Segmentation of Stochastic Images using ... - Jacobs University

Chapter 3 SPDEs and Polynomial Chaos Expansions
The multiplication of two polynomial chaos variables is more difficult. Since polynomials form
the basis, the naive multiplication of polynomial chaos variables results in a polynomial with twice
the degree of the factors. Thus, an additional projection step onto a polynomial with the same degree
as the factors of the multiplication is necessary. This projection step is done by using the Galerkin
or L 2 -projection, leading to a projection polynomial, whose error is orthogonal to the space spanned
by the polynomial chaos. The idea of the projection is to multiply the naive product c = a · b with an
element Ψ γ of the polynomial chaos basis, integrate over the stochastic dimensions and to compute
the coefficient of the multiplication one after another from this expression:
∫
Γ
N
∑
α=1
∫
c α Ψ α Ψ γ dΠ =
Γ
N
∑
α=1
N
∑
β=1
a α b β Ψ α Ψ β Ψ γ dΠ ⇒ c γ =
N
∑
α=1
N
∑
β=1
〈Ψ α Ψ β Ψ γ 〉
a α b β
〈(Ψ γ ) 2 . (3.34)
〉
} {{ }
C αβγ
Note that we omit denoting the dependence of Ψ α from ξ and ω to simplify the notation. The
quantity C αβγ is independent of the actual problem, it depends on the basis only. The values of C αβγ
can be precomputed in a lookup table. The next section describes the generation of this table.
The computation of the quotient of two random variables, a = c b
is possible, too. To do this, we
multiply the expression by b, yielding c = ab and use again the Galerkin projection for this equation:
c γ = ∑ N α=1∑ N β=1 C αβγb β a α = ∑ N α=0 A γαa α . (3.35)
This is a system of linear equations for the coefficients a α , which we solve by an iterative solver.
In a similar manner, we compute the square root b = √ a of a polynomial chaos variable. First, we
rewrite the equation in the form a = b 2 and then use the Galerkin projection to obtain
a γ = ∑ N α=1∑ N β=1 C αβγb α b β . (3.36)
This is a nonlinear system of equations for the unknown coefficients b α , which we solve using
Newton’s method to find a root of
f (b) = b 2 − a . (3.37)
The partial derivatives of this function are
∂ f α (b)
=
∂b β
N
∑
γ=1
C βγα b γ . (3.38)
As pointed out by Matthies and Rosic [98], it is possible to use a mild convergence criterion for
Newton’s method depending on the expected value and the variance of the polynomial chaos variable.
Using these building blocks, it is possible to construct numerical methods for nearly all possible
calculations, e.g. the exponential of a random variable in the polynomial chaos is
exp(a) = exp(a 1 )
(
1 +
K
∑
n=1
(
∑
N
α=2 a α Ψ α) )
n
n!
. (3.39)
With the methods from this section, it is also possible to construct finite difference schemes for
random variables. Chapter 4 investigates this further.
3.3.5 The Stochastic Lookup Table
We precompute the values of C αβγ in a lookup table to speed up the calculations in the polynomial
chaos. It is possible to replace the calculation of the multi-dimensional integrals ∫ Γ Ψα Ψ β Ψ γ dΠ
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