Abstract
This chapter and its sequels consider several spectral methods for uncertainty quantification. At their core, these are orthogonal decomposition methods in which a random variable stochastic process (usually the solution of interest) over a probability space \((\varTheta,\mathcal{F},\mu )\) is expanded with respect to an appropriate orthogonal basis of \(L^{2}(\varTheta,\mu; \mathbb{R})\).
The mark of a mature, psychologically healthy mind is indeed the ability to live with uncertainty and ambiguity, but only as much as there really is.
Julian Baggini
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Notes
- 1.
In the case that \(\mathcal{X}\) is compact, it is enough to assume that the covariance function is continuous, from which it follows that it is bounded and hence square-integrable on \(\mathcal{X}\times \mathcal{X}\).
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Sullivan, T.J. (2015). Spectral Expansions. In: Introduction to Uncertainty Quantification. Texts in Applied Mathematics, vol 63. Springer, Cham. https://doi.org/10.1007/978-3-319-23395-6_11
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