WIAS Preprint No. 2844, (2021)

Efficient approximation of high-dimensional exponentials by tensor networks


  • Eigel, Martin
    ORCID: 0000-0003-2687-4497
  • Farchmin, Nando
  • Heidenreich, Sebastian
  • Trunschke, Philipp

2020 Mathematics Subject Classification

  • 35R60 47B80 60H35 65C20 65N12 65N22 65J10


  • Uncertainty quantification, dynamical system approximation, Petrov--Galerkin, a posteriori error bounds, tensor product methods, tensor train format, holonomic functions, Bayesian likelihoods, log-normal random field




In this work a general approach to compute a compressed representation of the exponential exp(h) of a high-dimensional function h is presented. Such exponential functions play an important role in several problems in Uncertainty Quantification, e.g. the approximation of log-normal random fields or the evaluation of Bayesian posterior measures. Usually, these high-dimensional objects are intractable numerically and can only be accessed pointwise in sampling methods. In contrast, the proposed method constructs a functional representation of the exponential by exploiting its nature as a solution of an ordinary differential equation. The application of a Petrov--Galerkin scheme to this equation provides a tensor train representation of the solution for which we derive an efficient and reliable a posteriori error estimator. Numerical experiments with a log-normal random field and a Bayesian likelihood illustrate the performance of the approach in comparison to other recent low-rank representations for the respective applications. Although the present work considers only a specific differential equation, the presented method can be applied in a more general setting. We show that the composition of a generic holonomic function and a high-dimensional function corresponds to a differential equation that can be used in our method. Moreover, the differential equation can be modified to adapt the norm in the a posteriori error estimates to the problem at hand.

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