1\chapter{Reliability Methods}\label{uq:reliability} 2 3%This chapter explores local and global reliability methods in greater 4%detail than that provided in the Uncertainty Quantification chapter of 5%the User's Manual. 6 7 8\section{Local Reliability Methods}\label{uq:reliability:local} 9 10Local reliability methods include the Mean Value method and the family 11of most probable point (MPP) search methods. Each of these methods is 12gradient-based, employing local approximations and/or local 13optimization methods. 14 15 16\subsection{Mean Value}\label{uq:reliability:local:mv} 17 18The Mean Value method (MV, also known as MVFOSM in \cite{Hal00}) is 19the simplest, least-expensive reliability method because it estimates 20the response means, response standard deviations, and all CDF/CCDF 21response-probability-reliability levels from a single evaluation of 22response functions and their gradients at the uncertain variable 23means. This approximation can have acceptable accuracy when the 24response functions are nearly linear and their distributions are 25approximately Gaussian, but can have poor accuracy in other 26situations. 27 28The expressions for approximate response mean $\mu_g$ and approximate 29response variance $\sigma^2_g$ are 30\begin{eqnarray} 31\mu_g & = & g(\mu_{\bf x}) \label{eq:mv_mean1} \\ 32\sigma^2_g & = & \sum_i \sum_j Cov(i,j) \frac{dg}{dx_i}(\mu_{\bf x}) 33 \frac{dg}{dx_j}(\mu_{\bf x}) \label{eq:mv_std_dev} 34\end{eqnarray} 35where ${\bf x}$ are the uncertain values in the 36space of the original uncertain variables (``x-space''), $g({\bf x})$ 37is the limit state function (the response function for which 38probability-response level pairs are needed), and the use of a linear 39Taylor series approximation is evident. 40These two moments are then used for mappings from response target to 41approximate reliability level ($\bar{z} \to \beta$) and from 42reliability target to approximate response level 43($\bar{\beta} \to z$) using 44\begin{eqnarray} 45\bar{z} \rightarrow \beta: & ~ & 46\beta_{\rm CDF} = \frac{\mu_g - \bar{z}}{\sigma_g}, ~~~~~ 47\beta_{\rm CCDF} = \frac{\bar{z} - \mu_g}{\sigma_g} \label{eq:mv_ria} \\ 48\bar{\beta} \rightarrow z: & ~ & 49z = \mu_g - \sigma_g \bar{\beta}_{\rm CDF}, ~~~~~ 50z = \mu_g + \sigma_g \bar{\beta}_{\rm CCDF} \label{eq:mv_pma} 51\end{eqnarray} 52respectively, where $\beta_{\rm CDF}$ and $\beta_{\rm CCDF}$ are the 53reliability indices corresponding to the cumulative and complementary 54cumulative distribution functions (CDF and CCDF), respectively. 55 56With the introduction of second-order limit state information, MVSOSM 57calculates a second-order mean as 58\begin{equation} 59\mu_g = g(\mu_{\bf x}) + \frac{1}{2} \sum_i \sum_j Cov(i,j) 60\frac{d^2g}{dx_i dx_j}(\mu_{\bf x}) \label{eq:mv_mean2} 61\end{equation} 62This is commonly combined with a first-order variance 63(Equation~\ref{eq:mv_std_dev}), since second-order variance involves 64higher order distribution moments (skewness, kurtosis)~\cite{Hal00} 65which are often unavailable. 66 67The first-order CDF probability $p(g \le z)$, first-order 68CCDF probability $p(g > z)$, $\beta_{\rm CDF}$, and $\beta_{\rm CCDF}$ are 69related to one another through 70\begin{eqnarray} 71p(g \le z) & = & \Phi(-\beta_{\rm CDF}) \label{eq:p_cdf} \\ 72p(g > z) & = & \Phi(-\beta_{\rm CCDF}) \label{eq:p_ccdf} \\ 73\beta_{\rm CDF} & = & -\Phi^{-1}(p(g \le z)) \label{eq:beta_cdf} \\ 74\beta_{\rm CCDF} & = & -\Phi^{-1}(p(g > z)) \label{eq:beta_ccdf} \\ 75\beta_{\rm CDF} & = & -\beta_{\rm CCDF} \label{eq:beta_cdf_ccdf} \\ 76p(g \le z) & = & 1 - p(g > z) \label{eq:p_cdf_ccdf} 77\end{eqnarray} 78where $\Phi()$ is the standard normal cumulative distribution 79function, indicating the introduction of a Gaussian assumption on the 80output distributions. A common convention in the literature is to 81define $g$ in such a way that the CDF probability for a response level 82$z$ of zero (i.e., $p(g \le 0)$) is the response metric of interest. 83Dakota is not restricted to this convention and is designed to support 84CDF or CCDF mappings for general response, probability, and 85reliability level sequences. 86 87With the Mean Value method, it is possible to obtain importance 88factors indicating the relative contribution of the input variables to 89the output variance. The importance factors can be viewed as an 90extension of linear sensitivity analysis combining deterministic 91gradient information with input uncertainty information, \emph{i.e.} 92input variable standard deviations. The accuracy of the importance 93factors is contingent of the validity of the linear Taylor series 94approximation used to approximate the response quantities of interest. 95The importance factors are determined as follows for each of $n$ random 96variables: 97%, where we require uncorrelated input variables: 98\begin{equation} 99 {\rm ImportFactor}_i = \left[ \frac{\sigma_{x_i}}{\sigma_g} 100 \frac{dg}{dx_i}(\mu_{\bf x}) \right]^2, ~~~~ i = 1, \dots, n 101\end{equation} 102where it is evident that these importance factors correspond to the 103diagonal terms in Eq.~\ref{eq:mv_std_dev} normalized by the total 104response variance. %involve an attribution of the total response 105%variance among the set of input variables. 106In the case where the input variables are correlated 107resulting in off-diagonal terms for the input covariance, we can also 108compute a two-way importance factor as 109\begin{equation} 110 {\rm ImportFactor}_{ij} = 2 \frac{\sigma^2_{x_{ij}}}{\sigma^2_g} 111 \frac{dg}{dx_i}(\mu_{\bf x}) \frac{dg}{dx_j}(\mu_{\bf x}), 112 ~~~~ i = 1, \dots, n; ~~~~ j = 1, \dots, i-1 113\end{equation} 114These two-way factors differ from the Sobol' interaction terms that 115are computed in variance-based decomposition (refer to 116Section~\ref{uq:expansion:vbd}) due to the non-orthogonality of the 117Taylor series basis. Due to this non-orthogonality, two-way 118importance factors may be negative, and due to normalization by the 119total response variance, the set of importance factors will always sum 120to one. 121 122 123\subsection{MPP Search Methods}\label{uq:reliability:local:mpp} 124 125All other local reliability methods solve an equality-constrained nonlinear 126optimization problem to compute a most probable point (MPP) and then 127integrate about this point to compute probabilities. The MPP search 128is performed in uncorrelated standard normal space (``u-space'') since 129it simplifies the probability integration: the distance of the MPP 130from the origin has the meaning of the number of input standard 131deviations separating the mean response from a particular response 132threshold. The transformation from correlated non-normal 133distributions (x-space) to uncorrelated standard normal distributions 134(u-space) is denoted as ${\bf u} = T({\bf x})$ with the reverse 135transformation denoted as ${\bf x} = T^{-1}({\bf u})$. These 136transformations are nonlinear in general, and possible approaches 137include the Rosenblatt~\cite{Ros52}, Nataf~\cite{Der86}, and 138Box-Cox~\cite{Box64} transformations. The nonlinear transformations 139may also be linearized, and common approaches for this include the 140Rackwitz-Fiessler~\cite{Rac78} two-parameter equivalent normal and the 141Chen-Lind~\cite{Che83} and Wu-Wirsching~\cite{Wu87} three-parameter 142equivalent normals. Dakota employs the Nataf nonlinear transformation 143which is suitable for the common case when marginal distributions and 144a correlation matrix are provided, but full joint distributions are 145not known\footnote{If joint distributions are known, then the 146Rosenblatt transformation is preferred.}. This transformation occurs 147in the following two steps. To transform between the 148original correlated x-space variables and correlated standard normals 149(``z-space''), a CDF matching condition is applied for each of the 150marginal distributions: 151\begin{equation} 152\Phi(z_i) = F(x_i) \label{eq:trans_zx} 153\end{equation} 154where $F()$ is the cumulative distribution function of the original 155probability distribution. Then, to transform between correlated 156z-space variables and uncorrelated u-space variables, the Cholesky 157factor ${\bf L}$ of a modified correlation matrix is used: 158\begin{equation} 159{\bf z} = {\bf L} {\bf u} \label{eq:trans_zu} 160\end{equation} 161where the original correlation matrix for non-normals in x-space has 162been modified to represent the corresponding ``warped'' correlation in 163z-space~\cite{Der86}. 164 165The forward reliability analysis algorithm of computing CDF/CCDF 166probability/reliability levels for specified response levels is called 167the reliability index approach (RIA), and the inverse reliability 168analysis algorithm of computing response levels for specified CDF/CCDF 169probability/reliability levels is called the performance measure 170approach (PMA)~\cite{Tu99}. The differences between the RIA and PMA 171formulations appear in the objective function and equality constraint 172formulations used in the MPP searches. For RIA, the MPP search for 173achieving the specified response level $\bar{z}$ is formulated as 174computing the minimum distance in u-space from the origin to the 175$\bar{z}$ contour of the limit state response function: 176\begin{eqnarray} 177{\rm minimize} & {\bf u}^T {\bf u} \nonumber \\ 178{\rm subject \ to} & G({\bf u}) = \bar{z} \label{eq:ria_opt} 179\end{eqnarray} 180where ${\bf u}$ is a vector centered at the origin in u-space and 181$g({\bf x}) \equiv G({\bf u})$ by definition. For PMA, the MPP search 182for achieving the specified reliability level $\bar{\beta}$ or 183first-order probability level $\bar{p}$ is formulated as computing the 184minimum/maximum response function value corresponding to a prescribed 185distance from the origin in u-space: 186\begin{eqnarray} 187{\rm minimize} & \pm G({\bf u}) \nonumber \\ 188{\rm subject \ to} & {\bf u}^T {\bf u} = \bar{\beta}^2 \label{eq:pma_opt} 189\end{eqnarray} 190where $\bar{\beta}$ is computed from $\bar{p}$ using 191Eq.~\ref{eq:beta_cdf} or~\ref{eq:beta_ccdf} in the latter case of a 192prescribed first-order probability level. For a specified generalized 193reliability level $\bar{\beta^*}$ or second-order probability level 194$\bar{p}$, the equality constraint is reformulated in terms of the 195generalized reliability index: 196\begin{eqnarray} 197{\rm minimize} & \pm G({\bf u}) \nonumber \\ 198{\rm subject \ to} & \beta^*({\bf u}) = \bar{\beta^*} \label{eq:pma2_opt} 199\end{eqnarray} 200where $\bar{\beta^*}$ is computed from $\bar{p}$ using 201Eq.~\ref{eq:gen_beta} (or its CCDF complement) in the latter case of a 202prescribed second-order probability level. 203 204In the RIA case, the optimal MPP solution ${\bf u}^*$ defines the 205reliability index from $\beta = \pm \|{\bf u}^*\|_2$, which in turn 206defines the CDF/CCDF probabilities (using 207Equations~\ref{eq:p_cdf}-\ref{eq:p_ccdf} in the case of first-order 208integration). The sign of $\beta$ is defined by 209\begin{eqnarray} 210G({\bf u}^*) > G({\bf 0}): \beta_{\rm CDF} < 0, \beta_{\rm CCDF} > 0 \\ 211G({\bf u}^*) < G({\bf 0}): \beta_{\rm CDF} > 0, \beta_{\rm CCDF} < 0 212\end{eqnarray} 213\noindent where $G({\bf 0})$ is the median limit state response computed 214at the origin in u-space\footnote{It is not necessary to explicitly compute 215the median response since the sign of the inner product 216$\langle {\bf u}^*, \nabla_{\bf u} G \rangle$ 217can be used to determine the orientation of the optimal response with 218respect to the median response.} (where $\beta_{\rm CDF}$ = $\beta_{\rm CCDF}$ = 0 219and first-order $p(g \le z)$ = $p(g > z)$ = 0.5). In the PMA case, the 220sign applied to $G({\bf u})$ (equivalent to minimizing or maximizing 221$G({\bf u})$) is similarly defined by either $\bar{\beta}$ (for a specified 222reliability or first-order probability level) or from a $\bar{\beta}$ 223estimate\footnote{computed by inverting the second-order probability 224relationships described in Section~\ref{uq:reliability:local:mpp:int} at 225the current ${\bf u}^*$ iterate.} computed from $\bar{\beta^*}$ (for a 226specified generalized reliability or second-order probability level) 227\begin{eqnarray} 228\bar{\beta}_{\rm CDF} < 0, \bar{\beta}_{\rm CCDF} > 0: {\rm maximize \ } G({\bf u}) \\ 229\bar{\beta}_{\rm CDF} > 0, \bar{\beta}_{\rm CCDF} < 0: {\rm minimize \ } G({\bf u}) 230\end{eqnarray} 231where the limit state at the MPP ($G({\bf u}^*)$) defines the desired 232response level result. 233 234\subsubsection{Limit state approximations} \label{uq:reliability:local:mpp:approx} 235 236There are a variety of algorithmic variations that are available for 237use within RIA/PMA reliability analyses. First, one may select among 238several different limit state approximations that can be used to 239reduce computational expense during the MPP searches. Local, 240multipoint, and global approximations of the limit state are possible. 241\cite{Eld05} investigated local first-order limit state 242approximations, and \cite{Eld06a} investigated local second-order 243and multipoint approximations. These techniques include: 244 245\begin{enumerate} 246\item a single Taylor series per response/reliability/probability level 247in x-space centered at the uncertain variable means. The first-order 248approach is commonly known as the Advanced Mean Value (AMV) method: 249\begin{equation} 250g({\bf x}) \cong g(\mu_{\bf x}) + \nabla_x g(\mu_{\bf x})^T 251({\bf x} - \mu_{\bf x}) \label{eq:linear_x_mean} 252\end{equation} 253and the second-order approach has been named AMV$^2$: 254\begin{equation} 255g({\bf x}) \cong g(\mu_{\bf x}) + \nabla_{\bf x} g(\mu_{\bf x})^T 256({\bf x} - \mu_{\bf x}) + \frac{1}{2} ({\bf x} - \mu_{\bf x})^T 257\nabla^2_{\bf x} g(\mu_{\bf x}) ({\bf x} - \mu_{\bf x}) 258\label{eq:taylor2_x_mean} 259\end{equation} 260 261\item same as AMV/AMV$^2$, except that the Taylor series is expanded 262in u-space. The first-order option has been termed the u-space AMV 263method: 264\begin{equation} 265G({\bf u}) \cong G(\mu_{\bf u}) + \nabla_u G(\mu_{\bf u})^T 266({\bf u} - \mu_{\bf u}) \label{eq:linear_u_mean} 267\end{equation} 268where $\mu_{\bf u} = T(\mu_{\bf x})$ and is nonzero in general, and 269the second-order option has been named the u-space AMV$^2$ method: 270\begin{equation} 271G({\bf u}) \cong G(\mu_{\bf u}) + \nabla_{\bf u} G(\mu_{\bf u})^T 272({\bf u} - \mu_{\bf u}) + \frac{1}{2} ({\bf u} - \mu_{\bf u})^T 273\nabla^2_{\bf u} G(\mu_{\bf u}) ({\bf u} - \mu_{\bf u}) 274\label{eq:taylor2_u_mean} 275\end{equation} 276 277\item an initial Taylor series approximation in x-space at the uncertain 278variable means, with iterative expansion updates at each MPP estimate 279(${\bf x}^*$) until the MPP converges. The first-order option is 280commonly known as AMV+: 281\begin{equation} 282g({\bf x}) \cong g({\bf x}^*) + \nabla_x g({\bf x}^*)^T ({\bf x} - {\bf x}^*) 283\label{eq:linear_x_mpp} 284\end{equation} 285and the second-order option has been named AMV$^2$+: 286\begin{equation} 287g({\bf x}) \cong g({\bf x}^*) + \nabla_{\bf x} g({\bf x}^*)^T 288({\bf x} - {\bf x}^*) + \frac{1}{2} ({\bf x} - {\bf x}^*)^T 289\nabla^2_{\bf x} g({\bf x}^*) ({\bf x} - {\bf x}^*) \label{eq:taylor2_x_mpp} 290\end{equation} 291 292\item same as AMV+/AMV$^2$+, except that the expansions are performed in 293u-space. The first-order option has been termed the u-space AMV+ method. 294\begin{equation} 295G({\bf u}) \cong G({\bf u}^*) + \nabla_u G({\bf u}^*)^T ({\bf u} - {\bf u}^*) 296\label{eq:linear_u_mpp} 297\end{equation} 298and the second-order option has been named the u-space AMV$^2$+ method: 299\begin{equation} 300G({\bf u}) \cong G({\bf u}^*) + \nabla_{\bf u} G({\bf u}^*)^T 301({\bf u} - {\bf u}^*) + \frac{1}{2} ({\bf u} - {\bf u}^*)^T 302\nabla^2_{\bf u} G({\bf u}^*) ({\bf u} - {\bf u}^*) \label{eq:taylor2_u_mpp} 303\end{equation} 304 305\item a multipoint approximation in x-space. This approach involves a 306Taylor series approximation in intermediate variables where the powers 307used for the intermediate variables are selected to match information at 308the current and previous expansion points. Based on the 309two-point exponential approximation concept (TPEA, \cite{Fad90}), the 310two-point adaptive nonlinearity approximation (TANA-3, \cite{Xu98}) 311approximates the limit state as: 312\begin{equation} 313g({\bf x}) \cong g({\bf x}_2) + \sum_{i=1}^n 314\frac{\partial g}{\partial x_i}({\bf x}_2) \frac{x_{i,2}^{1-p_i}}{p_i} 315(x_i^{p_i} - x_{i,2}^{p_i}) + \frac{1}{2} \epsilon({\bf x}) \sum_{i=1}^n 316(x_i^{p_i} - x_{i,2}^{p_i})^2 \label{eq:tana_g} 317\end{equation} 318 319\noindent where $n$ is the number of uncertain variables and: 320\begin{eqnarray} 321p_i & = & 1 + \ln \left[ \frac{\frac{\partial g}{\partial x_i}({\bf x}_1)} 322{\frac{\partial g}{\partial x_i}({\bf x}_2)} \right] \left/ 323\ln \left[ \frac{x_{i,1}}{x_{i,2}} \right] \right. \label{eq:tana_pi_x} \\ 324\epsilon({\bf x}) & = & \frac{H}{\sum_{i=1}^n (x_i^{p_i} - x_{i,1}^{p_i})^2 + 325\sum_{i=1}^n (x_i^{p_i} - x_{i,2}^{p_i})^2} \label{eq:tana_eps_x} \\ 326H & = & 2 \left[ g({\bf x}_1) - g({\bf x}_2) - \sum_{i=1}^n 327\frac{\partial g}{\partial x_i}({\bf x}_2) \frac{x_{i,2}^{1-p_i}}{p_i} 328(x_{i,1}^{p_i} - x_{i,2}^{p_i}) \right] \label{eq:tana_H_x} 329\end{eqnarray} 330\noindent and ${\bf x}_2$ and ${\bf x}_1$ are the current and previous 331MPP estimates in x-space, respectively. Prior to the availability of 332two MPP estimates, x-space AMV+ is used. 333 334\item a multipoint approximation in u-space. The u-space TANA-3 335approximates the limit state as: 336\begin{equation} 337G({\bf u}) \cong G({\bf u}_2) + \sum_{i=1}^n 338\frac{\partial G}{\partial u_i}({\bf u}_2) \frac{u_{i,2}^{1-p_i}}{p_i} 339(u_i^{p_i} - u_{i,2}^{p_i}) + \frac{1}{2} \epsilon({\bf u}) \sum_{i=1}^n 340(u_i^{p_i} - u_{i,2}^{p_i})^2 \label{eq:tana_G} 341\end{equation} 342 343\noindent where: 344\begin{eqnarray} 345p_i & = & 1 + \ln \left[ \frac{\frac{\partial G}{\partial u_i}({\bf u}_1)} 346{\frac{\partial G}{\partial u_i}({\bf u}_2)} \right] \left/ 347\ln \left[ \frac{u_{i,1}}{u_{i,2}} \right] \right. \label{eq:tana_pi_u} \\ 348\epsilon({\bf u}) & = & \frac{H}{\sum_{i=1}^n (u_i^{p_i} - u_{i,1}^{p_i})^2 + 349\sum_{i=1}^n (u_i^{p_i} - u_{i,2}^{p_i})^2} \label{eq:tana_eps_u} \\ 350H & = & 2 \left[ G({\bf u}_1) - G({\bf u}_2) - \sum_{i=1}^n 351\frac{\partial G}{\partial u_i}({\bf u}_2) \frac{u_{i,2}^{1-p_i}}{p_i} 352(u_{i,1}^{p_i} - u_{i,2}^{p_i}) \right] \label{eq:tana_H_u} 353\end{eqnarray} 354\noindent and ${\bf u}_2$ and ${\bf u}_1$ are the current and previous 355MPP estimates in u-space, respectively. Prior to the availability of 356two MPP estimates, u-space AMV+ is used. 357 358\item the MPP search on the original response functions without the 359use of any approximations. Combining this option with first-order and 360second-order integration approaches (see next section) results in the 361traditional first-order and second-order reliability methods (FORM and 362SORM). 363\end{enumerate} 364 365The Hessian matrices in AMV$^2$ and AMV$^2$+ may be available 366analytically, estimated numerically, or approximated through 367quasi-Newton updates. The selection between x-space or u-space for 368performing approximations depends on where the approximation will be 369more accurate, since this will result in more accurate MPP estimates 370(AMV, AMV$^2$) or faster convergence (AMV+, AMV$^2$+, TANA). Since 371this relative accuracy depends on the forms of the limit state $g(x)$ 372and the transformation $T(x)$ and is therefore application dependent 373in general, Dakota supports both options. A concern with 374approximation-based iterative search methods (i.e., AMV+, AMV$^2$+ and 375TANA) is the robustness of their convergence to the MPP. It is 376possible for the MPP iterates to oscillate or even diverge. However, 377to date, this occurrence has been relatively rare, and Dakota contains 378checks that monitor for this behavior. Another concern with TANA is 379numerical safeguarding (e.g., the possibility of raising negative 380$x_i$ or $u_i$ values to nonintegral $p_i$ exponents in 381Equations~\ref{eq:tana_g}, \ref{eq:tana_eps_x}-\ref{eq:tana_G}, 382and~\ref{eq:tana_eps_u}-\ref{eq:tana_H_u}). Safeguarding involves 383offseting negative $x_i$ or $u_i$ and, for potential numerical 384difficulties with the logarithm ratios in Equations~\ref{eq:tana_pi_x} 385and~\ref{eq:tana_pi_u}, reverting to either the linear ($p_i = 1$) or 386reciprocal ($p_i = -1$) approximation based on which approximation has 387lower error in $\frac{\partial g}{\partial x_i}({\bf x}_1)$ or 388$\frac{\partial G}{\partial u_i}({\bf u}_1)$. 389 390\subsubsection{Probability integrations} \label{uq:reliability:local:mpp:int} 391 392The second algorithmic variation involves the integration approach for 393computing probabilities at the MPP, which can be selected to be 394first-order (Equations~\ref{eq:p_cdf}-\ref{eq:p_ccdf}) or second-order 395integration. Second-order integration involves applying a curvature 396correction~\cite{Bre84,Hoh88,Hon99}. Breitung applies a correction 397based on asymptotic analysis~\cite{Bre84}: 398\begin{equation} 399p = \Phi(-\beta_p) \prod_{i=1}^{n-1} \frac{1}{\sqrt{1 + \beta_p \kappa_i}} 400\label{eq:p_2nd_breit} 401\end{equation} 402where $\kappa_i$ are the principal curvatures of the limit state 403function (the eigenvalues of an orthonormal transformation of 404$\nabla^2_{\bf u} G$, taken positive for a convex limit state) and 405$\beta_p \ge 0$ (a CDF or CCDF probability correction is selected to 406obtain the correct sign for $\beta_p$). An alternate correction in 407\cite{Hoh88} is consistent in the asymptotic regime ($\beta_p \to \infty$) 408but does not collapse to first-order integration for $\beta_p = 0$: 409\begin{equation} 410p = \Phi(-\beta_p) \prod_{i=1}^{n-1} 411\frac{1}{\sqrt{1 + \psi(-\beta_p) \kappa_i}} \label{eq:p_2nd_hr} 412\end{equation} 413where $\psi() = \frac{\phi()}{\Phi()}$ and $\phi()$ is the standard 414normal density function. \cite{Hon99} applies further corrections to 415Equation~\ref{eq:p_2nd_hr} based on point concentration methods. At 416this time, all three approaches are available within the code, but the 417Hohenbichler-Rackwitz correction is used by default (switching the 418correction is a compile-time option in the source code and has not 419been exposed in the input specification). 420 421\subsubsection{Hessian approximations} \label{sec:hessian} 422 423To use a second-order Taylor series or a second-order integration when 424second-order information ($\nabla^2_{\bf x} g$, $\nabla^2_{\bf u} G$, 425and/or $\kappa$) is not directly available, one can estimate the 426missing information using finite differences or approximate it through 427use of quasi-Newton approximations. These procedures will often be 428needed to make second-order approaches practical for engineering 429applications. 430 431In the finite difference case, numerical Hessians are commonly computed 432using either first-order forward differences of gradients using 433\begin{equation} 434\nabla^2 g ({\bf x}) \cong 435\frac{\nabla g ({\bf x} + h {\bf e}_i) - \nabla g ({\bf x})}{h} 436\end{equation} 437to estimate the $i^{th}$ Hessian column when gradients are analytically 438available, or second-order differences of function values using 439\begin{equation} 440\begin{array}{l} 441\nabla^2 g ({\bf x}) \cong \frac{g({\bf x} + h {\bf e}_i + h {\bf e}_j) - 442g({\bf x} + h {\bf e}_i - h {\bf e}_j) - 443g({\bf x} - h {\bf e}_i + h {\bf e}_j) + 444g({\bf x} - h {\bf e}_i - h {\bf e}_j)}{4h^2} 445\end{array} 446\end{equation} 447to estimate the $ij^{th}$ Hessian term when gradients are not directly 448available. This approach has the advantage of locally-accurate 449Hessians for each point of interest (which can lead to quadratic 450convergence rates in discrete Newton methods), but has the 451disadvantage that numerically estimating each of the matrix terms can 452be expensive. 453 454Quasi-Newton approximations, on the other hand, do not reevaluate all 455of the second-order information for every point of interest. Rather, 456they accumulate approximate curvature information over time using 457secant updates. Since they utilize the existing gradient evaluations, 458they do not require any additional function evaluations for evaluating 459the Hessian terms. The quasi-Newton approximations of interest 460include the Broyden-Fletcher-Goldfarb-Shanno (BFGS) update 461\begin{equation} 462{\bf B}_{k+1} = {\bf B}_{k} - \frac{{\bf B}_k {\bf s}_k {\bf s}_k^T {\bf B}_k} 463{{\bf s}_k^T {\bf B}_k {\bf s}_k} + 464\frac{{\bf y}_k {\bf y}_k^T}{{\bf y}_k^T {\bf s}_k} \label{eq:bfgs} 465\end{equation} 466which yields a sequence of symmetric positive definite Hessian 467approximations, and the Symmetric Rank 1 (SR1) update 468\begin{equation} 469{\bf B}_{k+1} = {\bf B}_{k} + 470\frac{({\bf y}_k - {\bf B}_k {\bf s}_k)({\bf y}_k - {\bf B}_k {\bf s}_k)^T} 471{({\bf y}_k - {\bf B}_k {\bf s}_k)^T {\bf s}_k} \label{eq:sr1} 472\end{equation} 473which yields a sequence of symmetric, potentially indefinite, Hessian 474approximations. ${\bf B}_k$ is the $k^{th}$ approximation to 475the Hessian $\nabla^2 g$, ${\bf s}_k = {\bf x}_{k+1} - {\bf x}_k$ is 476the step and ${\bf y}_k = \nabla g_{k+1} - \nabla g_k$ is the 477corresponding yield in the gradients. The selection of BFGS versus SR1 478involves the importance of retaining positive definiteness in the 479Hessian approximations; if the procedure does not require it, then 480the SR1 update can be more accurate if the true Hessian is not positive 481definite. Initial scalings for ${\bf B}_0$ and numerical safeguarding 482techniques (damped BFGS, update skipping) are described in \cite{Eld06a}. 483 484 485\subsubsection{Optimization algorithms} 486 487The next algorithmic variation involves the optimization algorithm 488selection for solving Eqs.~\ref{eq:ria_opt} and~\ref{eq:pma_opt}. The 489Hasofer-Lind Rackwitz-Fissler (HL-RF) algorithm~\cite{Hal00} is a 490classical approach that has been broadly applied. It is a 491Newton-based approach lacking line search/trust region globalization, 492and is generally regarded as computationally efficient but 493occasionally unreliable. Dakota takes the approach of employing 494robust, general-purpose optimization algorithms with provable 495convergence properties. In particular, we employ the sequential 496quadratic programming (SQP) and nonlinear interior-point (NIP) 497optimization algorithms from the NPSOL~\cite{Gil86} and 498OPT++~\cite{MeOlHoWi07} libraries, respectively. 499 500 501\subsubsection{Warm Starting of MPP Searches} 502 503The final algorithmic variation for local reliability methods involves 504the use of warm starting approaches for improving computational 505efficiency. \cite{Eld05} describes the acceleration of MPP 506searches through warm starting with approximate iteration increment, 507with $z/p/\beta$ level increment, and with design variable increment. 508Warm started data includes the expansion point and associated response 509values and the MPP optimizer initial guess. Projections are used when 510an increment in $z/p/\beta$ level or design variables occurs. Warm 511starts were consistently effective in \cite{Eld05}, with greater 512effectiveness for smaller parameter changes, and are used by default 513in Dakota. %for all computational experiments presented in this paper. 514 515 516\section{Global Reliability Methods}\label{uq:reliability:global} 517 518Local reliability methods, while computationally efficient, have 519well-known failure mechanisms. When confronted with a limit state 520function that is nonsmooth, local gradient-based optimizers may stall 521due to gradient inaccuracy and fail to converge to an MPP. Moreover, 522if the limit state is multimodal (multiple MPPs), then a 523gradient-based local method can, at best, locate only one local MPP 524solution. Finally, a linear (Eqs.~\ref{eq:p_cdf}--\ref{eq:p_ccdf}) or 525parabolic (Eqs.~\ref{eq:p_2nd_breit}--\ref{eq:p_2nd_hr}) approximation 526to the limit state at this MPP may fail to adequately capture the 527contour of a highly nonlinear limit state. %For these reasons, 528%efficient global reliability analysis (EGRA) is investigated 529%in~\cite{bichon_egra_sdm}. 530 531%Global reliability methods include the efficient global reliability 532%analysis (EGRA) method. Analytical methods of reliability analysis solve a 533%local optimization problem to locate the most probable point of failure (MPP), 534%and then quantify the reliability based on its location and an approximation 535%to the shape of the limit state at this point. Typically, gradient-based 536%solvers are used to solve this optimization problem, which may fail to 537%converge for nonsmooth response functions with unreliable gradients or 538%may converge to only one of several solutions for response functions that 539%possess multiple local optima. In addition to these MPP convergence issues, 540%the evaluated probabilites can be adversely affected by limit state 541%approximations that may be inaccurate. Analysts are then forced 542%to revert to sampling methods, which do not rely on MPP convergence or 543%simplifying approximations to the true shape of the limit state. 544%However, employing such methods is impractical when evaluation of the 545%response function is expensive. 546 547A reliability analysis method that is both efficient when applied to 548expensive response functions and accurate for a response function of 549any arbitrary shape is needed. This section develops such a method 550based on efficient global optimization~\cite{Jon98} (EGO) to the 551search for multiple points on or near the limit state throughout the 552random variable space. By locating multiple points on the limit state, 553more complex limit states can be accurately modeled, resulting in a 554more accurate assessment of the reliability. It should be emphasized 555here that these multiple points exist on a single limit state. Because 556of its roots in efficient global optimization, this method of 557reliability analysis is called efficient global reliability analysis 558(EGRA)~\cite{Bichon2007}. The following two subsections describe two 559capabilities that are incorporated into the EGRA algorithm: importance 560sampling and EGO. 561 562\subsection{Importance Sampling}\label{uq:reliability:global:ais} 563 564An alternative to MPP search methods is to directly 565perform the probability integration numerically by sampling the 566response function. 567Sampling methods do not rely on a simplifying approximation to the shape 568of the limit state, so they can be more accurate than FORM and SORM, but they 569can also be prohibitively expensive because they generally require a large 570number of response function evaluations. 571Importance sampling methods reduce this expense by focusing the samples in 572the important regions of the uncertain space. 573They do this by centering the sampling density function at the MPP rather 574than at the mean. 575This ensures the samples will lie the region of interest, 576thus increasing the efficiency of the sampling method. 577Adaptive importance sampling (AIS) further improves the efficiency by 578adaptively updating the sampling density function. 579Multimodal adaptive importance sampling~\cite{Dey98,Zou02} is a 580variation of AIS that allows for the use of multiple sampling densities 581making it better suited for cases where multiple sections of the limit state 582are highly probable. 583 584Note that importance sampling methods require that the location of at 585least one MPP be known because it is used to center the initial sampling 586density. However, current gradient-based, local search methods used in 587MPP search may fail to converge or may converge to poor solutions for 588highly nonlinear problems, possibly making these methods inapplicable. 589As the next section describes, EGO is a global optimization method that 590does not depend on the availability of accurate gradient information, making 591convergence more reliable for nonsmooth response functions. 592Moreover, EGO has the ability to locate multiple failure points, 593which would provide multiple starting points and thus a good 594multimodal sampling density for the initial steps of multimodal AIS. 595The resulting Gaussian process model is accurate in the 596vicinity of the limit state, thereby providing an inexpensive surrogate that 597can be used to provide response function samples. 598As will be seen, using EGO to locate multiple points along the limit state, 599and then using the resulting Gaussian process model to provide function 600evaluations in multimodal AIS for the probability integration, 601results in an accurate and efficient reliability analysis tool. 602 603\subsection{Efficient Global Optimization}\label{uq:reliability:global:ego} 604 605Chapter \ref{uq:ego} is now rewritten to support EGO/Bayesian optimization theory. 606 607\subsubsection{Expected Feasibility Function}\label{uq:reliability:global:ego:eff} 608 609The expected improvement function provides an indication of how much the true 610value of the response at a point can be expected to be less than the current 611best solution. 612It therefore makes little sense to apply this to the forward reliability problem 613where the goal is not to minimize the response, but rather to find where it is 614equal to a specified threshold value. 615The expected feasibility function (EFF) is introduced here to provide an 616indication of how well the true value of the response is expected to satisfy 617the equality constraint $G({\bf u})\!=\!\bar{z}$. 618Inspired by the contour estimation work in~\cite{Ran08}, this 619expectation can be calculated in a similar fashion as Eq.~\ref{eq:eif_int} 620by integrating over a region in the immediate vicinity of the threshold value 621$\bar{z}\pm\epsilon$: 622\begin{equation} 623EF\bigl( \hat{G}({\bf u}) \bigr) = 624 \int_{z-\epsilon}^{z+\epsilon} 625 \bigl[ \epsilon - | \bar{z}-G | \bigr] \, \hat{G}({\bf u}) \; dG 626\end{equation} 627\noindent where $G$ denotes a realization of the distribution $\hat{G}$, as 628before. 629Allowing $z^+$ and $z^-$ to denote $\bar{z}\pm\epsilon$, respectively, this 630integral can be expressed analytically as: 631\begin{align} 632EF\bigl( \hat{G}({\bf u}) \bigr) &= \left( \mu_G - \bar{z} \right) 633 \left[ 2 \, \Phi\left( \frac{\bar{z} - \mu_G}{\sigma_G} \right) - 634 \Phi\left( \frac{ z^- - \mu_G}{\sigma_G} \right) - 635 \Phi\left( \frac{ z^+ - \mu_G}{\sigma_G} \right) 636 \right] \notag \\ & \ \ \ \ \ \ \ \ - 637 \sigma_G \left[ 2 \, \phi\left( \frac{\bar{z} - \mu_G}{\sigma_G} \right) \, - 638 \phi\left( \frac{ z^- - \mu_G}{\sigma_G} \right) \, - 639 \phi\left( \frac{ z^+ - \mu_G}{\sigma_G} \right) 640 \right] \notag \\ & \ \ \ \ \ \ \ \ + \ \ \, 641 \epsilon \left[ \Phi\left( \frac{ z^+ - \mu_G}{\sigma_G} \right) - 642 \Phi\left( \frac{ z^- - \mu_G}{\sigma_G} \right) 643 \right] \label{eq:eff} 644\end{align} 645where $\epsilon$ is proportional to the standard deviation of the GP 646predictor ($\epsilon\propto\sigma_G$). 647In this case, $z^-$, $z^+$, $\mu_G$, $\sigma_G$, and $\epsilon$ are all functions of the 648location ${\bf u}$, while $\bar{z}$ is a constant. 649Note that the EFF provides the same balance between exploration and 650exploitation as is captured in the EIF. 651Points where the expected value is close to the threshold 652($\mu_G\!\approx\!\bar{z}$) and points with a large uncertainty in the 653prediction will have large expected feasibility values. 654