Globally Convergent Numerical Methods for Coefficient Inverse Problems
Abstract
Coefficient Inverse Problems (CIPs) for Partial Differential Equations (PDEs) represent a very important tool for such needs of the Army as imaging of unknown targets hidden in cluttered heterogeneous backgrounds. The goal of this project is the development of globally convergent numerical methods for a wide class of CIPs. These methods are tested on mathematical models of the interest to the Army such as imaging of antipersonnel land mines and targets on battlefields covered by smogs and flames. In our definition "global convergence" entails: (1) a rigorous convergence analysis that does not depend on the quality of the initial guess, and (2) numerical simulations that confirm the advertised convergence property. A conventional way to solve a CIP is via the minimization of a least squares objective functional. This functional characterizes misfit between the data and the solution of that PDE with a "guess" for the unknown coefficient. However, it is well known to researchers working on computations of inverse problems that the phenomenon of multiple local minima of these functionals represents the major obstacle for the development of reliable numerical methods for multidimensional CIPs. This phenomenon in turn is caused by the above mentioned non-linearity and ill-posedness. Because of local minima, one should somehow guess in advance about a good approximation for the solution. Without the availability of a first good guess, however, there is no guarantee that the calculated coefficient is indeed close to the correct one. In our terminology these are locally convergent numerical methods. In other words, their convergence to the correct solution can be guaranteed only if the starting point is located in a small neighborhood of this solution. Because of local minima, conventional numerical methods for multidimensional CIPs are locally convergent ones. However, in many important applications the first good guess is unavailable.
Document Details
- Document Type
- Technical Report
- Publication Date
- Sep 23, 2008
- Accession Number
- ADA499357
Entities
People
- Michael Klibanov
Organizations
- University of North Carolina at Charlotte