HAL CCSD
A Priori Error Analysis and Spring Arithmetic
Chabert, Gilles
Jaulin, Luc
Laboratoire d'Informatique de Nantes Atlantique (LINA) ; Mines Nantes (Mines Nantes)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS)
Développement des Technologies Nouvelles (DTN) ; École Nationale Supérieure de Techniques Avancées Bretagne (ENSTA Bretagne)
WOS
International audience
ISSN: 1064-8275
SIAM Journal on Scientific Computing
Society for Industrial and Applied Mathematics
hal-00428952
https://hal.science/hal-00428952
https://hal.science/hal-00428952/document
https://hal.science/hal-00428952/file/69698.pdf
https://hal.science/hal-00428952
SIAM Journal on Scientific Computing, 2009, 31 (3), pp.2214-2230. ⟨10.1137/070696982⟩
DOI: 10.1137/070696982
info:eu-repo/semantics/altIdentifier/doi/10.1137/070696982
en
Error analysis
interval arithmetic
global optimization
AMS 65G40
[MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
info:eu-repo/semantics/article
Journal articles
Error analysis is defined by the following concern: bounding the output variation of a (nonlinear) function with respect to a given variation of the input variables. This paper investigates this issue in the framework of interval analysis. The classical way of analyzing the error is to linearize the function around the point corresponding to the actual input, but this method is local and not reliable. Both drawbacks can be easily circumvented by a combined use of interval arithmetic and domain splitting. However, because of the underlying linearization, a standard interval algorithm leads to a pessimistic bound, and even simply fails (i.e., returns an infinite error) in case of singularity. We propose an original nonlinear approach where intervals are used in a more sophisticated way through the so-called "springs". This new structure allows to represent an (infinite) set of intervals constrained by their midpoints and their radius. The output error is then calculated with a spring arithmetic in the same way as the image of a function is calculated with interval arithmetic. Our method is illustrated on two examples, including an application of geopositioning.
2009
info:eu-repo/semantics/OpenAccess