Abstract
Chapter 12 considers a spectral approach to UQ, namely Galerkin expansion, that is mathematically very attractive in that it is a natural extension of the Galerkin methods that are commonly used for deterministic PDEs and (up to a constant) minimizes the stochastic residual, but has the severe disadvantage that the stochastic modes of the solution are coupled together by a large system such as (12.15).
[W]hen people thought the Earth was flat, they were wrong. When people thought the Earth was spherical, they were wrong. But if you think that thinking the Earth is spherical is just as wrong as thinking the Earth is flat, then your view is wronger than both of them put together.
The Relativity of Wrong
Isaac Asimov
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Notes
- 1.
As usual, readers will lose little by assuming that \(\mathcal{U} = \mathbb{R}\) on a first reading. Later, \(\mathcal{U}\) should be thought of as a non-trivial space of time- and space-dependent fields, so that \(U(t,x;\theta ) =\sum _{k\in \mathbb{N}_{0}}(t,x)\varPsi _{k}(\theta )\).
- 2.
Indeed, many standard numerical linear algebra packages will readily solve the system (13.4) without throwing any error whatsoever.
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Sullivan, T.J. (2015). Non-Intrusive Methods. In: Introduction to Uncertainty Quantification. Texts in Applied Mathematics, vol 63. Springer, Cham. https://doi.org/10.1007/978-3-319-23395-6_13
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