTIFICATION

Abstract This paper develops an algorithm to reconstruct the locations of a collection of linear cracks inside a homogeneous electrical conductor from boundary measurements. We measure the boundary voltages induced by certain specified two-electrode current fluxes. The algorithm is based on a variation of Newtons method and it uses weighted averages of the measured boundary data. The algorithm adaptively changes the applied current fluxes at each iteration to maintain “maximal” sensitivity to the estimated locations of the cracks.

Making Do with Less: An Introduction to Compressed Sensing

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.

Impedance Imaging, Inverse Problems, and Harry Potter's Cloak

This article offers an accessible but rigorous and essentially self-contained account of some of the central ideas in compressed sensing, aimed at nonspecialists and undergraduates who have had linear algebra and some probability. The basic premise is first illustrated by considering the problem of detecting a few defective items in a large set. We then build up the mathematical framework of compressed sensing to show how combining efficient sampling methods with elementary ideas from linear algebra and a bit of approximation theory, optimization, and probability allows the estimation of unknown quantities with far less sampling of data than traditional methods.