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Polynomial Chaos Methods for Hyperbolic Partial Differential EquationsRandom Field Representation

Polynomial Chaos Methods for Hyperbolic Partial Differential Equations: Random Field Representation [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 𝒞f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{C}_{f}$$ \end{document}, and define a normed space that does not require its elements to be smooth functions. Let (Ω,ℱ,𝒫)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$(\varOmega,\mathcal{F},\mathcal{P})$$ \end{document} be a probability space with event space Ω, and probability measure 𝒫\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{P}$$ \end{document} defined on the σ-field ℱ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{F}$$ \end{document} of subsets of Ω. Let ξ={ξj(ω)}j=1N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\boldsymbol{\xi }=\{\xi _{j}(\omega )\}_{j=1}^{N}$$ \end{document} 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 2.1L2(Ω,𝒫)=Cf|fmeasurable w.r.t.𝒫;∫Ωf2d𝒫(ξ)<∞.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\displaystyle{ L^{2}(\varOmega,\mathcal{P}) = \left \{{\text{C}}_{ f}\vert f\text{ measurable w.r.t.}\mathcal{P};\int _{\varOmega }f^{2}d\mathcal{P}(\xi ) < \infty \right \}. }$$ \end{document} The inner product between two functionals a(ξ) and b(ξ) belonging to L2(Ω,𝒫)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$L^{2}(\varOmega,\mathcal{P})$$ \end{document} is defined by 2.2⟨a(ξ)b(ξ)⟩=∫Ωa(ξ)b(ξ)d𝒫(ξ).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\displaystyle{ \langle a(\xi )b(\xi )\rangle =\int _{\varOmega }a(\xi )b(\xi )d\mathcal{P}(\xi ). }$$ \end{document} This inner product induces the norm fL2(Ω,𝒫)2=⟨f2⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\left \|f\right \|_{L_{2}(\varOmega,\mathcal{P})}^{2} =\langle f^{2}\rangle$$ \end{document}.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

Polynomial Chaos Methods for Hyperbolic Partial Differential EquationsRandom Field Representation

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References (20)

Publisher
Springer International Publishing
Copyright
© Springer International Publishing Switzerland 2015
ISBN
978-3-319-10713-4
Pages
11 –21
DOI
10.1007/978-3-319-10714-1_2
Publisher site
See Chapter on Publisher Site

Abstract

[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 𝒞f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{C}_{f}$$ \end{document}, and define a normed space that does not require its elements to be smooth functions. Let (Ω,ℱ,𝒫)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$(\varOmega,\mathcal{F},\mathcal{P})$$ \end{document} be a probability space with event space Ω, and probability measure 𝒫\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{P}$$ \end{document} defined on the σ-field ℱ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{F}$$ \end{document} of subsets of Ω. Let ξ={ξj(ω)}j=1N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\boldsymbol{\xi }=\{\xi _{j}(\omega )\}_{j=1}^{N}$$ \end{document} 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 2.1L2(Ω,𝒫)=Cf|fmeasurable w.r.t.𝒫;∫Ωf2d𝒫(ξ)<∞.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\displaystyle{ L^{2}(\varOmega,\mathcal{P}) = \left \{{\text{C}}_{ f}\vert f\text{ measurable w.r.t.}\mathcal{P};\int _{\varOmega }f^{2}d\mathcal{P}(\xi ) < \infty \right \}. }$$ \end{document} The inner product between two functionals a(ξ) and b(ξ) belonging to L2(Ω,𝒫)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$L^{2}(\varOmega,\mathcal{P})$$ \end{document} is defined by 2.2⟨a(ξ)b(ξ)⟩=∫Ωa(ξ)b(ξ)d𝒫(ξ).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\displaystyle{ \langle a(\xi )b(\xi )\rangle =\int _{\varOmega }a(\xi )b(\xi )d\mathcal{P}(\xi ). }$$ \end{document} This inner product induces the norm fL2(Ω,𝒫)2=⟨f2⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\left \|f\right \|_{L_{2}(\varOmega,\mathcal{P})}^{2} =\langle f^{2}\rangle$$ \end{document}.]

Published: Sep 17, 2014

Keywords: Resolution Level; Haar Wavelet; Polynomial Basis; Stochastic Dimension; Polynomial Chaos

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