gPCE with numerically generated orthogonal polynomials

[sitadrost]

Following up on an earlier question I asked here, I was wondering how these numerically-generated orthogonal polynomials work:

Coefficients of Polynomial Chaos Expansion for R1:
        coefficient   u1   u2   u3   u4   u5   u6   u7
        ----------- ---- ---- ---- ---- ---- ---- ----
   1.7642338446e+00 Num0 Num0 Num0 Num0 Num0 Num0   P0
   7.8350046724e-04 Num1 Num0 Num0 Num0 Num0 Num0   P0
   4.5430569440e-03 Num0 Num1 Num0 Num0 Num0 Num0   P0
   8.4910279301e-01 Num0 Num0 Num1 Num0 Num0 Num0   P0
  -9.0134378903e-06 Num0 Num0 Num0 Num1 Num0 Num0   P0
   2.6503945379e+00 Num0 Num0 Num0 Num0 Num1 Num0   P0
  -1.3398933210e-02 Num0 Num0 Num0 Num0 Num0 Num1   P0
  -1.1126509255e-01 Num0 Num0 Num0 Num0 Num0 Num0   P1

u1 to u6 are bounded normal variables here, u7 has uniform distribution.

According to the User Manual:

Extended (default if no option is selected): The Extended option avoids the use of any nonlinear variable transformations by augmenting the Askey approach with numerically-generated orthogonal polynomials for non-Askey probability density functions. Extended polynomial selections with numerically-generated polynomials that are orthogonal to the prescribed probability density functions replace each of the sub-optimal Askey basis selections for bounded normal, lognormal, bounded lognormal, loguniform, triangular, gumbel, frechet, weibull, and bin-based histogram.

And from the theory section on Numerically generated orthogonal polynomials:

If all random inputs can be described using independent normal, uniform, exponential, beta, and gamma distributions, then Askey polynomials can be directly applied. If correlation or other distribution types are present, then additional techniques are required. One solution is to employ nonlinear variable transformations such that an Askey basis can be applied in the transformed space. This can be effective as shown in [EWC08], but convergence rates are typically degraded. In addition, correlation coefficients are warped by the nonlinear transformation [DKL86], and simple expressions for these transformed correlation values are not always readily available. An alternative is to numerically generate the orthogonal polynomials (using Gauss-Wigert [Sim78], discretized Stieltjes [Gau04], Chebyshev [Gau04], or Gramm-Schmidt [WB06] approaches) and then compute their Gauss points and weights (using the Golub-Welsch [GW69] tridiagonal eigensolution). These solutions are optimal for given random variable sets having arbitrary probability density functions and eliminate the need to induce additional nonlinearity through variable transformations, but performing this process for general joint density functions with correlation is a topic of ongoing research (refer to “Transformations to uncorrelated standard variables” for additional details).

Where can I find more information on how these numerically-generated orthogonal polynomials are applied in Dakota? Would it be possible for me to reconstruct the gPCE using the coefficients provided by Dakota?