1 What are inverse problems and why are they important?

We start with a definition:

Direct problem: cause effect
Inverse problem: effect cause

The cause - effect relation is well known from our daily experience. So to speak, it is provided by nature. Nature solves direct problems routinely. There is only one solution: the world at this very moment. This is the well-known principle of causality or determinism.

If it rains (cause), the street will be wet (effect). The direct problem is simple indeed. The inverse problem, finding the cause from the effect, is more difficult. Nature does not solve this problem for us. We must do it. It seems reasonable to conclude that, since the street is wet, it must have rained not too long ago. This conclusion, taken generally, is logically false but nevertheless practically correct in most cases, but not in all. In fact, for instance, a near-by waterpipe may have broken.

Inverse problems everywhere. A comprehensive example from everyday life. In the morning, the professor gets up (cause), starts the car at 7.30, and drives to his office to give his lecture at 8.15 a.m. (effect). The direct problem in this simple case may work in 90% of the cases. In the remaining 10% the effect is different: the professor doesn't show up.The corresponding inverse problem may have many solutions. Let us take an extreme case, a veritable nightmare. The professor gets into the car and turns the key, but the car won't start. The professor calls up the emergency mechanician, who fortunately comes within a few minutes and has to solve the first inverse problem: what is wrong? The solution is not difficult because the man is an experienced professional: the battery is dead. A new battery is installed, and the professor finally takes off. Soon hereafter, he has an accident which is fortunately not too bad. The police comes and has to solve the second inverse problem: what was the cause of the accident, whose guilt is it? The professor has survived without visible damages; his car is not too bad either, so he drives to the nearest hospital. The doctor wants to make sure and takes an X-ray (result: no damage), solving the third inverse problem this morning. Finally the professor arrives at the university garage but in confusion of the morning, he cannot find his chip card which opens the gate. After a frantic search the fourth inverse problem is solved, and the professor arrives ready for his second lecture at 11 a.m., to teach about inverse problems in geophysics.

In fact, a whole science, physics of the Earth's interior, consists almost completely of inverse problems! The reason, of course, is that one cannot penetrate very deeply directly into the Earth (Jules Verne' travel to the center of the Earth has never been repeated). Fortunately, the physical structure of the Earth's interior is a cause of many measurable effects on the Earth's surface and outside, where they can be measured on an airplane or by a satellite. So you can penetrate into the Earth's interior indirectly by a satellite, by solving inverse problems which are frequently very difficult mathematically. Thus the recent boom of measuring satellites has greatly contributed to the boom of the mathematical theory of inverse problems, as one sees by the great number of internet sites on both subjects.

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