An Approximately Globally Convergent Scheme For a Hyperbolic Coefficient Inverse Problem
Michael Klibanov
Abstract:
The question of constructing reliable numerical methods for Coefficient Inverse Problems (CIPs) arises in many applications. The standard technique of least squares regularization functionals suffers from a well known phenomenon of multiple local minima and ravines. This results in locally convergent numerical methods, e.g. Newton method. However, a good first guess for the solution is rarely known in applications. Therefore, an important problem is to construct such numerical methods for CIPs, which would deliver good approximations for true coefficients regardless of the availability of any information about small neighborhoods of those coefficients. It is not necessary that such algorithms would be able to start from any point. Rather, the starting point should be such that its choice would not rely on a knowledge of that small neighborhood. On the other hand, it is clear that the problem of constructing such algorithms is an enormously challenging one. Nonlinearity and ill-posedness of CIPs cause this challenge. Therefore, as it is always done in Mathematical Modeling, it is inevitable that some approximations should be made. The convergence analysis should be made within the frameworks of corresponding approximate mathematical models. Next, however, these approximations must be firmly verified on computationally simulated data and, if possible, on experimental data as well. Roughly, this is what we call "approximate global convergence" (we have a rigorous definition of this notion as well). In the past five years we have developed various versions of an approximate globally convergent numerical method for CIPs for a hyperbolic PDE with the data resulting from a single measurement event. The convergence theory is developed and numerical simulations confirm it. In addition, this method was tested on two types of experimental data. The case of blind real data was considered in both instances, i.e. the solution was unknown in advance. Results are summarized in the recently published book [1]. The most recent publication about blind real data is [2]. The above will be the main topic of my talk. If time will allow, I will also present logarithmic stability estimates for the problem of thermo-acoustic tomography in the case of an arbitrary elliptic operator [3].
[1] L. Beilina and M.V. Klibanov, Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems, Springer, New York, 2012.
[2] A.V. Kuzhuget, L. Beilina, M.V. Klibanov, A. Sullivan, L. Nguyen and M.A. Fiddy, Blind experimental data collected in the field and an approximately globally convergent inverse algorithm, Inverse Problems, 28, 095007, 2012.
[3] M.V. Klibanov, Thermoacoustic tomography with an arbitrary elliptic operator, Arxiv 1208.5187v1 [math-ph], 26 August 2012, submitted for publication.