Importance Sampling

Overview

The Importance Sampling (IS) is a useful Monte Carlo method which allows to estimate the probabilities of rare failure events $P_{f}$ for reliability problems with both simple and complex limit state functions $g(\vec{X})$. The IS method is based on the following reformulation of the general analytical expression for the probability of failure $P_{f}$:

\[P_{f} = P(\Omega_{f}) = \int_{\Omega_{f}} f_{\vec{X}}(\vec{x}) d\vec{x} = \int_{\mathbb{R}^{n}} \mathbb{I}(\vec{x}) f_{\vec{X}}(\vec{x}) = \int_{\mathbb{R}^{n}} \dfrac{\mathbb{I}(\vec{x}) f_{\vec{X}}(\vec{x})}{q(\vec{x})} q(\vec{x}) d\vec{x} = \mathbb{E}_{q}\left[\dfrac{\mathbb{I}(\vec{x}) f_{\vec{X}}(\vec{x})}{q(\vec{x})}\right]\]

where $f_{\vec{X}}(\vec{x})$ is the target joint probability density function of the input random vector $\vec{X}$, $q(\vec{x})$ is the proposal probability density function, $\Omega_{f} = \{\vec{X}: g(\vec{X}) \leq 0\}$ is the failure domain defined by the limit state function $g(\vec{X})$, and $\mathbb{I}(\vec{x})$ is the indicator function given by:

\[\mathbb{I}(\vec{x}) = \begin{cases} 1 & \text{if } \vec{x} \in \Omega_{f} \\ 0 & \text{otherwise} \end{cases}\]

Therefore, the probability of failure $P_{f}$ is defined as the expectation of $\mathbb{I}(\vec{x}) f_{\vec{X}}(\vec{x}) / q(\vec{x})$ evaluated with respect to the proposal probability density function $q(\vec{x})$. If samples of the input random vector $\vec{x}$ are generated numerically from the proposal probability density function $q(\vec{x})$, then the estimator of the probability of failure $\hat{P}_{f}$ is

\[\hat{P}_{f} = \dfrac{1}{N} \sum_{i = 1}^{N} \dfrac{\mathbb{I}(\vec{x}_{i}) f_{\vec{X}}(\vec{x}_{i})}{q(\vec{x}_{i})}\]

where $N$ is the number of generated sampled. The estimator $\hat{P}_{f}$ is unbiased, i.e., it correctly predicts the true probability of failure, such that $\mathbb{E}(\hat{P}_{f}) = P_{f}$.

If the proposal probability density function $q(\vec{x})$ is chosen to be such that has large values in the failure domain $\Omega_{f}$ (important region), then it is possible to relatively accurately estimate small probability of failure $P_{f}$ with a small number of samples $N$. The hard part is, of course, finding a "good" proposal probability density function $q(\vec{x})$. Typically, it is recommended to use a multivariate normal distribution with uncorrelated marginals centered at the design point $\vec{x}^{*}$, such that,

\[q \sim N(\vec{M} = \vec{x}^{*}, \Sigma = \sigma I)\]

API

solve(Problem::ReliabilityProblem, AnalysisMethod::IS; showprogressbar = false)

IS <: AbstractReliabililyAnalysisMethod

ISCache