5 Applications

Examples of direct and inverse problems

This table is largely self-explanatory. Land masses such as mountains or rocks, through their gravitational (Newtonian) attraction, produce a gravity field. Nature thus automatically solves a direct problem. The inverse problem, to determine the rock masses by measuring gravity, is a mathematical problem, apparently simple, in reality, however, surprisingly deep and difficult. Generally the problem arises to determine the interior from data measured outside, without penetrating into the body's interior. As we have just seen, the gravity problem is of this nature. In science, technology and medicine, such problems are freqently called tomography, for instance X-ray tomography in medicine or seismic tomography in geophysics. (Seismic waves are the geophysicist's X-rays!)

Generally, God (or however you like to put it) creates nature and thus solves a direct problem; the natural scientist, in exploring nature, has to solve a corresponding inverse problem. Such a conceptual structure occurs everywhere: the criminal commits a crime (direct problem), and the detective, investigating the crime, must solve an inverse problem. The scientist as detective of nature!

Another example. An expert programmer writes a program, say in the language C++, the source code (cause), the compiler, compiles it into an EXE-code, and it runs and gives the expected result (effect). This is clearly an example of a straightforward direct problem. (In practice, alas, it is frequently different: instead of a nice EXE code, the compiler provides several error messages. Obviously, finding and correcting the errors is a sometimes quite frustrating, but almost unavoidable inverse problem). Summarizing, we may say: source code EXE-code: direct problem, relative easy; EXE-code source code: inverse problem, in general unsolvable (this is why software companies make money by selling only the EXE-code).

We may even generalize. Logical deduction is a direct problem, but scientific inference is an incomparably more difficult inverse problem. Don't believe Sherlock Holmes or Hercule Poirot that they have solved their problem by pure logical deduction: they had to solve their criminalistic problem by the inverse problem of induction, which makes their solution so difficult, interesting and surprising.

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