Some historical aspects of error calculus by Dirichlet forms [chapter]

Nicolas Bouleau
2014 Festschrift Masatoshi Fukushima  
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
more » ... . 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 [54], Von Neumann [57], Turing [58], Wilkinson [78], etc.); b) The global evaluation of deterministic errors such as the interval calculus (cf. Moore [74], etc..). Some works of Laplace are related to this approach and also the paper of Cauchy [8]; 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). The error calculus by Dirichlet forms assumes the errors to be both small, actually infinitesimal, and probabilistic. These two characteristics imply a peculiar differential calculus for the propagation of errors through models. As we will see the part of the calculation related to what is called today the squared field operator or more often the carré du champ operator, is ancient and dates back to the turn of the XVIII-th and XIX-th centuries in connection with the birth of the least squares method. Let us note, however, that our purpose is not to make a history of the method of least squares, broad topic that would lead to decline all the benefits of optimization in L 2 and its developments in statistics and analysis. I refer in this regard to the historical work of Kolmogorov and Yushkevich [84], also to the book of Pearson [85] , and to the article of Sheynin [88] not always clear from a mathematical point of view probably because of an intrinsic ambiguity of the thought of the authors of the turn of the XVII-th and XIX-th centuries, but extremely well documented. Unfortunately it does not address at all the propagation of errors.
doi:10.1142/9789814596534_0005 fatcat:bxccky6itfa4fazqtucruuplce