Segmentation of Stochastic Images using ... - Jacobs University

3.3 Polynomial Chaos Expansions
H 1 (x) = 1
H 2 (x) = x
H 3 (x) = 1 √
2!
(x 2 − 1)
H 4 (x) = 1 √
3!
(x 3 − 3x)
H 5 (x) = 1 √
4!
(x 4 − 6x 2 + 3)
H 6 (x) = 1 √
5!
(x 5 − 10x 3 + 15x)
H 7 (x) = 1 √
6!
(x 6 − 15x 4 + 45x 2 − 15)
H 8 (x) = 1 √
7!
(x 7 − 21x 5 + 105x 3 − 105x)
H 9 (x) = 1 √
8!
(x 8 − 28x 6 + 210x 4 − 420x 2 + 105)
H 10 (x) = 1 √
9!
(x 9 − 36x 7 + 378x 5 − 1260x 3 + 945x)
Table 3.3: The first ten one-dimensional Hermite-polynomials. The construction of the multidimensional
polynomials up to degree 9 is based on these polynomials and (3.40).
Thus, it is necessary to select a vector of random variables ξ = (ξ 1 ,...,ξ n ) and the polynomial
degree p. Then, an approximation of a random variable in the polynomial chaos is
X(ω) ≈ ∑ N α=1 a αΨ α (ξ ) . (3.31)
The remaining part of this chapter deals with numerical methods for polynomial chaos expansions.
Although the presented material is valid for all polynomials from the Askey-scheme, the numerical
implementation is based on the Legendre-polynomials and uniform distributed random variables,
because the support of the Legendre-polynomials is compact. This is advantageous for algorithms,
especially when dealing with stochastic level sets. Chapter 4 discusses the combination of polynomial
chaos expansions and SPDEs. There the information presented for the polynomial chaos is
combined with finite element and finite difference schemes for the discretization of the equations.
3.3.4 Calculations in the Polynomial Chaos
To use the polynomial chaos in numerical schemes it is necessary to perform arithmetic operations
in the polynomial chaos. In this section, we review the development of the basic operations like
addition, subtraction, multiplication, division and the calculation of square roots. The presentation
is based on the work of Debusschere et al. [38]. For the remaining part of this section let
a = ∑ N α=1 a αΨ α (ξ ), b = ∑ N α=1 b αΨ α (ξ ), c = ∑ N α=1 c αΨ α (ξ ) (3.32)
be three polynomial chaos variables. We compute the sum and the difference of quantities in the
polynomial chaos by adding or subtracting the corresponding chaos coefficients, because the addition
or subtraction of polynomials results in a polynomial with the same degree at most:
c = a ± b = ∑ N α=1 a αΨ α (ξ ) ±∑ N α=1 b αΨ α (ξ ) = ∑ N α=1 (a α ± b α )Ψ α (ξ ) . (3.33)
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