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Low-Rank Tensor Methods for Model Order Reduction

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Handbook of Uncertainty Quantification

Abstract

Parameter-dependent models arise in many contexts such as uncertainty quantification, sensitivity analysis, inverse problems, or optimization. Parametric or uncertainty analyses usually require the evaluation of an output of a model for many instances of the input parameters, which may be intractable for complex numerical models. A possible remedy consists in replacing the model by an approximate model with reduced complexity (a so-called reduced order model) allowing a fast evaluation of output variables of interest. This chapter provides an overview of low-rank methods for the approximation of functions that are identified either with order-two tensors (for vector-valued functions) or higher-order tensors (for multivariate functions). Different approaches are presented for the computation of low-rank approximations, either based on samples of the function or on the equations that are satisfied by the function, the latter approaches including projection-based model order reduction methods. For multivariate functions, different notions of ranks and the corresponding low-rank approximation formats are introduced.

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Authors and Affiliations

  1. Department of Computer Science and Mathematics, GeM, Ecole Centrale Nantes, GeM, 1 rue de la Noë, 44321, Nantes, France

    Anthony Nouy

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  1. Anthony Nouy
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Correspondence to Anthony Nouy .

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Editors and Affiliations

  1. Department of Civil and Environmental Engineering, University of Southern California, Los Angeles, California, USA

    Roger Ghanem

  2. Social and Decision Analytics Laboratory, Virginia Bioinformatics Institute, Virginia Tech University, Arlington, Virginia, USA

    David Higdon

  3. Computing and Mathematical Sciences, California Institute of Technology, Pasadena, California, USA

    Houman Owhadi

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Nouy, A. (2017). Low-Rank Tensor Methods for Model Order Reduction. In: Ghanem, R., Higdon, D., Owhadi, H. (eds) Handbook of Uncertainty Quantification. Springer, Cham. https://doi.org/10.1007/978-3-319-12385-1_21

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