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- Bryan, K. Single Measurement Detection of a Discontinuous Conductivity.
Comm. in PDE, 15, (1990), pp. 503-514.
The inverse problem of detecting a perturbation in a body with known
electrical conductivity is considered. We prove results showing that certain
perturbations can be detected using a single measurement, and also prove some
continuous dependence results for the inverse problem.
- Bryan, K. Numerical Recovery of Certain Discontinuous Electrical
Conductivities. Inverse Problems, 7, (1991), pp. 827-840.
The inverse problem of recovering a conductive inhomogeneity in an
otherwise known two-dimensional conductor is considered.
A linearization of the forward problem is formed and used in a
least squares output method for approximately solving the inverse problem.
Convergence results are proved and some numerical results presented.
- Bryan, K. and Vogelius, M., A Uniqueness Result Concerning the
Identification of a Collection of Cracks from Finitely Many Electrostatic
Boundary Measurements. SIAM J. Math. Anal., 23, (1992), pp. 950-958.
We consider the problem of locating and identifying a collection of finitely
many cracks inside a planar domain from measurements of the electrostatic
boundary potentials induced by specified current fluxes. It is shown that a
collection of n or fewer cracks can be uniquely identified by measuring
the boundary potentials induced by n+1 specified current fluxes, consisting
entirely of electrode pairs.
- Bryan, K. and Vogelius, M., A Computational Algorithm to Determine
Crack Locations from Electrostatic Boundary Measurements: the Case of Multiple
Cracks. Int. J. Engng Sci., 32, (1994), pp. 579-603.
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-ele
ctrode current fluxes. The algorithm is based on a variation of Newton's 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
- Bryan, K., A Boundary Integral Method for an Inverse
Problem in Thermal Imaging. ICASE report 92-38, Journal of
Mathematical Systems, Estimation and Control (7), 1997, pp 1-27.
This paper examines an inverse problem in thermal imaging, that of recovering a
void in a material sample from the sample surface temperature response to external
heating. Uniqueness and continuous dependence results for the inverse problem
are demonstrated and a numerical method for its solution developed. This method
is based on an optimization approach, coupled with a boundary integral equation
formulation of the forward heat conduction problem. Some convergence results
for the method are proved and several
examples are presented using computationally generated data.
- Bryan, K., Liepa, V., and Vogelius, M., Reconstruction of Multiple
Cracks from Experimental, Electrostatic Boundary Measurements.
Inverse Problems and Optimal Design in Industry, Eds. H.W. Engl and J. McLaughlin, Teubner, Stuttgart,
1994, p. 147-167.
This paper describes an algorithm for recovering a collection of linear cracks
in a homogeneous electrical conductor from boundary measurements of voltages
induced by specified current fluxes. The technique is a variation of Newton's
method and is based on taking weighted averages of the boundary data. We
also describe an apparatus that was constructed specifically for generating
laboratory data on which to test the algorithm. We apply the algorithm to
a number of different test cases and discuss the results.
- Bryan, K., and Caudill, L. An Inverse Problem in Thermal
Imaging. SIAM J. of App. Math (59), 1996, pp 715-735.
In this paper uniqueness and stability results for an inverse problem
in thermal imaging are examined. The goal is to identify an unknown and
inaccessible portion of the boundary of an object
by applying a heat flux and measuring the induced temperature on some accessible
portion of boundary. A linearized version of the inverse problem is developed
that reduces the problem to that of solving a first kind Fredholm integral
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.
- Bryan, K. Structure Characterization with Thermal Wave Imaging.
Proceedings of Review of Progress in Quantitative Nondestructive Testing, UCSD,
San Diego, CA, 1992. Plenum Publishers, 1993.
In this paper the problem of detecting and identifying the location,
size and shape of an unknown internal void in a planar domain using thermal
methods is examined. The void could represent a defect in the material, or it
could be a feature that is supposed to be present, e.g., a conduit, whose
location or geometry is to be assessed. The focus is on the case in which
the thermal stimulus, an applied heat flux at the
boundary of the sample, is a periodic point heat source. Separating the temporal
and spatial variables leads to an inverse or domain identification problem
for an elliptic equation. This is solved with an optimization/least-squares approach.
The technique is applied to actual experimental data generated at NASA Langley.
- Bryan, K. An Inverse Problem in Thermal Nondestructive Testing.
Proceedings of Computation and Control III, University of Montana, Bozeman, MT,
July, 1992. Birkhauser, 1993.
In this paper the problem of detecting and identifying an unknown internal void in a
planar domain using thermal methods is examined. The void could
represent a defect in the material, or it could be a feature that is supposed to
be present, e.g., a conduit, whose location or geometry is to be assessed.
We examine the case in which the thermal stimulus, an applied heat flux at the
boundary of the sample, is a periodic point heat source. In this case one can
separate the temporal and spatial variables, which leads to an inverse problem
for an elliptic equation. We prove uniqueness and continuous dependence results
for the inverse problem. The problem is computationally with an optimization
approach and uses a boundary integral equation formulation to approximate the
heat conduction problem. We prove some numerical convergence results for this
approach and examine a number of computational test cases.
- Bryan, K. and Caudill, L. Stability and Resolution in Thermal Imaging.
Proceedings of the Symposium on Parameter Estimation at the 15th ASME Biennial
Conference on Vibration and Noise, Boston, 1995.
This paper examines an inverse problem that arises in thermal imaging. We
investigate the problem of detecting and imaging corrosion in a material sample
by applying a heat flux and measuring the induced temperature on the sample's
The goal is to identify the profile of some inaccessible portion of the boundary.
We study the case in which one has data at every point
on the boundary of the region, as well as 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.
- Anderson, C., Bryan, K., et. al. Competency Matrix
Assessment in an Integrated First-Year Curriculum in Science, Engineering, and
Mathematics, in the proceedings of the Frontiers in Education Conference,
November 6-9, 1996, Salt Lake City, Utah.
The Integrated, First-Year Curriculum in Science, Engineering, and
Mathematics (IFYCSEM) at Rose-Hulman Institute of Technology integrates
topics in calculus, mechanics, statics, electricity and magnetism, computer
science, general chemistry, engineering design, and engineering graphics
into a three course, twelve-credit-per-quarter sequence. In 1995-96, faculty
teaching IFYCSEM unanimously agreed to move toward a competency matrix
assessment approach advocated by Lynn Bellamy at Arizona State University.
This paper presents a detailed description of the "competency matrix" approach
to assessment. We discuss the advantages and disadvantages of the approach, and
include feedback from both faculty and students, and ideas for future improvement
of the system.
- Bryan, K. and Hariri, H. Teaching a Transport Phenomena Problem using
a Symbolic Algebra Package. Proceedings of the Illinois-Indiana meeting of the
ASEE, March, 1997.
The solution of an unsteady-state flow of a viscous fluid in a tube is presented
using the symbolic algebra package Maple. The mathematical modeling of transient
problems in fluid flow and
heat transfer often lead to partial differential equations that may be solved
by using separation of variables. It is shown how these types of problems may
can be solved very conveniently using a symbolic algebra package such as Maple.
It is also shown how the solution can be visualized graphically, and even animated.
- Bryan, K., and Vogelius, M. Effective Behavior of Clusters of Microscopic
Cracks inside a Homogeneous Conductor. Applications to Impedance Imaging.
Asymptotic Analysis, (16), 1998, pp 141-178.
We study the effective behaviour of a periodic array of microscopic cracks inside a
homogeneous conductor. Special emphasis in 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.
- Bryan, K. and Caudill, L. Uniqueness for a Boundary Identification Problem
in Thermal Imaging. Electronic Journal of Differential Equations, C-1 (1997), p. 23-39.
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
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.
- Bryan, K. and Caudill, L. Stability and Reconstruction for an Inverse
Problem for the Heat Equation. Inverse Problems, (14), 1998, pp.
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. 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.
- Bryan, K. and Vogelius, M. Singular Solutions to a Nonlinear Elliptic Boundary Value Problem
Originating from Corrosion Modeling, Quarterly of Applied Math, (60), 2002,
We consider a non-linear elliptic boundary value problem on a planar domain. The exponential
type non-linearity in the boundary condition is one that frequently appears in the modeling of
electrochemical systems. For the case of a disk we construct a family of exact solutions that
exhibit limiting logarithmic singularities at certain points on the boundary. Based on these
solutions we develop two criteria that we believe predict the possible locations of the boundary
singularities on quite general domains.
- Bryan, K. and Caudill, L. Solvability of a Parabolic Boundary Value Problem with Internal
Jump Condition. Preprint.
We examine a model for the propagation of heat through a one-dimensional object with an interior ``flaw.''
The flaw is modeled as a non-linear relationship between the heat flux and temperature jump
at an interior point of the object. Under realistic hypotheses, the resulting non-linear initial
boundary value problem is shown to have a globally unique and suitably smooth solution.
- Bryan, K. and Vogelius, M. A Review of Selected Works on Crack
Identification, , in "Geometric Methods in Inverse Problems
and PDE Control", IMA Volume 137, Springer-Verlag, 2004.
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.
Bryan, K., Ogborne, R., and Vellela, M. Reconstruction of Cracks
with Unknown Transmission Condition from Boundary Data,
Inverse Problems, 21, 2005, pp. 21-36.
This paper examines the problem of identifying both the location
and constitutive law governing electrical current flow across a
one-dimensional linear crack in a two-dimensional region when the
crack only partial blocks the flow of current. We develop a
constructive numerical procedure for solving the inverse problem
and provide computational examples.
- Bryan, K., and Caudill, L.,
Reconstruction of an Unknown Boundary Portion from Cauchy Data in n-dimensions ,
Inverse Problems, 21, 2005, pp. 239-256.
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
- Bryan, K., Krieger, R., and Trainor, N.,
Imaging of Multiple Linear Cracks Using Impedance Data ,
in the Journal of Computational and Applied Mathematics, March 2007, Vol. 200 (1),
This paper develops a fast, simple algorithm for locating one or more perfectly insulating
pair-wise disjoint linear cracks in a homogeneous two-dimensional conductor, using flux-potential
boundary measurements. We also explore the issue of what types of boundary inputs yield the most
- Bryan, K., and Leise, T. The $25,000,000,000 Eigenvector,
in the education secton of SIAM Review, August 2006.
This paper is aimed at undergraduates;
we examine the beautiful and simple linear algebra that underlies one
important facet of Google's PageRank algorithm.
- Bryan, K., Haugh, J, and McCune, D., Fast Imaging of Partially
Conductive Linear Cracks, in Inverse Problems 22 (2006), p. 1337-1358.
We develop two closely-related fast and simple numerical
algorithms to address the inverse problem of identifying a
collection of disjoint linear cracks in a two-dimensional
homogeneous electrical conductor from exterior boundary
voltage/current measurements. We allow the possibility that the
cracks are partially conductive. Our approach also allows us to
determine the actual number of cracks present, as well as make use
of one or multiple input fluxes. We illustrate our algorithms
with a variety of computational examples.
- Bryan, K., and Caudill, L., Algorithm-independent optimal input fluxes
for boundary identification in thermal imaging, in the proceedings of
the Applied Inverse Problems (AIP) conference, Vancouver BC, June 2007.
We consider an inverse boundary determination problem for a
parabolic model arising in thermal imaging. The focus is on intelligently choosing an effective input
heat flux, to maximize the practical effectiveness of an inversion algorithm. Three different methods,
based on different interpretations of the term "effective", are presented and analyzed, then illustrated with numerical examples.
- Bryan K., and Leise, T., Impedance Imaging, Inverse Problems, and Harry Potter's Cloak, in the Education Section of SIAM Review, Vol. 52, No. 2, May 2010.
In this article we provide an accessible account of the essential idea behind cloaking, aimed at nonspecialists and undergraduates who have had some vector calculus, Fourier series, and linear algebra. The goal of cloaking is to render an object invisible to detection from electromagnetic energy, by surrounding the object with a specially engineered ``metamaterial'' that redirects the energy around the object. We show how to cloak an object against detection from impedance tomography, an imaging technique of much recent interest, though the
mathematical ideas apply to much more general forms of imaging. We also include some exercises and ideas for undergraduate research projects.
- Bryan K., and Vogelius, M. Precise Bounds for Finite Time Blow-up of Solutions to Very General One Space-Dimensional Nonlinear Neumann Problems, in the Quarterly of Applied Math 69(1), March 2011, p. 57-78.
In this paper we analyze the asymptotic finite time blow-up of solutions to the heat equation
with nonlinear Neumann boundary flux in one space dimension. We perform a detailed examination
of the nature of the blow-up, which can occur only at the boundary, and we provide tight upper
and lower bounds for the blow-up rate for "arbitrary" nonlinear flux functions, subject to very
- Bryan K., and Vogelius, M., Transient Behavior of Solutions to a Class of Nonlinear Boundary Value Problems, in the Quarterly of Applied Math, 69(2), June 2011, p. 261-290.
In this paper we consider the asymptotic behavior in time of solutions
to the heat equation with certain nonlinear Neumann boundary conditions of the form du/dn=F(u),
where F is a function that grows superlinearly. In general, solutions exist for only a finite time before ``blowing up''. In particular, it is known that solutions with positive initial data
must blow up in finite time, but solutions with sign-changing initial data are less well understood.
We make a detailed examination of conditions under which sign-changing solutions with certain symmetries either blow-up or decay to zero. The analysis is carried out in one space dimension for rather general F and in two dimensions for F of a very special form.
- Bryan, K., A Tale of Two Masses, PRIMUS, 21(2), February 2011, p. 149-162. An issue to honor Brian Winkel and his 20 years of service to the journal. The article is aimed at undergraduates and those who teach undergraduates, and has a some simple analysis and computer explorations of tuned-mass dampers for stabilizing skyscrapers.
- Bryan, K., and Leise, T., Making Do With Less: The Mathematics of Compressed Sensing, submitted to the Education section of SIAM Review, June 2011.
This article offers an accessible but rigorous and essentially self-contained account of the main ideas in compressed sensing (also known as compressive sensing or compressive sampling), 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.