Abstract
Chapter 11 considered spectral expansions of square-integrable random variables, random vectors and random fields of the form
where \(U \in L^{2}(\varTheta,\mu;\mathcal{U})\), \(\mathcal{U}\) is a Hilbert space in which the corresponding deterministic variables/vectors/fields lie, and \(\{\varPsi _{k}\mid k \in \mathbb{N}_{0}\}\) is some orthogonal basis for \(L^{2}(\varTheta,\mu; \mathbb{R})\).
Not to be absolutely certain is, I think, one of the essential things in rationality.
Am I an Atheist or an Agnostic?
Bertrand Russell
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- 1.
Or, more generally, topological vector space.
- 2.
Readers familiar with tensor notation from continuum mechanics or differential geometry will see that M ijk is covariant in the indices i and j and contravariant in the index k, and thus is a (2, 1)-tensor; therefore, if this text were following standard tensor algebra notation and writing vectors as \(\sum _{k}u^{k}\varPsi _{k}\), then the multiplication tensor would be denoted M ij k. In terms of the dual basis \(\{\varPsi ^{k}\mid k \in \mathbb{N}_{0}\}\) defined by \(\langle \varPsi ^{k}\mathop{\vert }\varPsi _{\ell}\rangle =\delta _{ \ell}^{k}\), \(M_{ij}^{k} =\langle \varPsi ^{k}\mathop{\vert }\varPsi _{i}\varPsi _{j}\rangle\).
- 3.
Usually, but not always, the convention will be that \(\dim \mathcal{U}_{M} = M\); sometimes, alternative conventions will be followed.
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Sullivan, T.J. (2015). Stochastic Galerkin Methods. In: Introduction to Uncertainty Quantification. Texts in Applied Mathematics, vol 63. Springer, Cham. https://doi.org/10.1007/978-3-319-23395-6_12
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