Urban Cellular Automata: The Inverse Problem - Cellular Automata - page 350

2. The Inverse Problem in Modeling

We define inverse the problem faced by an observer who is dealing with a system
whose dynamics is unknown; he tries to understand how it works, recording outputs
without interacting with it and/or analyzing how outputs change if he perturbs the
system.
The adjective “inverse” means that the order of cause and effect is reverse: the
observer knows effects instead of causes and tries to deduce causes going backward;
the unknown quantity of this problem is a function, that is the set of rules that
determine the evolution of the system.
Solving this kind of problems is strictly dependent on the quality of the available
data, because it affects the uniqueness of solutions. A good general rule is that, in
order to determine a function of n variables, one should collect data that also depend
on n variables.
Often inverse problems solving is difficult because of the problem itself is ill
posed . In an ill posed problem small changes in the observations may correspond to
big changes in the phenomenon being observed. Ill posed problems are difficult to
deal with because the algorithms to solve them tend to be unstable , that means they
are likely to produce very different solutions if inputs are changed just a little (Fig. 1).
When the problem is ill posed accurate observations are not enough; it is important
to have prior information about the unknown system, derived from oneself
experience. As a beginning, you may consider a very simple set of rules, even if it
introduces strong simplifications, and then make corrections to make the model fit
reality.
When you deal with well posed problems (which are characterized by existence,
uniqueness and stable solutions) more data available can improve the reconstruction;
on the contrary, more data in solving an unstable problem can be a double-edge
sword. In fact, new data might generate a solution very different from the previous,
posing the observer in front of a choice; this is the problem of data consistency [2].
A general method of solution for this kind of problem, adopted in our case studies
as well, is called successive approximation method , and consists in changing
progressively the starting structure of the elements of the model, testing the new
model after each change through simulations [3]. This procedure addresses the full
non-linearity of the problem, even if it could be computationally intensive.

Fig. 1. Example of ill posed problem