How to specify an approximate numerical result

TL;DR

This paper explores how Dirichlet forms and stochastic calculus can be used to better understand and specify errors in numerical approximations, especially in strongly stochastic measurement scenarios.

Contribution

It introduces a new perspective on numerical errors using Dirichlet forms and stochastic calculus, emphasizing the strongly stochastic nature of measurement errors.

Findings

Errors due to graduated instrument measurements are strongly stochastic.

An Ito-like second order calculus is relevant for describing certain numerical errors.

Implications for specifying approximate numerical results are discussed.

Abstract

The Dirichlet forms methods, in order to represent errors and their propagation, are particularly powerful in infinite dimensional problems such as models involving stochastic analysis encountered in finance or physics, cf. [5]. Now, coming back to the finite dimensional case, these methods give a new light on the very classical concept of 'numerical approximation' and suggest changes in the habits. We show that for some kinds of approximations only an Ito-like second order differential calculus is relevant to describe and propagate numerical errors through a mathematical model. We call these situations strongly stochastic. The main point of this work is an argument based on the arbitrary functions principle of Poincar\'e-Hopf showing that the errors due to measurements with graduated instruments are strongly stochastic. Eventually we discuss the consequences of this phenomenon on the specification of an approximate numerical result.