Expansion with Multivariate Random Variable

Definition 2.2 provided a description of the orthogonal polynomial space for a scalar RV.

This subsection gives an extension to multiple dimensions by using the tensor product of one-dimensional grids. Thus, it holds for the N -dimensional orthogonal space (i.e., there are N RVs) [140, p. 252f.]:

Definition 2.3 (N -dimensional Continuous Orthogonal Space). An N -dimensional orthog-onal space is defined by a tensor grid expansion of one-dimensiorthog-onal orthogorthog-onal polynomial spaces in the following manner:

W_N^(D) ≡ ^O

|d|≤D

W^(di)

Here, |d| is a multi-index operator defined by |d| = ^PN

i=1

d_i.

Definition 2.3 defines a combination of one-dimensional orthogonal spaces in a manner that a tensor grid of maximum order D is achieved. Figure 2.12 shows a visualization of a two-dimensional tensor grid: Here, the red crosses give the positions of the values in the created tensor grid from the one-dimensional grids (θ₁ ∼ N (µ = 0, σ = 1) and θ₂ ∼ U (a = −1, b = 1)). These are depicted by the blue (first uncertain parameter) and green circles (second uncertain parameter) and provide the basic values. Here, the one-dimensional grid of the first uncertain parameter has a dimension of four (d₁ = 4), while the second one has a dimension of two (d₂ = 2). This shows the possibility to combine different order one-dimensional grids as well as how the tensor grid is created from them by applying Definition 2.3.

A general prerequisite of the gPC expansion in the N -dimensional case is now that a properly defined probability space is used, as given in e.g., Definition B.2: Here, all RVs must be mutually independent (Section B.5). Take into account that this independence can be achieved using e.g., a Karhunen-Loève transformation [68], if necessary. This transformation is essentially a principle axis transformation.

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−0.6

−0.4

−0.2 0 0.2 0.4 0.6

θ1

θ2

Tensor Grid (with 1D): d1 = 4, d2 = 2 Tensor Grid θ₁

θ₂

Figure 2.12: Visualization of two-dimensional tensor grid based on one-dimensional grids and full expansion.

Having defined a proper probability space as well as secured mutual independence, all PDFs of the scalar RVs and their respective supports can be multiplied according to (B.2) as follows [140, p. 249]:

ρΘ(θ) =

N

Y

i=1

ρ_Θi(θi) ∈ R, Ω =

N

Y

i=1

Ωi ⊂ R^N (2.59)

With (2.59), the N -dimensional orthogonal polynomial space is defined by multiplying all one-dimensional orthogonal polynomials as follows [140, p. 253]:

Φ^(m)(θ) =

N

Y

i=1

φ^(mi)(θ_i) ∈ R, m₁+ ... + m_N ≤ D (2.60) It should be noted that the multivariate expansion order of the orthogonal polynomial is D (see Definition 2.3). Additionally, the normalization constants of the orthogonal polynomials can also be multiplied like the orthogonal polynomials in (2.60).

Using (2.59) and (2.60), (2.52) can directly be expanded to the N -dimensional space as follows [140, p. 254]:

y (z; θ) ≈ P_N^(D)y = y^(D)_N (z; θ) =

M −1

X

m=0

ˆ

y^(m)(z) Φ^(m)(θ) , M − 1 = N + D N

!

(2.61)

Here, M is the expansion order of the gPC and ˆy^(m)(z) ∈ R^ny are the multi-dimensional expansion coefficients matching the size of the output vector. It should be noted that the full tensor grid, i.e., the multi-index operator in Definition 2.3 for the tensor grid

2.3 Generalized Polynomial Chaos

allows all polynomial order combinations, is also sometimes used [140, p. 253]. This yields an expansion order of M − 1 = D^N, which is normally much larger than the binomial coefficient in (2.61) and does not improve the accuracy significantly [140, p. 253ff.]. Thus, this thesis applies the tensor grid and multivariate gPC expansion with the corresponding expansion order as introduced in Definition 2.3 and (2.61) respectively.

Further take into account that, just like for the scalar case in (2.54), (2.61) can be rewritten as a matrix-vector operation:

y (z; θ)

Here, it is again clear that there exist M expansion coefficients with M corresponding or-thogonal polynomials. Further, it can be seen that each of the output expansion coefficients is multiplied using the same series of orthogonal polynomials because the uncertainties acting on the dynamic system are the same.

The expansion coefficients themselves are then also directly obtained by expand-ing (2.58) as follows [140, p. 254]:

ˆ

Take into account that the orthogonal space defined in Definition 2.3 grows rapidly with the number of uncertainties due to the fact that a standard tensor product is used (“curse of dimensionality”). To overcome this issue, sparse grids can be implemented that use a suitable subset of tensor grids to achieve a good approximation of the tensor grid with fewer required expansion coefficients. As a rule of thumb, a sparse grid is generally required starting from N ≥ 5 and the origins of this method date back to the Russian mathematician Smolyak. Within this thesis, a sparse grid is defined as follows [142, p. 1127]:

Definition 2.4 (Smolyak Sparse Grid). A sparse collocation grid, in the sense of Smolyak, is defined as a subset of standard tensor products in the following manner:

W_N^(D)=^X

Here, k_sg is the approximation level of the Smolyak grid. Again, |d| symbolizes the multi-index operator.

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−1

−0.8

−0.6

−0.4

−0.2 0 0.2 0.4 0.6 0.8 1

θ₁ θ2

Sparse vs. Tensor Grid with k = 3

Tensor (Q=169) Sparse (Q=37)

Figure 2.13: Visualization of two-dimensional sparse grid based on Smolyak rule in comparison to full tensor grid based on one-dimensional grids from sparse grid algorithm.

Definition 2.4 states that an arbitrary accurate representation of a tensor grid can be achieved by a sparse grid using a sufficiently large approximation level k_sg [142].

The sparse grid given in Definition 2.4 can also be visualized as shown in Figure 2.13.

The same definition for the uncertainties (θ₁ ∼ N (µ = 0, σ = 1); θ₂ ∼ U (a = −1, b = 1)) as in Figure 2.12 is applied. The approximation order of the sparse grid is chosen to be k_sg = 3. In Figure 2.13, the blue circles depict the created sparse grid (number of points: Q = 37), while the red crosses denote the tensor grid (number of points: Q = 169) that is created from the one-dimensional grids that are used to create the sparse grid.

A first observation in Figure 2.13 is that the sparse grid only requires 22% of the points that the tensor grid requires. Thus, the sparse grid is more efficient in the approximation. In addition, it is clear that the sparse grid mainly uses points that are on the coordinate axes or close to the coordinate axes. This is due to the fact that points close to the coordinate axes normally have an increased influence on the response surface compared to points far away. Thus, it is reasonable to “cluster” points around the center. Further examples and comparisons of sparse and tensor grids, especially regarding the efficiency with a large number of uncertainties, can also be found in study [142].

With the gPC expansion defined for the scalar as well as the multivariate case, the next subsection deals with the calculation of the expansion coefficients, which requires the ap-proximation of the integral in (2.63).

2.3 Generalized Polynomial Chaos