Computing the Polynomial Chaos Coefficients

The literature offers two main choices for the computation of the series coeffi- cientss_pin the context of non-intrusive gPC approximations, namely least squares (LS) regression [22, 24], also referred to as random discrete projection (RDP) in [105, 106], and pseudo-spectral projection [49, 88]. In this work, the adaptive methods detailed in Section 4.2 are based on the LS-based computation of coef- ficients, therefore, this approach will be the main focus of this section. However, we also present the pseudo-spectral projection approach, for completeness. We note that the two approaches are equivalent, if infinitely many samples are con-

sidered in the regression case. We also note that we use unweighted discrete LS [36, 40, 105, 106] in this work. Regarding weighted LS approximations, the interested reader is referred to [2, 41, 42].

Pseudo-spectral projection

We first multiply (4.13) with Ψq, whereq ∈ Λ and Λ is either the TP or TD multi-

index set. Then, we apply the expectation operator on both sides of the resulting equation. Under the orthogonality condition (4.8), the series coefficientssp are

given as s_p= E  gΨp  E ” Ψ_p2— = R

Ξg (y) Ψp(y) % (y) dy

R

ΞΨp(y) Ψp(y) % (y) dy

. (4.15)

In (4.15), the multivariate integrals of the numerator are typically computed by means of numerical integration [87], e.g. (quasi-) MC sampling or Gauss quadra- ture. For Wiener-Askey polynomials [164], the denominator can be determined analytically.

Least squares regression

We start by considering an experimental design, i.e. a set of realizations {yl}L_l=1

and the corresponding set of model evaluations {g (yl)}_l=1L . The gPC approximation

is obtained as the result of the minimization problem

eg = arg min π∈PΛ L X l=1 (g(yl) − π(yl))2. (4.16)

Assuming that a polynomial basis Ψ_{p p∈Λ} is available and has a cardinality #Λ = M, we introduce the design matrix D ∈ RL×M_{, with elementsD}

lm= Ψm(yl),

and the vector of model evaluationsg = (g(y1), g(y2) . . . , g(yL)). Collecting the un-

known gPC coefficients into a vectors = (s1, s2, . . . , sM), we form the minimization

problem

s = arg min

ˆs∈RM kDˆs − gk2. (4.17)

By transforming (4.17) into the normal equation, we obtain the linear system

D>_{Ds = D}>_{g, (4.18)}

where the system matrixD>_{D is called the information matrix. The system is solv-}

if_{L ≥ M, i.e. the corresponding system of equations in (4.18) is overdetermined,} and rank (D) = M, i.e. the design matrix has full rank.

Due to conditioning issues, it is generally not recommended to directly invert the information matrix in (4.18). A Cholesky decomposition of a symmetric positive definite information matrix can be employed for the solution of (4.18), instead. Alternatively, a QR decomposition of the design matrixD can be employed, such

that D = QR, with R =  eR 0  ∈ RL×M, (4.19)

whereQ ∈ RL×L _{is orthogonal and eR ∈ R}M×M _{is upper triangular. Then, (4.17) can}

be transformed into the equivalent problem

kDˆs − gk2= kQRˆs − QQ>gk2= kRˆs − Q>gk2, (4.20)

the solution of which is given as

ˆs = eR−1 _Qe>_{g
, (4.21)}

where eQ ∈ RL×M _{corresponds to theM first columns of Q. The QR decomposition}

of the design matrixD is employed as solution method in this work.

In [36, 40, 105, 106], the authors present several studies regarding the stabil- ity of the (unweighted) LS problem (4.16), equivalently, the conditioning of the system in (4.18). All relevant results have also been collected in the survey pa- per [42]. The condition number of the information matrix κ(D>_{D) is used as a}

stability measure. In [106] it is shown that lim

L→∞κ(D

>_{D) → 1, thus, the size of}

the experimental design is important for a well-conditioned LS problem. Assum- ing probability measures of the Jacobi type, e.g. uniform or Chebyshev ones, an RDP-based approximation is stable and optimally convergent under the relation L = C Mq_{, whereq ≥ ln 3/ ln 2 [42]. The equalities q = ln 3/ ln 2 and q = 2 hold}

for Chebyshev and uniform measures, respectively. In all other cases, it holds that q > 2.

Considering non-Chebyshev measures, the condition _{L = C M}q _{withq ≥ 2 has}

been found to be too pessimistic in practice, despite being theoretically optimal. Several numerical experiments indicate that a linear relation between the size of the experimental design and the number of basis terms is sufficient to obtain well- conditioned systems and near-optimal convergence, see e.g. [105, 106]. Therefore, a linear relation_{L = C M is typically employed, where the constant C > 1 is called} the oversampling coefficient. A choice of _{C ∈ [2, 5] most often results in suffi-} ciently accurate approximations, however, the extent of oversampling is generally

problem-dependent. We note that such linear relations have been theoretically proven for the stability and optimal convergence of weighted LS approximations [41, 42], however, weighted LS are not considered in this work.

In our adaptive algorithms, presented in Section 4.2, we also employ κ D>_D

as a measure of LS stability. However, as already pointed out, we use here a QR factorization of the design matrixD and solve problem (4.20), in order to avoid

the ill-conditioning issues related to the normal equation (4.18). Nevertheless, it is straightforward to show a direct relation between the condition numbers appearing in both cases. First, using the singular value decomposition (SVD) of the design matrixD, it is easy to show that

κ D>D
_{= (κ (D))}2. (4.22) Moreover, sinceD = QR with Q being an orthogonal matrix, it holds that

κ(D) = κ(QR) = κ(R). (4.23)

We observe that the condition number of the system matrix of the normal equation (4.18) is significantly worse than the one of the design matrix and, equivalently, of the system matrix obtained with the QR decomposition.