Propagation of uncertainties and applications in numerical modeling: tutorial

Abstract

Some inputs of computational models are commonly retrieved from external sources (handbooks, articles, dedicated measurements), and therefore are subject to uncertainties. The known experimental dispersion of the inputs can be propagated through the numerical models to produce samples of outputs. The stemming propagation of uncertainties is already significant in metrology but also has applications in optimization and inverse problem resolution of the modeled physical system. Moreover, the information on uncertainties can be used to characterize and compare models, and to deduce behavior laws. This tutorial gives tools and applications of the propagation of experimental uncertainties through models. To illustrate the method and its applications, we propose to investigate the scattering of light by gold nanoparticles, which also enables the comparison of the full Mie theory and the dipole approximation. The position of the localized surface plasmon resonance and the corresponding value of the scattering efficiency are more specifically studied.

Gold Nanoparticles as a Photothermal Agent in Cancer Therapy: The Thermal Ablation Characteristic Length

“…The intersections between the ablation threshold (horizontal line at ) and the curves of the temperature reveal that the TACL varies between 50 nm and 120 nm, even if the laser power is increased by 50% and the volume of the nanoparticle by 100%. Therefore, a classical calculation of the propagation of uncertainties may be adequate to evaluate the sensitivity of the TACL to the laser power and the nanoparticles’ diameters dispersion [ 25 ]. To check the validity of our claim, we also evaluate the variations of the TACL (Equation ( 5 )) at a central temperature of 47 C: .…”

“…Assuming a legitimate assumption of the dispersion of nanoparticle diameters and of laser power (due to the penetration in tissues), the uncertainty on the target maximum temperature in the nanoparticle helps to evaluate the sensitivity of the temperature elevation to the variations of the diameter and the illumination power. With such an approximation and expressing and as a function of and Q as a function of , the uncertainty over the maximum temperature can be deduced [ 25 ]: where are the relative uncertainties; for the values of laser power W with uncertainty mW, a nanoparticle diameter nm with dispersion nm and a target temperature value in the nanoparticle C, C. This value is small with reference to the target values . This first numerical result shows that a wide range of diameter and laser power can be used to reach the target values.…”

Optical metrology for digital manufacturing: a review

“…Ongoing research in evaluating uncertainty in point clouds represents an interesting new method of uncertainty evaluation [120,[132][133][134][135][136][137][138][139][140][141][142][143][144][145][146][147][148], particularly within the scope of optical in-line measurement. Future work will investigate how error in 3D point clouds may propagate through the algorithmic procedures commonly applied at the industrial level, to verify whether workpieces conform to geometric and dimensional specifications.…”

“…Concerning the evaluation of uncertainty associated with point clouds, conventional approaches (such as [145,146]) are not suitable, due to the multitude of possible error sources and the complexities of their interactions. These issues can lead to significant difficulties in the mathematical modelling of the aggregated errors failing to produce a comprehensive analytical representation of uncertainty.…”

Performance of Surface Plasmon Resonance Sensors Using Copper/Copper Oxide Films: Influence of Thicknesses and Optical Properties

“…Therefore, the resolution of a difficult inverse problem would require more than one metaheuristic to ensure the best result. The repeated realizations of the same algorithm and the use of a tolerance threshold for the values multi-objective function (here 25%) help assess the stability of the methods and the relevance of the outcome [55]. Using this careful approach leads to better results than those found in Reference [15].…”