Reduction of some physical problems to finding the minimum of the maximum deviation

Abstract

THE PROBLEM of the approximation of experimental curves by functions specified analytically, the choice of which depends on a set of parameters, is considered. The choice of those values of these parameters which ensure the best approximation leads to the problem of the minimization of the maximum and meansquare deviations. It is proved that the perturbations of the optimal values of the parameters caused by errors of experiment and other factors resulting from the passage to the minimization problem, can be made arbitrarily small. In an entire series of physical problems the necessity arises of simulating experimental curves by means of analytically specified functions. As a rule these functions depend on a set of parameters describing internal properties or the state of the physical system studied. It is then natural to use those values of the parameters for which the experimental curves are best approximated by functions of a chosen form. The problem of best approximation is one of the most common of the problems reducing to the problem of minimizing the maximum deviation. However, in the given case the characteristics of the approximation problem are such that the “validity” of the reduction of its solution to finding the minimum of some function (of the maximum deviation) requires additional validation. Indeed, if A y ( x, y) is a curve obtained in an ideal experiment, that is, one where the experimental errors ϵ ( x) are equal to zero (here y ϵ R m describes physical conditions in which the experiment takes place, and x is the argument of the “function” A y ( x, y), time, frequency and the like), the problem of the approximation of the experimental curves A y ( x, y) by functions of the form a t ( x, y, a) on some segment of the x-axis ([ x 1, x 2]) in the domain of physical conditions Δ′ consists of finding on some compact Ω ⊂ R n of solutions of the equation max |A y (x,y) − A x (x,y α) |= x∈[x t,x 2] y∈δ′ = min max | A y (x,y)− A T(x,y,α)|, α∈Ω x∈[X 1,X 2 y∈δ′ and we are immediately forced to replace it by the equation max |A y (x,y) − A x (x,y α)|= x∈[x t,x 2] (x,y) ∈ M = min max | A y (x,y)− A T(x,y,α)|, α∈Ω (x,y)∈M where M is some finite subset of the set Δ = [ x 1, x 2] × Δ′ , since we can have at our disposal values of A y ( x, y) only for a finite number of y ϵ Δ′ and x ϵ [ x 1, x 2]. Moreover, at points of the set M we actually obtain not values of A y ( x, y), but values of a z ( x, y) = A y ( x, y) + ϵ( x) (we recall that ϵ( x) is a random function, describing the experimental errors), and this leads to the equation max | A y (x,y) + ϵ (x) − A T(x,y,α) | = (x,y) ∈ M = min max |A y(x,y) + ϵ (x)− A T(x,y,α) |, α∈Ω (x,y)∈M From this it is easy to draw the conclusion that validation of the passage from the physical problem to the problem of finding the minimum of the maximum deviation is necessary and consists of establishing the connection between the solutions of equations (0.1) and (0.3). The consequences of replacing equation (0.1) by (0.2) are studied in section 1 of the present paper, section 2 is devoted to questions of the passage from (0.1) and (0.2) to (0.3). In section 3 we consider questions similar to those considered in sections 1 and 2, but for the case where the criterion of approximation is not the maximum, but the mean-square deviation. Before considering the fundamental results of the paper, we mention that in [1, 2] the author in collaboration with A. G. Pokrovskii posed the question of the reduction to the problem of finding the minimum of the maximum deviation of the problem of determining the values of the quantum-mechanical parameters of the absorption spectra of atmospheric gases in the infrared region. The question of the reduction of this problem to the finding the mean-square deviation was raised in [3].