Asymptotology

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essay = September 2008
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Asymptotology – is the art of handling with applied mathematical systems in limiting cases (M.Kruskal); is the science about the synthesis of simplicity and exactness by means of localization (R.G.Baranzev).

The first time one encounters asymptotic in the school geometry dealing with the idea of an asymptote, which is defined as a line approaching a certain curve when tending to infinity. The word ἀσύμπτωτος (asymptotos) in Greek means „non-coincident“. It puts strong emphasis on the point that approximation does not turn into coincidence. It is salient feature of asymptotics, but the property alone does not cover entirely the idea of asymptotics and, etymologically, the term seems to be quit insufficient.

In physics and other fields of science one frequently comes across the problems of asymptotic nature, such as damping, orbiting, stabilization of a perturbed motion, etc. Asymptotic analysis (perturbation theory) is widely used in the modern applied mathematics, mechanics and physics. But asymptotic methods put a claim on being more then a part of classical mathematics. K.Friedrichs said: “Asymptotic description is not only a convenient tool in the mathematical analysis of nature, it has some more fundamental significance”. M.Kruskal introduced a special term “asymptotology” and defined it as an art of handling applied mathematical systems in limiting cases. He called for a formalization of the accumulated experience to convert the art of asymptotology to science.

An apt generalizing term is capable of possessing significant heuristic value. H.Poincaré wrote: “It is hard to believe what a huge amount of thinking a proper chosen word can save. Often it is enough to invent one new word to make it creative itself. Mathematics is the art of giving a variety of names to similar things. Certain facts become significant, when some shrewd thinker picks up the similarities brought up by the fact, and symbolically names it by a certain term”. “The success of ‘cybernetics’, ‘attractors’ and ‘catastrophe theory’ illustrates the fruitfulness of word creation as scientific research” (V.Arnol’d).

Almost every physical theory, formulated in the most general manner, is very difficult from the mathematical point of view. Therefore both at the genesis of the theory and its further development the simplest limit cases, which allow analytical solutions, are of particular importance. In those limits the number of equations usually decreases, their order reduces, nonlinear equations can be replaced by linear ones, the initial system becomes averaged in a certain sense, and so on. All these idealizations, different as they may seem, imply a high degree of symmetry, which is matching limit cases of the mathematical model of the phenomenon under consideration. The asymptotic approach to a complex problem in essence consists in treating the insufficiently symmetrical governing system as close to a certain symmetrical one. It is very important that the determination of correctness, which takes into account deflections from the limit case, is much simpler than direct investigation of the governing system. At first site, the possibilities of such an approach seem restricted to a narrow range of variation of the parameters, which determine the system. However, experience in the investigation of different physical problems shows that, if the system’s parameters have changed sufficiently and the system has far deviated from the symmetrical limit case, there can be found another limit system, often with less obvious symmetries, to which an asymptotic analysis is also applicable. This allows one to describe the system’s behavior on the basis of a small number of limit cases over the whole range of variation of the parameters. Such an approach corresponds to the maximum level of intuition and promotes its developments, and eventually leads to the formulation of new physical concepts. It is also important that asymptotic methods help to establish the connection between different physical theories. The aim of the asymptotic approach is to simplify the object. This simplification is attained by decreasing the vicinity of the singularity under consideration. It is typical that the accuracy of asymptotic expansions grows with localization. Exactness and simplicity are commonly regarded as antagonistic and complementary notions. When tending to simplicity we sacrifice exactness, and trying to achieve exactness we expect no simplicity. Under localization, however, the antipodes converge, the contradiction is resolved in a synthesis which is called asymptotics. In other words, simplicity and exactness are coupled by “uncertainty principle” relation while the domain size serves as a small parameter - measure of uncertainty. Let us illustrate “asymptotic uncertainty principle”. Let us take an expansion of the function f(x) in an asymptotic sequence {φn(x)}:f(x) = A partial sum of the series is designated by SN(x) and the exactness of approximation at a given N is estimated by ∆N(x) = │f(x) - SN(x) │. Simplicity is characterized here by the number N, and the locality by the length of interval x. Now let us consider in pairs the interrelation of values x, N, Δ basing ourselves on known properties of asymptotic expansion. At a fixed x the expansion initially converges, i.e., the exactness increases at the cost of simplicity. If we fix N, the exactness and the interval size begin to compete. The smaller the interval, the simpler the given value of Δ is reached.

Let us illustrate these regularities using a simple example. We consider the integral exponential functionEi(y) = y < 0. Integrating by parts we obtain the following asymptotic expansion Ei(y) ~ ey y → - .Put f(x) = -e-y Ei(y), y = -x-1. By calculating the partial sums of this series and the value ΔN(x) and f(x) for different x one obtains:

x f(x) Δ1 Δ2 Δ3 Δ4 Δ5 Δ6 Δ7 1/3 0.262 0.071 0.040 0.034 0.040 0.060 0.106 0.223 1/5 0.171 0.029 0.011 0.006 0.004 0.0035 0.0040 0.0043 1/7 0.127 0.016 0.005 0.002 0.001 0.0006 0.0005 0.0004

Thus at a given x the exactness firstly grows with the growth of N and then decreases (so one has asymptotic expansion). For given N one may observe an improvement of exactness with diminishing x.

Last, but not least. Is it worth using asymptotic methods if computers and numerical procedures have been developed so much? As D.Crighton mentioned, “Design of computational or experimental schemes without the guidance of asymptotic information is wasteful at best, dangerous at worst, because of possible failure to identify crucial (stiff) features of the process and their localization in coordinate and parameter space. Moreover, all experience suggests that asymptotic solutions are useful numerically far beyond their nominal range of validity, and can often be used directly, at least at a preliminary product design stage, for example, saving the need for accurate computation until the final design stage where many variables have been restricted to narrow ranges”. We can reformulate Galileo famous sentence: "The book of Nature is written in the language of asymptotology".

References

Kruskal M.D. Asymptotology. Proceedings of Conference on Mathematical Models on Physical Sciences. Englewood Cliffs, HJ: Prentice-Hall, 1963, 17-48.

Barantsev R.G. Asymptotic versus classical mathematics // Topics in Math. Analysis. Singapore e.a.: 1989, 49-64.

Friedrichs K.O. Asymptotic phenomena in mathematical physics // Bull. Amer. Math. Soc., 1955, 61, 485-504.

Segel L.A.The importance of asymptotic analysis in Applied Mathematics// Amer. Math. Monthly, 1966, 73, 7-14.

Andrianov I.V., Manevitch L.I. Asymptotology: Ideas, Methods, and Applications. Dordrecht, Boston, London: Kluwer Academic Publishers, 2002.

Dewar R.L. Asymptotology – a cautionary tale. ANZIAM J., 2002, 44, 33-40.

Crighton D.G. Asymptotics – an indispensible complement to thought, computation and experiment in Applied Mathematical modelling//Seventh Eur. Conf. Math. in Industry (March 2-6,1993, Montecatini Terme). A.Fasano, M.Primicerio (eds.). Stuttgart: B.G.Teubner, 3-19.