6 Conclusions

Direct and inverse problems again. Fore most problems which we have considered so far, the inverse problem is considerably more difficult than the corresponding direct problem. This is true in most cases which are considered by applied mathematicians at the present time. It is a pragmatic rather than logical distinction. Logically, one could easily convert an inverse problem into a direct problem by interchanging the operators A and A* (see Section 3). Pragmatically, the direct problem is the problem which has a well-known mathematical structure or which can be solved more easily.

Frequently, an inverse problem is not only much more difficult to solve, but the solution is also less stable and less smooth. Cf. the mass-from-gravity problem in Section 5. Inverse problems are frequently "improperly posed", see Section 4.

An elementary partial counterexample from calculus: differentiation of a function is usually considered the direct problem, for good reasons (it is always directly calculable by an algorithm, etc.), so the integration is the inverse problem. In fact, integration can frequently not be performed by an algorithm and its solution may even not be possible by elementary functions. On the other hand, differentiation usually roughens, and thus destabilizes, a function, wheras integration smoothes and thus stabilizes it..... So matters are not that easy.

Measurement space. In many cases it is useful to consider measurements as partial projections of reality. If in modern jargon, one speaks of "measurement space" as having a much lower dimensionality than the high (or even infinite) dimensional space of reality (of course, we mean the many parameters occurring in natural sciences rather than our usual geometrical three-dimensional space). It is quite easy to "project" a landscape into a photographic picture (direct), but it is in general impossible to reconstruct the landscape from a single photograph (inverse).

Philosophical implications. All we know about nature is obtained by observation and thinking about them. Scientists speak of scientific inference or induction, as opposed to purely logical deduction. (In practice one frequently starts from a "working hypothesis", deduces from it the values for the observables and  accepts or rejects the hypothesis depending on the agreement or less of pre-computated and observed values.) Anyway, we have again: deduction is the direct  and induction is the inverse problem.

A much more detailed discussion will be found in the main paper, where also a detailed list of references can be found.

Here we mention only our books:

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