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Approximation of Probability Density Functions for PDEs with Random Parameters Using Truncated Series Expansions

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Abstract

The probability density function (PDF) of a random variable associated with the solution of a partial differential equation (PDE) with random parameters is approximated using a truncated series expansion. The random PDE is solved using two stochastic finite element methods, Monte Carlo sampling and the stochastic Galerkin method with global polynomials. The random variable is a functional of the solution of the random PDE, such as the average over the physical domain. The truncated series are obtained considering a finite number of terms in the Gram–Charlier or Edgeworth series expansions. These expansions approximate the PDF of a random variable in terms of another PDF, and involve coefficients that are functions of the known cumulants of the random variable. To the best of our knowledge, their use in the framework of PDEs with random parameters has not yet been explored.

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Acknowledgements

MG and HW thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the program Uncertainty Quantification for Complex Systems: Theory and Methodologies where work on this paper was undertaken. This work was supported by UK EPSRC grant numbers EP/K032208/1 and EP/R014604/1. GC and MG were supported in part by the US Air Force Office of Scientific Research grant FA9550-15-1-0001 and by the US Department of Energy Office of Science grant DE-SC0016591.

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Authors and Affiliations

  1. Department of Scientific Computing, Florida State University, Tallahassee, FL, 32306, USA

    Giacomo Capodaglio & Max Gunzburger

  2. Computational Physics and Methods Group - Los Alamos National Laboratory, Los Alamos, NM, USA

    Giacomo Capodaglio

  3. The London School of Economics and Political Science, London, WC2A 2AE, UK

    Henry P. Wynn

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  1. Giacomo Capodaglio
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Correspondence to Max Gunzburger.

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Capodaglio, G., Gunzburger, M. & Wynn, H.P. Approximation of Probability Density Functions for PDEs with Random Parameters Using Truncated Series Expansions. Vietnam J. Math. 49, 685–711 (2021). https://doi.org/10.1007/s10013-020-00465-5

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