What are Inverse Problems?

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.

Backward or retrospective problem
The initial conditions are to be found. This is exactly the case illustrated above in the particle dynamics example.
Coefficient inverse problem
This is the classical parameter estimation problem where a constant multiplier in a governing equation is to be found. In the particles dynamics example above, this corresponds to estimation of the gravitational acceleration "g" based on all other conditions (including the initial conditions) being known.
Boundary inverse problem
Some missing information at the boundary of a domain is to be found. Note this can be a function estimation problem when this boundary condition changes with time. This does not have an analog in the particle dynamics example, as the example is only an initial value problem and has no "boundary" information. However, there is a classic example of a boundary inverse problem in the Inverse Heat Conduction Problem, where the unknown thermal action at the boundary of the object is to be found based on observations (measurements) of the temperature on the interior of the object.


Keith A. Woodbury
May, 1995