Polynomial Chaos Expansion
Description
Polynomial Chaos Expansion (PCE) is a mathematical technique used to approximate the behavior of a system that contains random variables. It is commonly used in the field of Uncertainty Quantification (UQ) to model the effects of uncertainty on a system’s behavior.
The basic idea behind PCE is to represent the solution of a problem as a combination of polynomials. These polynomials are chosen based on the probability distribution of the random variables in the system. The coefficients of the polynomials are also random variables, and their values are determined by solving the problem for different values of the random inputs. The PCE method can be used to obtain statistical information about the solution generated by the model, such as its mean, variance, and probability distribution. This allows user to analyze the effects of uncertainty on the system’s behavior.
Advantages
One advantage of PCE over other methods is that it can provide more precise approximation of the solution, especially when the input variables are represented by a polynomial expansion. PCE often has faster convergence rates than other methods, meaning that it requires fewer sample points to achieve a desired level of accuracy. PCE is also more flexible than other UQ methods, as it can handle non-Gaussian and correlated random variables, thus it can be adapted to different types of models and applications.
When comparing PCE with other methods, it should be noted that Stochastic Collocation can have a lower convergence rate and can be highly sensitive to the selection of collocation points, thus it may require a large number of points to achieve the desired level of accuracy, while Monte Carlo method, based on random sampling, may produce less precise results with the comparable number of samples.
Disadvantages
One significant disadvantage of PCE is its limitation in handling models with discontinuous or non-smooth response surfaces. PCE uses orthogonal polynomials to approximate the response surface, and therefore, the accuracy of the method can be compromised in cases where the response surface is not smooth. In such cases, other UQ methods such as Monte Carlo and Stochastic Collocation may be more suitable.
Furthermore, PCE as well as SC can be computationally expensive for multivariate problems. In such cases, the Monte Carlo method may be more efficient. Therefore, it is common practice to perform a coarse sensitivity analysis using the Monte Carlo method to detect a few parameters of the highest importance and then perform a detailed analysis using PCE or SC for only these key parameters.
Finally, while PCE is flexible in handling different types of probability distributions, it may not always be the best option for modeling complex distributions. In some cases, other UQ methods such as Monte Carlo and Stochastic Collocation can capture the distribution more efficiently, especially when the distribution has significant skewness, kurtosis, or heavy tails. Despite these limitations, PCE remains a powerful UQ tool that has been widely used in various applications in engineering, science, and finance.
Parameters
Polynomial order
The polynomial order refers to the degree of the polynomials that are used to represent the uncertain quantities in the system. A higher-order polynomial allows for a more detailed representation of the uncertainty, but also requires more computational resources to evaluate. In general, the polynomial order should be chosen to provide a trade-off between good approximation of the true behavior of the system and resource consumption.
Regression
Regression is a sampler parameter describing the building process of PCE. There are two main approaches to constructing the PCE - projection and regression.
- Projection method (regression=False): In the projection method, the solution of the problem is projected onto a set of orthogonal polynomials. The projection coefficients are then used to construct the PCE.
- Regression method (regression=True): In the regression method, the PCE is constructed by fitting the solution of the problem to a set of orthogonal polynomials. This is done by finding the polynomial coefficients that minimize the mean square error between the PCE and the solution.
Quadrature
Quadrature is a sampler parameter for setting the quadrature method or sequence sampler.
When projection-variant PCE is constructed, then Quadrature parameter is simply a quadrature method. There are multiple different quadratures that can be used, including the default Gaussian (“G”), 1D Clenshaw–Curtis (“C” or “clenshaw_curtis”), Gauss-Patterson (“gauss_patterson”) and Newton-Cotes (“newton_cotes”).
When the regression-variant PCE is constructed, then Quadrature parameter defaults to Hammersley sequence sampler.
Grid type
The grid type allows defining whether tensor product grid or sparse grid should be used for the grid generation in the projection-variant of the method.
Number of produces samples
- When using pseudo-spectral projection method with tensored quadrature: $$ Ns = (p + 1)^d $$
- When using pseudo-spectral projection method with sparse grid quadrature: $$ Ns = O((p + 1)*log(p + 1)^{(d-1)}) $$
- When using regression method: $$ Ns = 2*(p + d)!/(p!*d!) $$ Where: $p$ is the polynomial degree and $d$ is the number of uncertain parameters.