WIAS Preprint No. 2907, (2021)

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


  • Eigel, Martin
    ORCID: 0000-0003-2687-4497
  • Gruhlke, Robert
    ORCID: 0000-0003-3129-9423
  • Moser, Dieter

2020 Mathematics Subject Classification

  • 15A69 35R13 65N12 65N22 65J10 74B05 97N50


  • Fuzzy-stochastic partial differential equations, possibility, polymorphic uncertainty modeling, uncertainty quantification, low-rank hierarchical tensor formats, parametric partial differential equations, linear elasticity, homogenisation, tensor trains




We develop a new fuzzy arithmetic framework for efficient possibilistic uncertainty quantification. The considered application is an edge detection task with the goal to identify interfaces of blurred images. In our case, these represent realisations of composite materials with possibly very many inclusions. The proposed algorithm can be seen as computational homogenisation and results in a parameter dependent representation of composite structures. For this, many samples for 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 effective material tensor map, an appropriate diffeomorphism is constructed to generate a family of meshes reflecting the possible material realisations. In the application, the uncertainty model is propagated through distance maps with respect to consecutive symmetry class tensors. Additionally, the efficacy of the best/worst estimate analysis of the homogenisation map as a bound to the average displacement for chessboard like matrix composites with arbitrary star-shaped inclusions is demonstrated.

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