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Math Help - Modeling inaccuracies as a broad topic

  1. #1
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    Modeling inaccuracies as a broad topic

    Greetings,

    This is my first question in these forums. It is very general, for which I apologize in advance. If you think this is not the right place, I would appreciate directions to the appropriate site/forum where it would be better addressed.

    My math: some linear and abstract algebra, calculus/analysis, some mathematical logic, probability/statistics. I have had some more exotic courses in my curriculum, but not in such relevance to my question here. I am (try to be) a programmer (software engineer if I try to be fancy).

    Here is my problem. Many mathematical statements depend on the faithfulness of the model/theory, i.e. the compliance of the modeled entities with the axioms / the applicability of a system of constraints. But in practice this is never the case entirely and every observation provides evidence to the contrary to some extent. If I allow myself to make a profane guess, I am under the impression that even stochastic solutions use rather broad strokes, i.e. based on probability distributions. I am not sure how much qualification such solutions can subsume before quantification. (May be?)

    I want to give one example just to illustrate my confusion. A shape that resembles a ball, also truly solid(?), generally is expected to have small perturbations of its surface. Consequently it may have a surface area that is considerably bigger. This is the contact surface between the physical object and its environment, which I imagine, directly determines the dynamics of many physical processes. There will be a considerable deviation. In practice, because formulas are specified with many parameters for calibration, the result will be adjusted accordingly, but not because the math portion of the theory demands it. I can similarly advocate that there is no tangent plane at any point of the surface because the secant planes approaching the exterior do not converge. (These correspond to secant constructions at ever bigger magnifications of the physical object).

    My question --- Are there fields that address the subject of inaccurate modeling and its effects in detail? For example, in algebraic theory I imagine morphism-like functions between the elements of two algebraic structures. Say, instead the homomorphism between two groups, the theory studies functions that map between the group operations with certain loss of precision, or with some random error. Vector spaces, rings, fields, etc. A vector space may be allowed slight inaccuracy of the distributive law. The morphism between such pseudo-vector space and a real one could be studied. How much loss of structure results in how much loss of applicability of solutions to problems constructed in the "fully" structured space. I wonder if such studies could yield some prescriptive results. In analytic geometry, shapes and surfaces may be modified by various disturbances. Similarly in calculus/analysis functions that are not entirely smooth may be associated with smooth counterparts (or collections of smooth counterparts). What properties justify "approximate" integration and differentiation? (May be this is handled by numerical analysis to some extent.) Is it possible to study structural deviations in this highly abstract manner, or are they to be handled primarily on a case by case basis?

    I realize I don't have concrete idea what I am looking for. Anything you can point me to of that nature will be appreciated.

    Thanks and regards,
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  2. #2
    A Plied Mathematician
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    I can point you to one or two things of interest. In modeling differential equations, there is an entire field of inquiry entitled "sensitivity analysis". The idea there is to compute derivatives of outputs with respect to various inputs, all dependent on the DE's governing the system. That field can quantify to some extent the possible errors introduced by what are, essentially, fractals (surface area computed by somehow integrating around each molecule, for instance!).

    One other source I am aware of is the dependence of solutions on initial conditions in the field of ODE's. This has been handled in Coddington and Levinson (as, in fact, many things have been handled!).

    Although it sounds like you're more interested in extensive changes to a model, you might also check out perturbation theory.

    Hope this is at least somewhat helpful. I, alas, know very little about any of these fields, so this is as much as I can say, pretty much. Good luck!
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  3. #3
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    Thanks Ackbeet.

    I am not familiar with the fields of sensitivity analysis and perturbation theory at all. Reading from the wikipedia articles left me with some impression on what perturbation theory might look like, but I can't say the same for the article on sensitivity analysis. What I gathered from the overall use of terms is that it would be extremely supportive for problems in control engineering. I have had a course in ODE (fortunately) and I think I understand your reference to the study of initial conditions.

    These could be good fields for me to delve into, not because they fit the bill to my inquiry entirely (although, for all I know, they might), but because they study imperfect models and that makes me happy. Either way, may be the subject I am looking for is considered somewhat philosophical and the practitioners handle the problems sufficiently well without formalization.

    In fact, the ridiculous example with the imperfect ball I have given above and also, say, the state-of-practice to differentiate objects that are not differentiable is a frequent mind-game for me and one of the things bothering me. I have had an introductory course on generalized functions, but they addressed (IMHO) a rather different issue. It bothers me because of the apparent simplicity of such deviations, as though they couldn't have practical consequences. But of course they do, as strictly speaking nothing is without quantitative consequences (; as the surface area effect from my original post.) I thought that a relatively fundamental field exists that researches such intricacies. That is, such issues are fundamental to mathematics in general, not just to engineering, social sciences and applied math.

    Either way, thank you very much Ackbeet. I feel I have enough to explore and at least it raises my morale. I leave the question open for now, just in case. I mean, may be it is an open question.

    Thanks and regards,
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  4. #4
    A Plied Mathematician
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    You're welcome. Have a good one!
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