P REGUNTAS QUE SE FORMULAN

Two approaches are developed using the gPC methodology and the maximum likelihood function. The first method develops a gPC symbolic model to integrate multiple sources of known information and estimate the unknown parameters, i.e., means and variances of x2 in Equation B.1.

Algorithm 1 – For this method, the uncertainty quantification step on the measured variables with Galerkin

project is skipped in the optimization problem of Equation B.17. Instead the samples associated with each basic variable ξi in Equation B.5 are directly used to perform Monte Carlo simulations, while maximizing the likelihood

Equation B.17. The benefit is that the samples are randomly chosen from a prior standard distribution and tend to perform better in capturing the global structure. The Algorithm 1 involves a series of steps as follows.

Inputs initialization:

(1) Input the samples of known parameters x1 and the available data of the measured variables Ŷ.

(2) Choose the order of polynomials (p) used to approximate the unknown parameters x2 in gPC model,

decide the polynomial basis function Φk, and then formulate the gPC symbolic approximations of x2.Once again, the counterpart of unknown parameters κ in Equation B.4 can be calculated with the

gPC coefficients by using Equations B.10) and B.11.

(3) Substitute the gPC approximation of the unknown parameters x2 into the nonlinear first principles’

model, and generate a new gPC symbolic model with respect to the unknown gPC coefficients. (4) Decide the number of samples (l) for each random variable ξi, and generate samples from the

standard basis distributions (ξ).

(5) Initialize the initial guesses for {α_i,k[0]} in Equation B.5, i.e., the gPC coefficients for each unknown stochastic parameter.

Optimization with Equation (B.17):

i. Use each of the input samples of known parameters x1 and the initial values {α_i,k[0]} to perform l

Monte Carlo simulations with the nonlinear gPC symbolic model and the samples generated in (4), thus l model predictions are obtained for each input sample of x1.

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iii. Calculate the Euclidean distance between the mean values in (ii) and the corresponding measurements of the measured variables Ŷ.

iv. Solve the optimization problem B.17 to obtain the optimum of the unknown gPC coefficients {α_i,k[*]}. Each optimization iteration entails the repeated evaluation of the gPC model and updates the prediction mean and the Euclidean distance in Steps (ii) and (iii).

The use of the gPC expansions and the samples of random basic variables ξ in the Algorithm 1 significantly improve the efficiency while taking the probabilistic uncertainties into account, as compared with the standard Monte Carlo simulations. For nonlinear models, a major disadvantage of the Monte Carlo type sampling based method is the requirement for appropriate samples. To ensure samples prediction converges to the theoretical value, a large number of simulations are often required, which in turn may increases the computation burden, especially for high dimensional problems. In this method, however, the samples are generated from the random basic distribution of ξ, which can release the requirement on the number of samples and improve the computational efficiency.

The Algortihm 1 cannot provide an explicit expression of the measured variables. To mathematically propagate and quantify the effect of parametric uncertainty onto the measured variables in a computational efficient fashion, the polynomial chaos quadrature (PCQ) is used in the current work. As discussed in Section B.3.2, all moments of random variables, i.e., x2 and Y, are just functions of their gPC expansion coefficients. Hence, the optimization

problem Equation B.14 can be reformulated with the statistical moments calculated from the measurement data and the gPC coefficients of the measured variables.

Algorithm 2 – For the purpose of estimating the unknown stochastic parameters and their confidence interval, as

well as approximating the variation on measured variables introduced by unknown parameters and measurement noise, the PCQ is used to calculate the analytical gPC expression of the measured quantities. To this objective, the mean values and the variances of unknown parameters are obtained from optimizing a modified joint PDF fY(Ŷ,κ) of measured variables as:

𝐽 = max Ω2 ∏ ∑ Kh((ν1,k - γ1,k) 2 _{+ (ν} 2,k - γ2,k)2) n k=1 nobs m=1 (B.18)

where n is the number of known samples of x1, Kh is a Gaussian kernel function, ν1 and ν2 are the predicted mean

and variance of the measured variables that are calculated with the gPC models. Using Equations B.10 and B.11, these values can be explicitly computed. γ1,k and γ1,k are the mean and variance computed with the measurements,

and Ω2 is the decision variables vector consisting of the gPC coefficients for the unknown stochastic parameters x2. To solve the optimization as Equation B.18, the following procedures are preceded.

(1) Input the samples of known parameters x1 and the available data of the measured variables Ŷ.

(2) Choose the order of polynomials (p) used to approximate the unknown parameters x2 in Equation

B.5, determine the polynomial basis function Φk, and then formulate the gPC approximations for

both unknown parameters x2 and measured variables Y.

(3) Substitute the gPC approximations in Step (2) into the nonlinear first principles’ model , and generate a new gPC symbolic model by using polynomial chaos quadrature (PCQ), which transform the original stochastic model into a set of coupled deterministic equations (gPC symbolic model). (4) Set initial values of {α_i,k[0]} in Equation B.5, i.e., the gPC coefficients for each unknown parameter.

Optimization with Equation (B.15):

i. Substitute each input sample of parameters x1 and the initial values {α_i,k[0]} into the gPC symbolic

model generated in Initialization Step (3).

ii. Solve the gPC coefficients for the measured variables from the gPC symbolic model.

iii. Using Equations B.10) and B.11, calculate the mean and variance of the measured variables with the gPC coefficients in Step (ii).

iv. Calculate the Euclidean distance between the mean value in (iii) and the mean value computed from the collected measurements of the measured variables Ŷ.

v. Calculate the Euclidean distance between the variance in (iii) and the variance computed from the collected measurements of the measured variables Ŷ.

vi. Solve the optimization Equation B.18 to obtain the optimum of the unknown gPC coefficients {α_i,k[*]}. Each optimization iteration entails the repeated evaluation of the gPC expansion and the Euclidean distance as in Steps (iv) and (v).

As compared with the Algorithm 1, this method provides an explicit gPC expression of the measured variables, while estimating the unknown uncertain parameters. It can be further employed to evaluate how uncertainties of a dynamical system’s parameters manifest the effect on the measured variables.

Gram-Schmidt orthogonalization – The Gram-Schmidt polynomial chaos can be applied to both approaches

above, if the probability distribution of stochastic unknown parameters is outside of the Wiener-Askey scheme. A few more procedures can be performed to replace the Step (2) in the Inputs initialization for both algorithms, which involve as per the following steps. (i) Determine the weighting function w(ξ) in Equation (9); (ii) Compute the polynomial basis function {Φk(ξ)} with respect to a pre-assigned weighting function w(ξ) in (i), using the

Gram-Schmidt algorithm; (iii) Choose the order of polynomials (p) used to approximate the unknown parameters

x2 in Equation B.5, (iv) Formulate the gPC approximations for unknown parameters x2 in the Algorithm 1 or

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The rest parts of algorithm follow the same procedures as described in the proposed methods. The employment of Gram-Schmidt orghogonalization algorithm thus extends our algorithms to estimate an unknown parametric input for any type of probability distribution.