Let us assume that we have measured M times the geometric variations and that each of these observations can be described by anN-dimensional vector ym=ym,1 ym,2 . . . ym,N>(m = 1, . . . , M). And let us further assume that all components are normally distributed but possibly mutually correlated. The vector µ = {µi} denotes the expectation and is computed from the M measurement samples.
We aim to convert the observations into a (possibly smaller) set of uncorrelated variables. The first step is the collection of all the observations in the matrixT ∈ RM×N
T = y1 y2 . . . yM>, (4.5)
whose columns are the random variables and whose rows are the measured observations. The covariance matrix is the matrix Σ ∈ RN×N whose elements c
i j are defined as
ci j:= cov yi, yj = E(yi− µi)(yj− µj) i, j = 1, . . . , N (4.6) The main idea [37] is to construct a decomposition of the covariance matrix such that
Σ = ZZ>. (4.7)
There are many possible choices for matrixZ (for example the Cholesky factorisation), but we focus on
the eigendecomposition
Σ = VDV>, (4.8)
where the columns of V are the orthonormal right eigenvectors of Σ, and D = diag(d1, d2, . . . , dN) is a diagonal matrix whose elements are the corresponding eigenvalues. It is easy to prove that Σ is a symmetric positive definite matrix, thus the eigenvalues are all positive and real valued. Without loss of generality we can also assume that the columns of D and V are ordered in such a way that
d1≥ d2≥ · · · ≥ dN.
Equation (4.8) corresponds to choosing
ZKL:= VD1/2 (4.9)
and it is easy to see that the decomposition introduced can be exploited to express any random vector with mean µ and variance σ2 as
˜y = µ + ZKLε(θ), (4.10)
where ε(θ) is an N-dimensional vector whose components are independent and identically distributed (i.i.d.) samples ∼ N (0, 1). Expression (4.10) is often referred to as discrete Karhunen–Loève expansion and ˜y can be interpreted as a first order approximation of the randomness of the measured variables.
One useful property of eigendecomposition (4.8) is that the stronger the correlation between the variables, the faster the eigenvalues decay. It is then possible to introduce a further approximation by truncating the decomposition to a dimensionNt< N
ZtKL:= VtD1t/2 (4.11)
where ZtKL is now a rectangular matrix of dimensions N × Nt. The matrix Vt consists of the first Nt eigenvectors only, while the matrixDtis theNt× Ntnorth-west block ofD.
Now we can introduce the truncated discrete Karhunen–Loève expansion
˜yt= µ + ZtKLεt(θ ) ≈ y, (4.12)
where the vector εt(θ ) is now of reduced dimension Nt. The random variable ˜yt will still have the mean
value µ but covariance matrix
Σt:= VtDtV>t ≈ Σ. (4.13)
The faster the eigenvalues of Σ decay, the smaller Nt can be selected with respect to N for the same
accuracy. In practice,Ntis often chosen so that the considered eigenvalues represent at least95 % of the total, i.e. Nt is such that
Nt X i=1 di≥ 95 100 N X i=1 di. (4.14)
In many cases, instead of extracting the reduced system from the complete eigendecomposition, the matricesVt andDtare obtained through a truncated Singular Value Decomposition (SVD).
The process introduced can be used to significantly reduce the number of random inputs for the UQ analysis which, as it will be shown in the following sections, can have a huge impact on the performance. 4.3 Numerical Integration of Stochastic Moments
One way to evaluate the statistical moments 4.4 is numerical integration. Here we present two approaches that are commonly used.
4.3.1 Monte Carlo Sampling
A first approach to numerically estimate integrals (4.4), is the well-known Monte Carlo (MC) sampling method [67]. In this case problem (2.28) is solvedM times, for M random realizations ym(see Fig. 4.1a). The results obtained for the angular velocity ωn and the electric field En are then used to estimate expectation values of the desired quantity of interest by sample averages:
E(Q) ≈ 1 M M X i=1 Q ω(yi), E(yi). (4.15)
Among the advantages of MC methods there is the fixed convergence rate to the exact values that scales asM1/2regardless of the number of random parameters. However, the convergence is rather slow even when the solution has very smooth variation with respect to the parameter changes. More sophisticated methods like quasi Monte Carlo [42] or multilevel Monte Carlo [54] are available to mitigate this issue.
−1 0 1 −1 −0.5 0 0.5 1 x y
(a)Monte Carlo points.
−1 0 1 −1 −0.5 0 0.5 1 x y
(b)Gaussian tensor product points.
−1 0 1 −1 −0.5 0 0.5 1 x y
(c)Sparse Clenshaw-Curtis grid of level 2.
Figure 4.1: Some examples of different choices of collocation points in a two dimensional random para- meter space: Monte Carlo points (a), Gaussian tensor product points (b), level 2 Clenshaw-Curtis points (c).
4.3.2 Stochastic Collocation
The main idea of stochastic collocation methods, in an analogous way of that which is performed for classical numerical quadrature, is to choose pointsym in the parameter space in a more clever way than just randomly like in Monte Carlo, e.g. Gaussian points. By exploiting the solution regularity one can increase the speed of convergence by approximating the mapping from parameter to solution space by polynomials of degreep. Depending on the context, this method may also be referred to as generalised Polynomial Chaos (gPC) [100].
Depending on the distribution of the input variables (uniformly, normally, etc...), the approximation is constructed by creating a basis of orthonormal polynomials {ψi(y)}p0 and a grid of pointsyi in the parameter space called collocation points [100], i.e.
Q(ω(y), E(y)) ≈ p X
i=0
Q ω(yi), E(yi) ψi(y). (4.16)
The integrals in (4.4) can then be computed as E(Q) ≈X
i
Q ω(yi), E(yi)wi (4.17a)
var(Q) ≈X i
Q ω(yi), E(yi) − E(Q)2wi, (4.17b)
where to each collocation pointyi we have associated a quadrature weight wi. If the exact solution has a sufficiently smooth dependency with respect to its parameters this method converges exponentially [100].
The construction of these collocation points typically follows a tensor product approach, i.e. they are chosen independently for each variable and then multiplied in the multivariate case (see Fig. 4.1b). The direct consequence of this is the exponential growth of the quadrature points with the number of random variable N (the so called curse of dimensionality). One possible way to reduce this effect is the use of sparse grids [28] which construct the collocation points in a more clever way. Many different rules for constructing such grids have been proposed in literature based on the Clenshaw-Curtis rule (see Fig. 4.1c). We refer the interested reader to [5, 28, 75, 76].
Each collocation point requires the solution of the deterministic eigenvalue problem (2.28) and the number of points grows rapidly withN. As explained in section 4.2, the use of truncated Karhunen–Loève expansion can reduce the numerical cost by reducing the number of random dimensions.