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Representation of Gaussian fields in series with independent coefficients

by C. Gittelson

(Report number 2010-15)

Abstract
The numerical discretization of problems with stochastic data or stochastic parameters generally involves the introduction of coordinates that describe the stochastic behavior, such as coefficients in a series expansion or values at discrete points. The series expansion of a Gaussian field with respect to any orthonormal basis of its Cameron--Martin space has independent standard normal coefficients. A standard choice for numerical simulations is the Karhunen--Lo\`eve series, which is based on eigenfunctions of the covariance operator. We suggest an alternative basis that can be constructed directly from the covariance kernel. The resulting basis functions are often well localized, and the convergence of the series expansion seems to be comparable to that of the Karhunen--Lo\`eve series. We provide explicit formulas for particular cases, and general numerical methods for computing exact representations of such bases. Finally, we relate our approach to numerical discretizations based on replacing a random field by its values on a finite set.

Keywords:

@Techreport{G10_49,
  author = {C. Gittelson},
  title = {Representation of Gaussian fields in series with independent coefficients},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2010-15},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2010/2010-15.pdf },
  year = {2010}
}

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