diff --git a/docs/source/reliability/index.rst b/docs/source/reliability/index.rst index d3010039..51362cfb 100644 --- a/docs/source/reliability/index.rst +++ b/docs/source/reliability/index.rst @@ -2,19 +2,20 @@ Reliability =========== Reliability of a system refers to the assessment of its probability of failure (i.e the system no longer satisfies some -performance measures), given the model uncertainty in the structural, environmental and load parameters. Given a vector -of random variables :math:`\textbf{X}=\{X_1, X_2, \ldots, X_n\} \in \mathcal{D}_\textbf{X}\subset \mathbb{R}^n`, where -:math:`\mathcal{D}` is the domain of interest and :math:`f_{\textbf{X}}(\textbf{x})` is its joint probability density -function then, the probability that the system will fail is defined as +performance measure), given the model uncertainty in the structural, environmental and load parameters. Given a vector +of random variables :math:`\textbf{X}=[X_1, X_2, \ldots, X_n]^T \in \mathcal{D}_\textbf{X}\subset \mathbb{R}^n`, where +:math:`\mathcal{D}_\textbf{X}` is the domain of interest and :math:`f_{\textbf{X}}(\textbf{x})` is its joint probability density +function, then the probability that the system will fail is defined as -.. math:: P_f =\mathbb{P}(g(\textbf{X}) \leq 0) = \int_{D_f} f_{\textbf{X}}(\textbf{x})d\textbf{x} = \int_{\{\textbf{X}:g(\textbf{X})\leq 0 \}} f_{\textbf{X}}(\textbf{x})d\textbf{x} +.. math:: P_f =\mathbb{P}(g(\textbf{X}) \leq 0) = \int_{\mathcal{D}_f} f_{\textbf{X}}(\textbf{x})d\textbf{x} = \int_{\{\textbf{X}:g(\textbf{X})\leq 0 \}} f_{\textbf{X}}(\textbf{x})d\textbf{x} -where :math:`g(\textbf{X})` is the so-called performance function. The reliability problem is often formulated in the +where :math:`g(\textbf{X})` is the so-called performance function and :math:`\mathcal{D}_f=\{\textbf{X}:g(\textbf{X})\leq 0 \}` is the failure domain. +The reliability problem is often formulated in the standard normal space :math:`\textbf{U}\sim \mathcal{N}(\textbf{0}, \textbf{I}_n)`, which means that a nonlinear isoprobabilistic transformation from the generally non-normal parameter space -:math:`\textbf{X}\sim f_{\textbf{X}}(\cdot)` to the standard normal is required (see the :py:mod:`.transformations` module). +:math:`\textbf{X}\sim f_{\textbf{X}}(\cdot)` to the standard normal space is required (see the :py:mod:`.transformations` module). The performance function in the standard normal space is denoted :math:`G(\textbf{U})`. :py:mod:`.UQpy` does not require this transformation and can support reliability analysis for problems with arbitrarily distributed parameters. diff --git a/docs/source/reliability/subset.rst b/docs/source/reliability/subset.rst index 5da7def8..555b76af 100644 --- a/docs/source/reliability/subset.rst +++ b/docs/source/reliability/subset.rst @@ -2,27 +2,27 @@ Subset Simulation ------------------- In the subset simulation method :cite:`SubsetSimulation` the probability of failure :math:`P_f` is approximated by a product of probabilities -of more frequent events. That is, the failure event :math:`G = \{\textbf{x} \in \mathbb{R}^n:G(\textbf{x}) \leq 0\}`, +of more frequent events. That is, the failure event :math:`G = \{\textbf{X} \in \mathbb{R}^n:g(\textbf{X}) \leq 0\}`, is expressed as the of union of `M` nested intermediate events :math:`G_1,G_2,\cdots,G_M` such that :math:`G_1 \supset G_2 \supset \cdots \supset G_M`, and :math:`G = \cap_{i=1}^{M} G_i`. The intermediate failure events -are defined as :math:`G_i=\{G(\textbf{x})\le b_i\}`, where :math:`b_1>b_2>\cdots>b_i=0` are positive thresholds selected -such that each conditional probability :math:`P(G_i | G_{i-1}), ~i=2,3,\cdots,M-1` equals a target probability value +are defined as :math:`G_i=\{g(\textbf{X})\le b_i\}`, where :math:`b_1>b_2>\cdots>b_M=0` are non-negative thresholds selected +such that each conditional probability :math:`P(G_{i+1} | G_{i}),\ i=1,2,\cdots,M-1` equals a target probability value :math:`p_0`. The probability of failure :math:`P_f` is estimated as: -.. math:: P_f = P\left(\cap_{i=1}^M G_i\right) = P(G_1)\prod_{i=2}^M P(G_i | G_{i-1}) +.. math:: P_f = P\left(\bigcap_{i=1}^M G_i\right) = P(G_1)\prod_{i=1}^{M-1} P(G_{i+1} | G_{i}) where the probability :math:`P(G_1)` is computed through Monte Carlo simulations. In order to estimate the conditional -probabilities :math:`P(G_i|G_{i-1}),~j=2,3,\cdots,M` generation of Markov Chain Monte Carlo (MCMC) samples from the -conditional pdf :math:`p_{\textbf{U}}(\textbf{u}|G_{i-1})` is required. In the context of subset simulation, the Markov +probabilities :math:`P(G_{i+1}|G_i),~i=1,2,\cdots,M-1` generation of Markov Chain Monte Carlo (MCMC) samples from the +conditional pdf :math:`p_{\textbf{X}}(\textbf{x}|G_i)` is required. In the context of subset simulation, the Markov chains are constructed through a two-step acceptance/rejection criterion. Starting from a Markov chain state -:math:`\textbf{x}` and a proposal distribution :math:`q(\cdot|\textbf{x})`, a candidate sample :math:`\textbf{w}` is -generated. In the first stage, the sample :math:`\textbf{w}` is accepted/rejected with probability +:math:`\textbf{X}` and a proposal distribution :math:`q(\cdot|\textbf{X})`, a candidate sample :math:`\textbf{W}` is +generated. In the first stage, the sample :math:`\textbf{W}` is accepted/rejected with probability -.. math:: \alpha=\min\bigg\{1, \frac{p(\textbf{w})q(\textbf{x}|\textbf{w})}{p(\textbf{x})q(\textbf{w}|\textbf{x})}\bigg\} +.. math:: \alpha=\min\bigg\{1, \frac{p_\textbf{X}(\textbf{w})q(\textbf{x}|\textbf{W})}{p_\textbf{X}(\textbf{x})q(\textbf{w}|\textbf{X})}\bigg\} -and in the second stage is accepted/rejected based on whether the sample belongs to the failure region :math:`G_j`. -:class:`.SubSetSimulation` can be used with any of the available (or custom) :class:`.MCMC` classes in the -:py:mod:`sampling` module. +and in the second stage is accepted/rejected based on whether the sample belongs to the failure region :math:`G_i`. +:class:`.SubsetSimulation` can be used with any of the available (or custom) :class:`.MCMC` classes in the +:py:mod:`Sampling` module. SubsetSimulation Class ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^