Random Field Representation

  • Mass Per Pettersson
  • Jan Nordström
  • Gianluca Iaccarino
Part of the Mathematical Engineering book series (MATHENGIN)


Nonlinear conservation laws subject to uncertainty are expected to develop solutions that are discontinuous in spatial as well as in stochastic dimensions. In order to allow piecewise continuous solutions to the problems of interest, we follow [7] and broaden the concept of solutions to the class of functions equivalent to a function f, denoted \(\mathcal{C}_{f}\), and define a normed space that does not require its elements to be smooth functions. Let \((\varOmega,\mathcal{F},\mathcal{P})\) be a probability space with event space Ω, and probability measure \(\mathcal{P}\) defined on the σ-field \(\mathcal{F}\) of subsets of Ω. Let \(\boldsymbol{\xi }=\{\xi _{j}(\omega )\}_{j=1}^{N}\) be a set of N independent and identically distributed random variables for ω ∈ Ω. We consider second-order random fields, i.e., we consider f belonging to the space
$$\displaystyle{ L^{2}(\varOmega,\mathcal{P}) = \left \{{\text{C}}_{ f}\vert f\text{ measurable w.r.t.}\mathcal{P};\int _{\varOmega }f^{2}d\mathcal{P}(\xi ) < \infty \right \}. }$$
The inner product between two functionals a(ξ) and b(ξ) belonging to \(L^{2}(\varOmega,\mathcal{P})\) is defined by
$$\displaystyle{ \langle a(\xi )b(\xi )\rangle =\int _{\varOmega }a(\xi )b(\xi )d\mathcal{P}(\xi ). }$$
This inner product induces the norm \(\left \|f\right \|_{L_{2}(\varOmega,\mathcal{P})}^{2} =\langle f^{2}\rangle\).


Resolution Level Haar Wavelet Polynomial Basis Stochastic Dimension Polynomial Chaos 
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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Mass Per Pettersson
    • 1
  • Jan Nordström
    • 2
  • Gianluca Iaccarino
    • 3
  1. 1.Uni ResearchBergenNorway
  2. 2.Department of Mathematics Computational MathematicsLinköping UniversityLinköpingSweden
  3. 3.Department of Mechanical Engineering and Institute for Computational and Mathematical EngineeringStanford UniversityStanfordUSA

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