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3.3 Polynomial Chaos Expansions
Theorem 3.3. The Wiener chaos representation of any random variable X ∈ L 2 (Ω) converges in the
L 2 (Ω)-sense to X. This means, if X is any functional for which
∫
|X(ω)| 2 dω < ∞ , (3.24)
Ω
then
∫
lim |X(ω) −∑ N N→∞
α=1 a αΨ α (ξ (ω))| 2 dω = 0 . (3.25)
Ω
The Fourier-Hermite coefficient a α is
∫
a α = X(ω)Ψ α (ξ (ω))dω . (3.26)
Ω
The Cameron-Martin theorem ensures that every random variable with finite variance has a representation
in the Wiener chaos, but gives no information about the convergence rate of the representation.
The convergence rate is important when the series expansion is cut after a finite number of terms.
This is necessary for numerical algorithms dealing with polynomial chaos expansions. In fact, [160]
showed that the convergence rate of the Wiener chaos is substantially less the optimal, exponential,
convergence rate. The development of other chaos types leads to expansions that have better convergence
properties. This is the topic of the next section, which introduces the generalized polynomial
chaos expansion, originally proposed by Xiu and Karniadakis [160].
3.3.3 Generalized Polynomial Chaos
Xiu and Karniadakis [160] generalized the idea of the representation of random variables in an orthogonal
basis formed by polynomials in random variables with known distribution. They proposed
to use polynomials whose weighting functions correspond to the PDF of the underlying random variables.
It turns out that these polynomials are the polynomials from the Askey-scheme [16]. Table 3.2
shows the correspondence between important random variables and the associated polynomials. To
summarize, a random variable with finite variance has a representation in the polynomial chaos by
X(ω) = ∑ ∞ α=1 a αΨ α (ξ ) , (3.27)
where the multi-dimensional polynomials are selected from the Askey-scheme [16]. The multidimensional
polynomials are constructed from one-dimensional polynomials via
ψ α = ∏ n i=1 H α i
(ξ i ) , (3.28)
whereas α is the index corresponding to the multiindex (α 1 ,...,α n ) and H αi , i = 1,...,n are polynomials
in one random variable. Fig. 3.1 shows the first one-dimensional polynomials for the Legendrechaos
and Fig. 3.3 the polynomials for the Hermite-chaos. We rescaled the Legendre- and Hermitepolynomials
to get an orthonormal basis of L 2 (Ω) with respect to the weighted scalar product, i.e.
〈Ψ α ,Ψ β 〉 = δ αβ , (3.29)
because the weighting functions for the random variables are 0.5 and 1 √
2π
exp −x2
2 , respectively.
Ernst et al. [50] proved that the polynomial chaos expansion converges in quadratic mean, i.e. in
the L 2 (Ω) sense [73], if and only if the basic random variables have finite moments of all orders and
the probability density of the basic random variables is continuous. Furthermore, the moment problem
(cf. [50]), i.e. the identification of the measure from the moments, has to be uniquely solvable.
Nouy [116] showed that multimodal random variables are hard to approximate in a onedimensional
polynomial chaos expansion. He solved this problem by introducing a special kind
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