Numerical upscaling of parametric microstructures in a possibilistic uncertainty framework with tensor trains

A fuzzy arithmetic framework for the efficient possibilistic propagation of shape uncertainties based on a novel fuzzy edge detection method is introduced. The shape uncertainties stem from a blurred image that encodes the distribution of two phases in a composite material. The proposed framework employs computational homogenisation to upscale the shape uncertainty to a effective material with fuzzy material properties. For this, many samples of a linear elasticity problem have to be computed, which is significantly sped up by a highly accurate low-rank tensor surrogate. To ensure the continuity of the underlying mapping from shape parametrisation to the upscaled material behaviour, a diffeomorphism is constructed by generating an appropriate family of meshes via transformation of a reference mesh. The shape uncertainty is then propagated to measure the distance of the upscaled material to the isotropic and orthotropic material class. Finally, the fuzzy effective material is used to compute bounds for the average displacement of a non-homogenized material with uncertain star-shaped inclusion shapes

Fuzzy Interpolation Systems and Applications

Fuzzy inference systems provide a simple yet effective solution to complex non-linear problems, which have been applied to numerous real-world applications with great success. However, conventional fuzzy inference systems may suffer from either too sparse, too complex or imbalanced rule bases, given that the data may be unevenly distributed in the problem space regardless of its volume. Fuzzy interpolation addresses this. It enables fuzzy inferences with sparse rule bases when the sparse rule base does not cover a given input, and it simplifies very dense rule bases by approximating certain rules with their neighbouring ones. This chapter systematically reviews different types of fuzzy interpolation approaches and their variations, in terms of both the interpolation mechanism (inference engine) and sparse rule base generation. Representative applications of fuzzy interpolation in the field of control are also revisited in this chapter, which not only validate fuzzy interpolation approaches but also demonstrate its efficacy and potential for wider applications

On the Inversion of High Energy Proton

Inversion of the K-fold stochastic autoconvolution integral equation is an elementary nonlinear problem, yet there are no de facto methods to solve it with finite statistics. To fix this problem, we introduce a novel inverse algorithm based on a combination of minimization of relative entropy, the Fast Fourier Transform and a recursive version of Efron's bootstrap. This gives us power to obtain new perspectives on non-perturbative high energy QCD, such as probing the ab initio principles underlying the approximately negative binomial distributions of observed charged particle final state multiplicities, related to multiparton interactions, the fluctuating structure and profile of proton and diffraction. As a proof-of-concept, we apply the algorithm to ALICE proton-proton charged particle multiplicity measurements done at different center-of-mass energies and fiducial pseudorapidity intervals at the LHC, available on HEPData. A strong double peak structure emerges from the inversion, barely visible without it.Comment: 29 pages, 10 figures, v2: extended analysis (re-projection ratios, 2D

Rough-fuzzy rule interpolation

AbstractFuzzy rule interpolation forms an important approach for performing inference with systems comprising sparse rule bases. Even when a given observation has no overlap with the antecedent values of any existing rules, fuzzy rule interpolation may still derive a useful conclusion. Unfortunately, very little of the existing work on fuzzy rule interpolation can conjunctively handle more than one form of uncertainty in the rules or observations. In particular, the difficulty in defining the required precise-valued membership functions for the fuzzy sets that are used in conventional fuzzy rule interpolation techniques significantly restricts their application. In this paper, a novel rough-fuzzy approach is proposed in an attempt to address such difficulties. The proposed approach allows the representation, handling and utilisation of different levels of uncertainty in knowledge. This allows transformation-based fuzzy rule interpolation techniques to model and harness additional uncertain information in order to implement an effective fuzzy interpolative reasoning system. Final conclusions are derived by performing rough-fuzzy interpolation over this representation. The effectiveness of the approach is illustrated by a practical application to the prediction of diarrhoeal disease rates in remote villages. It is further evaluated against a range of other benchmark case studies. The experimental results confirm the efficacy of the proposed work