|Prof. Oleg Mikailivitch Alifanov,
the great Russian proponent of Inverse Methods,
probably said it best: Solution of an inverse problem
entails determining unknown causes based on observation
of their effects. This is in contrast to the corresponding
direct problem, whose solution involves finding effects based
on a complete description of their causes.
A simple illustration may highlight the distinction. In particle dynamics, the motion of a mass in a gravitational field depends completely on the initial position and velocity of the object. The physical description of the process (F=mg) and the corresponding initial conditions (position x0 and velocity V0) constitute the causes of the ensuing motion. If these causes are fully described, the resulting motion can be found. This motion (described as the vector x(t)) is the effect of these causes.
Now one inverse problem can be seen. Suppose we know the mass of the object and the strength of the gravitational field in which it moves. By observation (experiment), we also acquire knowledge of the position and/or velocity of the object at several known instants of time. An inverse problem can now be formulated in the form of a question: Can the initial position and velocity of the body be determined?
Classes of Inverse Problems
Of course, inverse problems may be classified arbitrarily. One means of sorting them is by the type of information that is being sought in the solution procedure.