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Production sequence of Si-spheres and interferometrical determination of the sphere volume

Systematic simulation as a practical alternative to the Monte Carlo method for uncertainty determination



As an alternative to the Monte-Carlo method, a GUM-compatible approach is developed and investigated, which simulates the uncertainty contributions of measurements in a as systematic and separate way as possible. The aim of this approach is to estimate measurement uncertainties simulatively in a significantly reduced time compared to the Monte-Carlo method and at the same time offering a higher flexibility for everyday work. 

If a measurement process, as e.g. in the case of complex atomic force microscope measurements, cannot be represented by a functional equation but in the form of algorithms, its measurement uncertainty is often estimated with the aid of the Monte-Carlo method through a large number of simulations. If the consideration of one or more influencing variables in these simulations is computationally expensive, such as the consideration of the exact tip geometry in atomic force microscope measurements, the Monte-Carlo method can quickly lead to simulation times of several weeks. However, if the superposition principle is approximately fulfilled for the influence quantities, they can each be propagated separately with a significantly reduced number of systematic simulations in which the probability density functions of the influence quantities are sampled in fixed quantiles. With approximate techniques such as kernel density estimators, the propagated probability density functions of the influence quantities can be determined from these simulations. The propagated probability density functions can be combined by convolution to approximate the probability density function of the measurand. In particular, the number of time-consuming simulations can be significantly reduced, but also the total number of simulations itself, can be reduced as well with this approach compared to the Monte-Carlo method. Initial investigations of the approach indicate that the uncertainties of complex atomic force microscope measurements, for example, can be estimated many times faster with the new approach than by using the classical Monte-Carlo approach.

Figure: Schematic structure of the systematic approach. The probability density distributions (PDF) of the influence quantities (EG) of the measurement process are individually numerically propagated and their propagated probability density distributions are approximated. These contain the information about the contribution of the influence quantities to the measurement uncertainty (MU) of the measurand and can be convoluted to determine the probability density distribution of the measurand.