The philosophy of the approximate global convergence for multidimensional coefficient inverse problems

Abstract

Both the most important and the most challenging question in the numerical treatment of a multidimensional coefficient inverse problem for a partial differential equation is the following: How to obtain a point in a small neighbourhood of the exact solution without any a priori knowledge of this solution? The recent numerical experience of the authors shows that in order to develop a truly efficient algorithm addressing this question, it is necessary to make some reasonable approximations. Although these approximations cannot be rigorously justified, numerical studies show that corresponding algorithms work quite well. The authors call this approach ‘approximate global convergence’. The goal of this article is to present a short illustrative review of this philosophy.

Keywords:

• philosophy
• coefficient inverse problems
• approximate global convergence

AMS Subject Classifications::

• 15A15
• 15A09
• 15A23

Acknowledgements

This research was supported by US Army Research Laboratory and US Army Research Office grant W911NF-11-1-0399, the Swedish Research Council, the Swedish Foundation for Strategic Research (SSF) in Gothenburg Mathematical Modelling Centre (GMMC) and by the Swedish Institute, Visby Program. The authors are grateful to Professor Roman Novikov for a number of useful discussions.