The diffusion equation can easily be solved forward in time, typically by finite element and/or finite difference methods. It might be thought that to go backwards in time, all that we need to do is to take negative time steps. However, all such methods are unstable and all that is returned is numerical 'noise'.
The method demonstrated here is one that I termed an 'operator-splitting' method in a research paper that I had published in 2002. It requires repeated application of the forward operator in order to reach an answer. It works as well with non-linear diffusion operator too.
The Guide on the webpage link gives links to this paper and to other useful publications. The Inverse Diffusion Excel Spreadsheet in Microsoft Excel (using macros) can also be downloaded.
The method was introduced in the paper:
Solution of Inverse Diffusion Problems by
operator-splitting methods
by Stephen Kirkup* and Mike Wadsworth,
* John Tyndall Nuclear Research Institute
School of Computing Engineering and Physical Sciences
University of Central Lancashire - Westlakes Campus
Samuel Lindow Building
West Lakes Science and Technology Park
Whitehaven
Cumbria CA24 3JY
United Kingdom