Stochastic Collocation
Description
Stochastic Collocation (SC) is a numerical method used in the field of uncertainty quantification to approximate the behavior of a complex system that exhibits uncertainty. It is based on the idea of sampling the uncertain quantities in the system at a set of carefully chosen points, known as collocation points, and then using these samples to construct a polynomial approximation of the system’s behavior.
This method is widely used for Sensitivity Analysis, and it’s one of the most popular methods. Using stochastic collocation, it’s possible to calculate statistical moments of the solution. These moments represent a variety of things, e.g., mean represents the average value of the solution, while the variance represents the spread or dispersion of the solution around the mean. Higher-order moments, such as the skewness or kurtosis provide additional information about the shape of the distribution of the solution.
Advantages
One of the advantages of SC is that it can provide a more accurate approximation of the solution, especially for problems with complex underlying probability distributions.
In contrast to other methods, SC can efficiently handle a wide range of probability distributions, including distributions that are not easily represented by a polynomial. For example, PCE is best suited to problems with random variables that can be represented by a polynomial expansion thus it may produce poor quality results for non-linear or discontinuous distributions. Monte Carlo, instead, is based on random sampling, which can result in a less accurate approximation of the solution, at least with acceptable number of samples: when the input variables have complex probability distributions, Monte Carlo will require more samples in regions where the function has high variation; without the extensions to Monte Carlo method (like adaptive sampling), this can lead to large number of samples being required.
Disadvantages
One potential disadvantage of using SC for uncertainty quantification is that it can be less accurate for systems with highly non-linear or non-smooth behavior, or for systems with very large amounts of uncertainty. In such cases, other methods may be more appropriate.
The SC method, just like the PCE, can be computationally expensive for the problems with multiple input parameters. The efficiency of Monte Carlo method is less sensitive to a number of parameters, and therefore this method may be more efficient for such cases. Consequently, a common practice is to perform a coarse sensitivity analysis using the Monte Carlo method to quantify the importance of all parameters, and then perform a detailed analysis using PCE or SC to discover an influence of only a few key parameters.
Parameters
Polynomial order
The polynomial order refers to the degree of the polynomials that are used within SC to generate collocation points. 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.
Quadrature
Quadrature refers to a method for approximating the definite integral of a function. In SC quadrature is used for proper selecting collocation points to ensure good performance of the method.
There are multiple different quadratures that can be used, but usually the default Gaussian (“G”) is a good suit. Other quadratures are: 1D Clenshaw–Curtis (“C” or “clenshaw_curtis”), Gauss-Patterson (“gauss_patterson”) and Newton-Cotes (“newton_cotes”).
Grid type
The grid type allows defining whether tensor product grid or sparse grid should be used for the collocation points selection.
Number of produced samples
When using tensor product grid: $$ Ns = (p + 1)^d $$
When using sparse grid: $$ Ns = O((p + 1)*log(p + 1)^{(d-1)}) $$ Where: $p$ is the polynomial degree and $d$ is the number of uncertain parameters.