7 Concluding Discussion

The general operator-splitting method of section 5.1 provides a vesatile means of solving inverse diffusion problems. The determination of the approximate inverse operator B places the onus is on the method-developer particularly to define the range of the operator in order to pre-determine the nature of acceptable solutions.

A general error analysis of the operator-splitting method is given in subsection 5.1. The error in the n^th iterate is Āⁿ f, but the fact that Ā is not necessarily a contraction simply reflects the ill-posedness of the original problem. In the case of the simple diffusion equation and provided f that can be written as a finite sum of the first m Fourier components, it is shown that

||Ā|| = r_m = (1-e^{m² p² T}) < 1

and hence, in this situation, Ā is a contraction. Figure 2 demonstrates the recovery of the original f when m=3.

In practice g is measured data from a practical experiment and the forward operator used in the inversion method is a model and hence not a precise representation of the operator K. An extension of the analysis of this paper, examining the effect of an imprecise operator K, would also be of value.

The Jacobi-like operator splitting method is a simple and remarkably effective method for solving inverse diffusion problems. In effect the method is similar to the method of partial eigenfunction expansion considered in subsection 2.3, except for the important distinction that the former is applicable to non-linear as well as linear problems. In this paper results from a number of test problems have been given demonstrating the effectiveness of the operator-splitting method with B = Į. The operator-splitting method has been applied by the authors to a practical problem in reference [6].

Acknowledgement

The authors would like to thank Professor D. G. Armour of the Department of Physics for his continuing interest and encouragement and their colleagues Dr S. Amini and Mr. P. Caine for some useful discussions on aspects of this work. The first author (SMK) is funded by by the EPSRC and DRA (RSRE, Malvern) grant GR/J35160.