Some historical aspects of error calculus by Dirichlet forms

Abstract

Abstract.We discuss the main stages of development of the error calculation since the beginning of XIX-th century by insisting on what prefigures the use of Dirichlet forms and emphasizing the mathematical properties that make the use of Dirichlet forms more relevant and efficient. The purpose of the paper is mainly to clarify the concepts. We also indicate some possible future research.I. Introduction. There are several kinds of error calculations which have not followed the same historical development. The error calculus by Dirichlet forms that we will explain and trace the origins has to be distinguished from the following calculations:a) The calculus of roundoff errors in numerical computations which appeared far before the representation of numbers in floating point be implemented on computers, and which possesses its specific difficulties. It has been much studied during the development of the numerical analysis for matrix discretization methods (cf. Hotelling c) The calculus of finite probabilistic errors where the errors are represented by random variables, which has been used by a very large number of authors to begin an argument and then, often, modified by supposing the errors to be small or gaussian in order to be able to pursue the calculation further (cf. Bienaymé [24], Birge [48], Bertrand [35], etc.) because the computation of image probability distributions is concretely inextricable what, in the second half of the XX-th century, justified the development of simulation methods (Monte-Carlo and quasi-Monte-Carlo).