Talk:Inverse problem

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What isn't Inverse Problems research about? For a general (non-scientist, non-mathematician) reader, the topic of inverse problems comes across as quite striking. But it is described in ways sometimes so general (the aim of obtaining parameters from data) that it's a little hard to see why, for instance, statistical theory, at least in its inferential aspects, isn't regarded as a subfield of inverse problems research. Is that in fact the implication? Somehow I get the feeling that such an implication is not intended. Is the phrase "inverse problems" short for something? It doesn't seem to be short for "inverse variational or optimizational problems," but, even after visiting various of the links, I'm unsure. (Still, thank you to whoever wrote the article, I was glad to find it). - Ben Udell

In my opinion, as someone who works in the field, it is not a very well defined term. "Ill-posed problem" is a better term, but I think perhaps the use of "inverse problem" arose so as not to seem that we were doing something impossible. However among those who work in the area, there are definately a collection of problems we all agree are inverse problems, and I expect some that we would not all agree on as well. It overlaps with statistical inference, and indeed there is a statistical approach to IP, but there many who treat IP from a non-statistical point of view. I'm sorry but there it is. It is a vaugue term. Perhaps User:Tarantola can explain better? Billlion 17:49, 6 June 2006 (UTC)

Thank you. The picture I'm getting is that it's a question of going from incomplete data to estimate or adjust parameters of a larger set. That's so general that it seems to include statistical inference without necessarily being limited to it. I see nothing objectionable in the idea of something's including statistical inference and inverse optimization as well, in that way. After all, statistical theory deals with a problem inverse to that of probability theory. One seems to discern a whole family of areas dealing with problems inverse to those of deductive maths of optimization, probability, information, even logic. But I just get the feeling from the article that such a general sense of the phrase "inverse problem" would be considered too general for what the article is referring to -- for instance, you speak of "overlap with" statistical inference rather than "inclusion" or "encompassment" of it. Anyway, I've rambled enough, if my question isn't clear at this point, then it's my fault, I'll need to do more than surf around the 'Net, maybe go out and buy a book or something! - Ben Udell

Update: Well, I haven't bought a book, though I have read some of Tarantola's online discussion. As far as I can discern through the mists of my own ignorance, there are basically two issues involved in distinguishing the inverse problem as discussed in the Wikipedia article (hereinafter capitalized "the Inverse Problem") from the seemingly more general terrain of problems which are "inverse" to those studied in deductive math theories of optimization, probability, information, & logic.

1. The Inverse Problem pertains to an actually developing interdisciplinary discipline of research for which it could be counterproductive to mount too ambitiously general a definition, e.g., such as to encompass among its subdivisions the whole of statistical theory, statistical theory of stochastic processes, etc.

2. Inverse Problem research is not always carried out in a "general" way but instead is often in the form of research into specific physical questions, where the generation of the explanatory content of hypotheses looms larger as a desideratum than it does, say, in general research in statistics. A field which understands itself as dealing with a problem inverse to that of probability or deductive logic will tend to understand itself as drawing conclusions usually in the form of inductive generalizations (statistically based or otherwise, though, to be sure, I don't mean mathematical induction); Inverse Problem research is not prepared to limit its conclusions to inductive generalizations, statistically based or otherwise, such as tend to answer problems the inverse of strictly deductive math problems of optimization, probability, information, logic -- instead, Inverse Problem research is concerned with "ill-posed" problems generally (or with some cross-section thereof), such that Inverse Problem research includes the generation of explanatory content of hypotheses.

Are 1. & 2., more or less, the issues? - Ben Udell

[edit] Bibliography too long

I suggest we need a severe cull of references, there are simply to many. It might be appropriiate to put them back in more specific articles (eg Inverse problems in geophysics)? Please discuss Billlion 09:48, 20 August 2006 (UTC)