Why Using Tolerance Solutions Is a Good Idea

Abstract

In many practical situations, we are interested in the values of physical quantities x=(x_1,...,x_n) which are difficult (or even impossible) to measure directly. In this case, a natural idea is to find auxiliary quantities y_1,...,y_m which are related to the desired quantities xi by a known dependence y_i=f(a_i,x), measure the values y_i, and then reconstruct xi from the measurement results. In the ideal case when measurement are absolutely accurate and when we know the exact values of the corresponding parameters a_i, we can find the values x_i by solving the corresponding system of equations y_i=f(a_i,x). In practice, due to measurement errors, the values y_i are somewhat different from the ideal values y_i=f(ai,x) and for similar reasons, the values a_i are also only known with some uncertainty. In some practical situations, we do not even know the accuracy of measuring y_i. To handle such situations, V. Golodov and A. V. Panyukov proposed to use a special method of ＂tolerance solutions.＂ This semi-heuristic method has been very successful in several applications. In this paper, we provide a possible justification for this success.