The Nineteenth Israeli Mini-Workshop in Applied and Computational Mathematics

Adi Ditkowski (Tel Aviv)

New Approach for Using the Generalized Polynomial Chaos Method with Dependent Parameters

The generalized polynomial chaos methods had proven to be an efficient method to analyze uncertainty quantification problems. When stochastic problems are formulated using a set to dependent random parameters, these parameters are usually transformed into a set of independent ones. A way to accomplish this goal using the Rosenblatt transformation [1]. This transformation, however, may be ill conditioned. Another approach is to consider the problem as an epistemic uncertainty, see e.g. [2], however, since the exact distributions are not used, the convergence rate may be slow.

In this work we present an approach for using the Generalized Polynomial Chaos method with dependent parameters. We demonstrate the exponential rate of convergence even for non-smooth distribution functions.

These results are applicable not only for uncertainty quantification problems but also to spectral, or orthogonal polynomials, approximations and numerical integration is complex multidimensional domains.

  1. M. Rosenblatt, Remark on a multivariate transformation, J. Ann. Math. Stat. 23(3), 470472, (1953).
  2. D. Xiu, Numerical methods for stochastic computations: a spectral method approach, Princeton University Press, 2010.
Joint work with Rami Kats.

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