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Physical Sciences and Mathematics Commons™
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Articles 1 - 5 of 5
Full-Text Articles in Physical Sciences and Mathematics
Reconstruction Of An Unknown Boundary Portion From Cauchy Data In N- Dimensions, Kurt Bryan, Lester Caudill
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.
Stability And Reconstruction For An Inverse Problem For The Heat Equation, Kurt Bryan, Lester Caudill
Stability And Reconstruction For An Inverse Problem For The Heat Equation, Kurt Bryan, Lester Caudill
Department of Math & Statistics Faculty Publications
We examine the inverse problem of determining the shape of some unknown portion of the boundary of a region Ω from measurements of the Cauchy data for solutions to the heat equation on Ω. 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 Bryan, Lester Caudill
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.
An Inverse Problem In Thermal Imaging, Kurt Bryan, Lester Caudill
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.
A Direct Method For The Inversion Of Physical Systems, Lester Caudill, Herschel Rabitz, Attila Askar
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.