2.1. gPC Method
The generalized chaotic polynomial method was used to solve the uncertainty of electromagnetic radiation. First, the original model was expressed as a functional
Y, and then the functional model was expanded in the form of a generalized chaotic polynomial. The expanded result is:
In the above formula,
is the nth-order representation of the mixed orthogonal polynomial, which is a function of the multidimensional standard random variable
.
is composed of the product of one-dimensional orthogonal polynomial basis functions. Each basis function corresponds to a different random variable.
Table 1 lists the orthogonal polynomial basis functions corresponding to different distribution types. Each distribution type has its own unique orthogonal polynomial basis functions. The variables in this paper showed normal distribution and uniform distribution.
Formula (1) shows that the chaotic polynomial expansion contains infinite items but cannot handle infinite items in the actual calculation. In order to improve the calculation efficiency, it is usually necessary to truncate the chaotic polynomial, and the truncation order is the
P order. Therefore, the generalized chaotic polynomial expansion model can be expressed by truncating the expansion (2):
is a polynomial coefficient. After the P-order truncation of the generalized chaotic polynomial expansion, the expansion contains the
Q terms, and the
p polynomial expansion terms remains after the first term is removed. The specific value of
Q is related to the truncation order
P and the dimension
d, which can be expressed as:
The chaotic polynomial in Formula (3) should satisfy:
In Formula (4),
represents the inner product operation in multidimensional space, and its weight function
is the product of the one-dimensional weight function corresponding to each dimension random variable
, which is the Kronecker function
:
Then, we used the random response surface method to solve chaotic polynomials.
2.2. Random Response Surface Method
In the calculation process of the PC method, calculating the expansion coefficient is a crucial step. Isukapalli first proposed a method based on linear regression to solve the PC coefficient [
20], which mainly constructs the response surface based on random probability space and can only use the Hermite orthogonal polynomial to construct the chaotic polynomial expansion model at the earliest. In this paper, it was extended to the gPC expansion model, and the calculation process of the stochastic response surface method adopted in this paper was as follows:
- a.
Construction of the generalized chaotic polynomial expansion model
Through the determination of the input variables and output variables of the system, according to their different distribution types and truncation processing, we obtained a chaotic polynomial model that can be expressed in a truncated polynomial form; the model is
To distinguish it from Equation (2), the polynomial coefficients to be found are expressed as , where is a set of D-dimensional standard random variables .
- b.
Estimation of the coefficient of interest
(1) A certain number of sample points were selected by the MC method, and n valid samples were selected from the standard random space, where each sample point was denoted as S, i.e., is .
(2) The sample points were transformed from the standard random space to the original random space, obtaining:
represents the value of the one-dimensional sample point
in the
j-th sample point in the original random space. For example,
in the formula obeys a normal distribution, and
is a standard normal random variable, then:
and represent the mean and standard deviation of the one-dimensional random input variable c, respectively.
(3) The real response function value obtained after calculating the selected effective sample points through the original model to obtain the response function
g(X) can be expressed as:
indicates the true response function value of the Nth sample point.
(4) Using least-squares regression to estimate the coefficients and then substituting the sample
and its true response function value
into the gPC expansion model, respectively, we obtained the following expressions:
The above formula can be abbreviated as
Ab = G, where the matrix A can be expressed as:
Finally, the coefficients of the polynomial chaotic could be obtained by the Formula:
The above process completed the solution of the gPC coefficients, and the analytical solution obtained by solving the proxy model could then be used to estimate the statistical properties of the output Y = g(X) of the constructed gPC proxy model using the MC method.
2.3. Global Sensitivity Analysis
The combination of the generalized chaotic polynomial and the Sobol global sensitivity analysis method was used to realize a fast calculation of the Sobol global sensitivity index. By combining the coefficients of the chaotic polynomial, the global sensitivity index of different random input variables can effectively obtained, and the first-order sensitivity index is obtained as:
represents the corresponding standard normal random variable
,
represents the set of polynomials
containing only the random variable
, and
represents the coefficient of the polynomial
.
can be calculated by the following formula:
where
n is the dimension of the random input variables in the model. Through the above analysis and using the coefficients of the chaotic polynomial, the total sensitivity index of the input parameters
can be expressed as:
indicates to add all items not related to x and the corresponding Sobol global sensitivity index. The Sobol first-order sensitivity index and the total sensitivity index of the random input variables can be obtained directly by using the generalized chaotic polynomial method, which is faster and more efficient than the Monte Carlo method. In addition, with the increase of the truncated order of the generalized chaotic polynomial, the calculation accuracy will be improved accordingly. Therefore, the generalized chaotic polynomial method is feasible to solve the sensitivity analysis problem.