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Piecewise Polynomial Approximation of Probability Density Functions with Application to Uncertainty Quantification for Stochastic PDEs

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Quantification of Uncertainty: Improving Efficiency and Technology

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 137 ))

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Abstract

The probability density function (PDF) associated with a given set of samples is approximated by a piecewise-linear polynomial constructed with respect to a binning of the sample space. The kernel functions are a compactly supported basis for the space of such polynomials, i.e. finite element hat functions, that are centered at the bin nodes rather than at the samples, as is the case for the standard kernel density estimation approach. This feature naturally provides an approximation that is scalable with respect to the sample size. On the other hand, unlike other strategies that use a finite element approach, the proposed approximation does not require the solution of a linear system. In addition, a simple rule that relates the bin size to the sample size eliminates the need for bandwidth selection procedures. The proposed density estimator has unitary integral, does not require a constraint to enforce positivity, and is consistent. The proposed approach is validated through numerical examples in which samples are drawn from known PDFs. The approach is also used to determine approximations of (unknown) PDFs associated with outputs of interest that depend on the solution of a stochastic partial differential equation.

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Notes

  1. 1.

    In the partial differential equation (PDE) setting, what we refer to as bins are often referred to as grid cells or finite elements or finite volumes. We instead refer to the subdomains \(\{{\mathcal B}_{\ell }\}_{{\ell }=1}^{N_{bins}}\) as bins because that is the notation in common use for histograms which we use to compare to our approach. Furthermore, in Sect. 4, we also use finite element grids for spatial discretization of partial differential equations, so that using the notation “bins” for parameter domain subdivisions helps us differentiate between subdivisions of parameter and spatial domains. For the same reason, we use δ instead of h to parametrize parameter bin sizes because h is in common use to parametrize spatial grid sizes.

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Acknowledgements

This work was supported by US Air Force Office of Scientific Research grant FA9550-15-1-0001 and by the Sandia National Laboratories contract 1985151.

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Authors and Affiliations

  1. Department of Scientific Computing, Florida State University, Tallahassee, FL, USA

    Giacomo Capodaglio & Max Gunzburger

Authors
  1. Giacomo Capodaglio
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  2. Max Gunzburger
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Correspondence to Max Gunzburger .

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Editors and Affiliations

  1. Computational Multiscale, Sandia National Laboratories, Albuquerque, NM, USA

    Marta D'Elia

  2. Dept of Scientific Computing, Florida State Univ, Tallahassee, FL, USA

    Max Gunzburger

  3. SISSA mathLab, Mathematics Area, International School for Advanced Studie, Trieste, Italy

    Gianluigi Rozza

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Capodaglio, G., Gunzburger, M. (2020). Piecewise Polynomial Approximation of Probability Density Functions with Application to Uncertainty Quantification for Stochastic PDEs. In: D'Elia, M., Gunzburger, M., Rozza, G. (eds) Quantification of Uncertainty: Improving Efficiency and Technology. Lecture Notes in Computational Science and Engineering, vol 137 . Springer, Cham. https://doi.org/10.1007/978-3-030-48721-8_5

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