A virtual experiment simulates a real measurement process by means of a numerical model. The numerical model produces virtual data whose properties reflect those of the data observed in the real experiment. In this work, we explore how the results of a virtual experiment can be employed in the context of uncertainty evaluation for a corresponding real experiment. The uncertainty evaluation was based on the Guide to the Expression of Uncertainty in Measurement (GUM), which defines the de facto standard for uncertainty evaluation in metrology. We show that, under specific assumptions about model structure and variance of the data, virtual experiments in combination with a Monte Carlo method lead to an uncertainty evaluation for the real experiment that is in line with Supplement 1 to the GUM. In the general case, a GUM-compliant uncertainty evaluation in the context of a real experiment can no longer be based on a corresponding virtual experiment in a simple way. Nevertheless, virtual experiments are still useful in order to increase the reliability of an uncertainty analysis. Simple generic examples as well the case study of a virtual coordinate measuring machine are presented to illustrate the treatment.

Stochastic Galerkin methods for non-affine coefficient representations are known to cause major difficulties from theoretical and numerical points of view. In this work, an adaptive Galerkin FE method for linear parametric PDEs with lognormal coefficients discretized in Hermite chaos polynomials is derived. It employs problem-adapted function spaces to ensure solvability of the variational formulation. The inherently high computational complexity of the parametric operator is made tractable by using hierarchical tensor representations. For this, a new tensor train format of the lognormal coefficient is derived and verified numerically. The central novelty is the derivation of a reliable residual-based a posteriori error estimator. This can be regarded as a unique feature of stochastic Galerkin methods. It allows for an adaptive algorithm to steer the refinements of the physical mesh and the anisotropic Wiener chaos polynomial degrees. For the evaluation of the error estimator to become feasible, a numerically efficient tensor format discretization is developed. Benchmark examples with unbounded lognormal coefficient fields illustrate the performance of the proposed Galerkin discretization and the fully adaptive algorithm.

Virtual experiments (VEs) and digital twins (DTs), pivotal for realizing European strategic policies on sustainability and digitalization within Industry 4.0 and the European Green Deal, simulate physical systems and characteristics in a virtual environment, with DTs incorporating dynamic inputs from and outputs to the real-world counterpart. To ensure confidence in their use and outcomes, traceability and methods to evaluate measurement uncertainty are needed, topics that are hardly covered by the literature so far. This paper provides a harmonized definition of VEs and DTs and introduces a framework for evaluating measurement uncertainty. Furthermore, it discusses how to propagate the uncertainty of the contributions coming from the different parts of the DT. For the core part of the DT, the framework derived for VEs can be used. For the physical-to-virtual (P2V) connection and the virtual-to-physical (V2P) connection, additional sources of uncertainty need to be considered. This paper provides a metrological framework for taking all these uncertainty contributions into account while describing a framework to establish traceability for DTs. Two case studies are presented to demonstrate the proposed methodology considering industrially relevant measuring instruments and devices, namely, a coordinate measuring machine (CMM) and a collaborative robot arm (cobot).

Vimentin intermediate filaments, together with actin filaments and microtubules, constitute the cytoskeleton in cells of mesenchymal origin. The mechanical properties of the filaments themselves are encoded in their molecular architecture and depend on their ionic environment. It is thus of great interest to disentangle the influence of both the ion type and their concentration on vimentin assembly. We combine small angle X-ray scattering and fluorescence microscopy and show that vimentin in the presence of the monovalent ions, K⁺ and Na⁺, assembles into “standard filaments” with a radius of about 6 nm and 32 monomers per cross-section. In contrast, di- and multivalent ions, independent of type and valency, lead to the formation of thicker filaments associating over time into higher order structures. Hence, our results may indeed be of relevance for living cells, as local ion concentrations in the cytoplasm during certain physiological activities may differ considerably from average intracellular concentrations.

A sampling-free approach to Bayesian inversion with an explicit polynomial representation of the parameter densities is developed, based on an affine-parametric representation of a linear forward model. This becomes feasible due to the complete treatment in function spaces, which requires an efficient model reduction technique for numerical computations. The advocated perspective yields the crucial benefit that error bounds can be derived for all occuring approximations, leading to provable convergence subject to the discretization parameters. Moreover, it enables a fully adaptive a posteriori control with automatic problem-dependent adjustments of the employed discretizations. The method is discussed in the context of modern hierarchical tensor representations, which are used for the evaluation of a random PDE (the forward model) and the subsequent high-dimensional quadrature of the log-likelihood, alleviating the ‘curse of dimensionality’. Numerical experiments demonstrate the performance and confirm the theoretical results.

The Guide to the Expression of Uncertainty in Measurement (GUM) has led to a harmonization of uncertainty evaluation throughout metrology. The simplicity of its employed methodology has fostered the broad acceptance of the GUM among metrologists. However, this simplicity also compromises best practice and does not provide state-of-the-art data analysis. Specifically, metrologists often possess useful prior knowledge about the measurand which cannot be accounted for by the GUM. Bayesian uncertainty evaluation, on the other hand, takes prior knowledge about the measurand as its starting point. While this technique has been successfully applied in several instances in metrology, its broad access for metrologists is still lacking. One reason for this is that application of Bayesian methods and their computation, e.g. by means of Markov chain Monte Carlo (MCMC) methods, usually requires statistical skills. In this work, we propose a simple method for Bayesian uncertainty evaluation applicable to measurement models that depend linearly on a single input quantity for which Type A information is available, and several input quantities for which Type B information is given. This category of measurement models covers many uncertainty evaluations in metrology. The approach utilizes a specific class of prior distributions to encode the prior information about the measurand. Explicit guidance is provided on how to choose the parameters of the prior in the light of one’s prior knowledge. MCMC methods are not required for the calculation of results; instead, a simple Monte Carlo procedure is developed that allows to draw uncorrelated samples from the posterior distribution, which greatly simplifies convergence assessments. The proposed Bayesian approach allows the treatment of small numbers of observations for a Type A uncertainty evaluation, including the case of only a single observation. Examples are provided that illustrate the proposed approach and corresponding open source Python software is made available.