Globally Convergent Inverse Algorithms via Carleman Weight Functions: Theory, Numerical Studies and Experimental Verifications
Abstract
The target application of this project is in the standoff detection and identification of explosive devices, e.g. land mines and improvised explosive devices (IEDs). By solving Coefficient Inverse Problems (CIPs), new globally convergent inverse numerical methods will provide accurate images of all three components of interest of buried targets: dielectric constants, shapes and sizes. The support letter in one of appendices states that this information "is very intriguing to the radar community for landmine and IED detection applications". This is an interdisciplinary project. All resulting algorithms for CIPs will be verified on both computationally simulated and real data. Experimental data in time domain are collected already by a microwave device, which is located in Grigg Hall of University of North Carolina at Charlotte. Experimental data in frequency domain will be collected by another microwave device, which is located in the same place as the above one. The following question is both crucial and the most challenging one in a numerical treatment of any CIP: How to obtain a good approximation for the unknown coefficient without any advanced knowledge of a small neighborhood of the solution? A numerical method providing such an approximation is called globally convergent. The project will address the above key question via both analytical development and computational verification of two new globally convergent numerical methods for CIPs for a hyperbolic PDE. In the first method globally strongly convex cost functionals will be constructed for those CIPs. The main element of each of these functionals will be the presence of a Carleman Weight Function, i.e. the one, which is involved in the Carleman estimate for an associated certain Partial Differential Operator. The second method will be an extension of the current globally convergent numerical technique, which works in the "Laplace transform domain", to the frequency domain. Frequency domain data are more sensitive than the data resulting from the Laplace transform of a time dependent signal. The most important property of above new numerical methods is that they address the above crucial question. Indeed, the global strong convexity of that cost functional implies the global convergence of the gradient method of the minimization of such a functional. In the case of noiseless data it converges to the exact solution, and in the case of noisy data it delivers points in a small neighborhood of the exact solution. The size of that neighborhood is defined by the noise level, and it is small as long as the noise level is small. In addition to the data produced by the above devices, experimental data collected in the field by Forward Looking Radar of Sensor and Electronic Device Directorate (SEDD) of US Army Research Laboratory will also be treated by new algorithms. As a result, new joint publications with SEDD engineers Drs. Lam Nguyen and Anders Sullivan are anticipated, in addition to current three joint publications with these people about imaging of targets using their data. The resulting user friendly software will be delivered to SEDD. A minor topic will be the development of numerical methods for solution of Inverse Scattering Problems without the phase information. The PI has recently pioneered this topic. In fact, above globally convergent numerical methods will be form a part of this new technique. "Without the phase information" means that only the modulus of the complex valued wave field is measured outside of scattereres. But phase is not measured. This topic has broad applications in imaging of nanostructures.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Jan 12, 2017
- Source ID
- W911NF1510233
Entities
People
- Michael Klibanov
Organizations
- Army Contracting Command
- United States Army
- University of North Carolina at Chapel Hill