WIAS Preprint No. 2580, (2019)

Low rank surrogates for polymorphic fields with application to fuzzy-stochastic partial differential equations


  • Eigel, Martin
  • Grasedyck, Lars
  • Gruhlke, Robert
    ORCID: 0000-0003-3129-9423
  • Moser, Dieter

2010 Mathematics Subject Classification

  • 15A69 35R13 35R60 60H35 65C20 65N12 65N22 65J10 74B05 97N50


  • Fuzzy-stochastic partial differential equations, possibility, polymorphic uncertainty modeling, uncertainty quantification, low-rank hierachical tensor formats, parameteric partial differential equations, polymorphic domain




We consider a general form of fuzzy-stochastic PDEs depending on the interaction of probabilistic and non-probabilistic ("possibilistic") influences. Such a combined modelling of aleatoric and epistemic uncertainties for instance can be applied beneficially in an engineering context for real-world applications, where probabilistic modelling and expert knowledge has to be accounted for. We examine existence and well-definedness of polymorphic PDEs in appropriate function spaces. The fuzzy-stochastic dependence is described in a high-dimensional parameter space, thus easily leading to an exponential complexity in practical computations. To aleviate this severe obstacle in practise, a compressed low-rank approximation of the problem formulation and the solution is derived. This is based on the Hierarchical Tucker format which is constructed with solution samples by a non-intrusive tensor reconstruction algorithm. The performance of the proposed model order reduction approach is demonstrated with two examples. One of these is the ubiquitous groundwater flow model with Karhunen-Loeve coefficient field which is generalized by a fuzzy correlation length.

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