Inverse problems are a crucial component to many critical scientific and engineering applications including Non-Destructive Evaluation and Parameter Identification. In this minisymposium we will address theoretical and computational issues that arise in applications to electromagnetics and biology.
Speakers are named first.
8:45-9:10 Nathan L. Gibson, North Carolina State University
Gap Detection with Electromagnetic Terahertz Signals
We apply an inverse problem formulation to determine characteristics of a defect from a perturbed electromagnetic interrogating signal. A defect (gap) inside of a dielectric material causes a disruption, via reflections and refractions at the material interfaces, in the windowed interrogating signal. We model the electromagnetic waves inside the material with Maxwell's equations. Using simulations as forward solves, our Newton-based, iterative optimization scheme resolves the dimensions and location of the defect. Numerical results, including standard errors, will be presented, and computational issues will be addressed. Our research is supported by NASA with the ultimate goal of designing devices for damage detection for use in applications such as preventing delamination of foam on the space shuttle fuel tanks.
9:15-9:40 Vrushali A. Bokil, North Carolina State University
Parameter Identification for Dispersive Dielectrics using Acoustooptic Material Interrogation
We consider an electromagnetic interrogation technique in two dimensions for identifying the dielectric parameters of a Debye medium. In this technique a traveling acoustic pressure wave in the Debye medium is used as a virtual reflector for an interrogating microwave electromagnetic pulse that is generated in free space. The reflections of the microwave pulse from the air-Debye interface and from the acoustic pressure wave are recorded at a remote antenna. The data is used in an inverse problem to estimate the locally pressure dependent dielectric parameters of the Debye medium. We present a time domain formulation that is solved using finite differences (FDTD) in time and in space. Perfectly matched layer (PML) absorbing boundary conditions are used to absorb outgoing waves at the finite boundaries of the computational domain, preventing spurious reflections from reentering the domain. Using the method of least squares for the parameter identification problem, we compare two different algorithms (the gradient based Levenberg-Marquardt method, and the gradient free, simplex based Nelder-Mead method) in solving an inverse problem to calculate estimates for two or more dielectric parameters. Finally we use statistical error analysis to construct confidence intervals for all the presented estimates, thereby providing a probabilistic statement about the computational procedure with uncertainty aspects of estimates.
9:45-10:10 Michele L. Joyner, State University of West Georgia
Issues in Implementing the POD Methodology in NDE
Previously, we have demonstrated the viability of using the Proper Orthogonal Decomposition (POD) (also called Principal Component Analysis) method in conjunction with eddy current nondestructive evaluation techniques to reduce the total computational time to detect damages when using either simulated or experimental data. In this talk we will examine two techniques (POD/Galerkin method and POD/Interpolation method) used to form the reduced order solution to the forward problem. In addition, we explore which factors impact the effectiveness of the POD method in the NDE damage detection problem (inverse problem). We examine whether or not the number of snapshots used to form the POD basis affects the accuracy of the estimation of the damage parameter. We also consider whether or not it is necessary to vary the damage parameters in the set of snapshots incrementally or whether it is possible to utilize random damage parameters and result in the same order of accuracy in the damage estimation. In addition, we examine how much relative error in the reduced order POD approximation we can sustain and still obtain fairly precise results in the inverse problem. Finally, we consider whether or not the answers to the above questions are the same for different damage parameters or consistent regardless of which parameter we wish to estimate.
2:00-2:25 Hoan K. Nguyen, North Carolina State University
Sensitivity with respect to Probability Densities in Inverse Problems
We consider general nonlinear dynamical systems in a Banach space with dependence on parameters in a second Banach space. An abstract theoretical framework for sensitivity equations is developed. An application to measure dependent delay differential systems arising in a class of HIV models is presented.
2:30-2:55 Shuhua Hu, North Carolina State University
Parameter Estimation in a Coupled System of Nonlinear Size-Structured Populations
A least-squares technique is developed for identifying unknown parameters in a coupled system of nonlinear size-structured populations. Convergence results for the parameter estimation technique are established. Ample numerical simulations and statistical evidence are provided to demonstrate the feasibility of this approach.
3:00-3:25 C. W. Groetsch, Department of Mathematical Sciences, University of Cincinnati, D.A. French and W. Krantz, College of Engineering, University of Cincinnati, and R. Flannery and S.J. Kleene, College of Medicine, University of Cincinnati
Numerical Approximation of Solutions of a Nonlinear Inverse Problem Arising in Olfaction Experimentation
A pair of nonlinear partial differential equations and a constrained Fredholm integral equation of the first kind are proposed to model experiments aimed at identifying gross features of the distribution of ion channels along cilia that extend from olfactory receptor neurons. A simple numerical method, based on a fine-grid finite difference scheme for the pdes and a course-grid discretization of the integral equation, is derived and used to obtain estimates of the spatial distribution of ion channels along the length of the cilium. Simulations using experimental data indicate that these channels have a non-uniform distribution.