Inverse problem

A mathematical problem is called, if one wants to infer from an observed or desired effect of a system on the effect of underlying cause of an inverse problem. Inverse problems are often very difficult or impossible to solve. The opposite of an inverse problem is ( partially also referred to as forward problem ), in which it is desired to derive from the known starting the operation of the system cause a direct problem.

A distinction is well-off and ill-posed inverse problems. In addition to questions of the mathematical and physical stability is often also those of the numerical stability to be observed ( for example, large normal equation systems). To improve the numerical stability regularization used.

Explanatory example

This abstract declaration and the difficulty of inverse problems can be illustrated with an example: A submarine boat located at a point x at a depth t in the sea. The drive will send sound waves ( engine and propeller noise). Knowing the properties of these sound waves ( intensity, frequency) and the transmitting medium (water), one can easily calculate how well the microphone y can hear the submarine at a remote location. This is an easy to solve, direct problem. It closes on the cause (noise at location x in the depth t ) to the effect ( acoustic signal at the microphone). As part of the submarine tracking would be reversed know from the measured at the location y engine noise, where and at what depth the submarine is. This is the associated inverse problem where you want to include from the effect to the cause. The localization problem is much more difficult to solve. If one has in the received signal, no information from which direction the sound is coming, the problem is unsolvable, because even if we know the properties of the radiation emitted from the submarine sound waves, one can only deduce the distance of the submarine to the receiver, not but the direction and depth.