"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." [Keith A. Woodbury, Univ. of Alabama] Inverse problems are ones in which the result is known but not all the specifics of preceding circumstances. They are the murder mysteries of mathematics and science.

They can be very simple: A number was added to 7 resulting in 37. What was the number?

Or, the price of a new TV was $359.99, but is on sale for 25% off. If the total after the discount and including tax was $283.49, what was the tax percentage? (Determine the answer. Raise you hand when you think you know. The first correct answer will win a prize!)

They can also be more complicated: Suppose you know the action of a function f on one element (or you know a set of images). Can you recover the entire function f?

Or, you know the derivative of a function, but need the actual function? Can you determine it just from information about its rate of change?

In applications, you may 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?

Or, in tomography where waves of various types are sent through a medium to determine its makeup. Here you record the results of the waves' travels and then recontrust the medium they travelled through. Successful use of seismography, CAT scans and sonograms are commonplace because these types of inverse tomography problems have been solved.

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