Polynomial chaos expansion
Polynomial chaos expansion (PCE) received much attention in applied sciences in recent years. As a novel method to determine the evolution of uncertainty, which origin from the probabilistic uncertainty in the system parameters, in a complex system, PCE has been found to be an attractive and promising numerical method to uncertain quantification.
The concept of PCE is to use a series of weighted orthogonal polynomials to approximate a function.
where φi is the mode function and βi is the model strength. The expansion in the above is a PCE. The strength is calculated as
where the notation <, > means the inner product of two functions with respect to the prob- ability density function of the parameter Ξ. The selection of polynomials associated to different distributions of Ξ:
An example application:
where ε follows an uniform distribution on [-1, 1] and ζ follows a normal distribution N(5,1). The expected value of x and its standard deviation is
···
