Segmentation of Stochastic Images using ... - Jacobs University

Chapter 4 Discretization of SPDEs
4.3.1 Discretization of the Spaces V and S
We approximate the deterministic space V using the classical finite element approach. That means
every u ∈ V ⊗ S is approximated by
u(x,ω) ≈ ∑ n i=1 u i(ω)P i (x) , (4.12)
where u i ∈ S and {P i } i=1,...,n is a basis of a finite dimensional subspace V h ⊂ V . We identify the
space V h with IR n , because we have to store the coefficients for the basis elements only.
We approximate the stochastic space S in two steps. First, we choose a finite set of random
variables ζ =(ζ 1 ,...,ζ m ), span(ζ 1 ,...,ζ m )=S m ⊂S with finite variance and approximate u i ∈ S by
u i (ω) ≈ ∑ m k=1 uk i ζ k (ω) , (4.13)
where the coefficients u k i are deterministic coefficients for the random variables ζ k . Numerical calculations
cannot use the space S m . Hence, we approximate the space S m by using the generalized
polynomial chaos [160], cf. Section 3.3. We approximate the random variables ζ i with unknown
distribution in the polynomial chaos by the same number of random variables and a prescribed polynomial
degree p:
u ∈ S m,p ⊂ S p : u = ∑ N i=1 u iΨ i (ξ ) . (4.14)
The dimension of the space S m,p is N = ( )
m+p
m .
For the finite dimensional subspace IR n ⊗ S p , the problem (4.9) is rewritten as
E(v T Au) = E(v T b) ∀v ∈ IR n ⊗ S p , (4.15)
where A ∈ Sp
n×n is a stochastic matrix.
Using the polynomial chaos basis, i.e. the space S m,p for the stochastic space and V h for the
deterministic space we end up with a huge deterministic equation system to approximate the solution
of ∇ · (a∇u) = f given by
∑ N α=1
where the matrices M α,β and L α,β are
(M α,β ) i, j = E (Ψ α Ψ β ) ∫
(
L α,β ) i, j = ∑
k
(
L α,β ) U α = ∑ N α=1 Mα,β F α , (4.16)
(
∑E
γ
D
P i P j dx
Ψ α Ψ β Ψ γ) a k γ
∫
D
∇P i · ∇P j P k dx .
(4.17)
In (4.16) we used the notation F α = ( f i α) for the polynomial chaos representation of the quantities.
4.4 Generalized Spectral Decomposition
Selecting suitable subspaces of S m,p ⊗IR n and a special basis, which captures the dominant stochastic
effects, we achieve a significant speed-up of the solution process and an enormous reduction of
the memory requirements. In the generalized spectral decomposition (GSD) [113], we approximate
the solution u ∈ L 2 (Ω) ⊗ H 1 (D) by
u(x,ξ ) ≈ ∑ K j=1 λ j(ξ )V j (x) , (4.18)
where V j is a deterministic function, λ j a stochastic function and K the number of modes of the
decomposition. Thus, the GSD computes a solution where the deterministic and the stochastic basis
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