COMSOL 6.4 - Uncertainty Quantification Theory

The Latin hypercube sampling (LHS) method is used to generate sample data with good space filling in the input parameter space. In addition, sequential LHS is used to generate additional sample points based on existing samples in an uncertainty quantification (UQ) study that requires more quantities of interest (QoIs).

The Morris sampling method is used to generate sample points for the Morris one-at-a-time (MOAT) method, where the next sample point only varies in one dimension from the current point of the input parameter space within each trajectory.

The importance sampling method is used to do Monte Carlo analysis with a multimodal sampling density created near the region where the criteria for reliability analysis are satisfied in the input parameter space.

Input distributions are used to set up the input parameters in a UQ study. This module supports a wide range of continuous independent distributions.

A key feature of the Uncertainty Quantification Module is its ability to train and use a so-called surrogate model, also known as a metamodel, for a particular UQ analysis to save computation resources. A surrogate model is a compact mathematical model constructed to represent and evaluate the QoIs in the domain of interest defined by the input parameters. This model is completely independent of the underlying COMSOL model and can, when trained properly, be used instead of the COMSOL model to predict values for the QoIs for other values of the input parameters than those solved for. The surrogate model evaluation (or prediction) has a very low cost. This is of paramount importance for UQ analysis when a Monte Carlo-type analysis is being used because a large number of evaluations are often required to achieve high accuracy. This is particularly true for more realistic problems where a COMSOL model evaluation might require significant resources and where the UQ analysis involves several parameters.

The sparse polynomial chaos expansion (SPCE) and adaptive sparse polynomial chaos expansion (ASPCE) models are used to model QoIs where each QoI is expanded with multivariate orthonormal polynomials that are specified for a particular type of probability distribution of input parameters. SPCE and ASPCE models are most commonly used for sensitivity analysis where the Sobol indices can be directly computed from the coefficients in the surrogate model.

The Gaussian process (GP) and adaptive Gaussian process (AGP) are popular surrogate models widely used for sensitivity analysis, uncertainty propagation, and reliability analysis. GP and AGP are probabilistic in that they provide the variance of the prediction at each point sampled from the input parameter space.

The Morris one-at-a-time (MOAT) method is a lightweight global screening method that gives a qualitative measure of the importance of each input parameter. The method is purely sample based and does not require a surrogate model. This is an ideal method when the number of input parameters is too large to allow the application of more computationally expensive UQ studies.

The Correlation method is a widely used method for determining the linear relationship between each input parameter and the QoI. The method is purely sample based and does not require a surrogate model. There are four types of correlations available: bivariate correlation (also known as Pearson’s correlation), rank bivariate correlation (also known as Spearman’s rank correlation), partial correlation, and rank partial correlation.

The Sobol method (or variance decomposition method, analysis of variance (ANOVA)) looks at the entire input parameter distribution and decomposes the variance of each QoI into a sum of contributions from the input parameters and their interactions. In general, the Sobol method is a sample-based method and requires a large number of samples to achieve good accuracy. Here, the first-order Sobol index and the total Sobol index based on surrogate models are computed.

Uncertainty propagation, also known as forward propagation of uncertainty, maps uncertainty of the input parameters to uncertainty of the QoIs. A Monte Carlo analysis is conducted based on a surrogate model to generate the approximated probability density function (PDF) of each QoI.

Reliability analysis computes the probability that the QoIs satisfy the reliability criteria defined with thresholds corresponding to the QoIs. The efficient global reliability analysis (EGRA) method is a global reliability analysis method that uses an AGP model to reduce the number of COMSOL model evaluations by only constructing a model that accurately separates the region where the reliability criteria is satisfied from the region where such criteria is not satisfied.

Inverse uncertainty quantification computes the posterior distribution of input parameters — the calibration parameters that best reflects their distribution based on the experimental data and the prior knowledge on the calibration parameters. The Markov chain Monte Carlo (MCMC) method infers the posterior distribution of the calibration parameters by constructing Markov chains whose stationary distribution is the posterior distribution. This analysis is called inverse because it propagates information of experimental data backward to gain information on the calibration parameters.