Optimization Under Uncertainty (OUU)

Reliability-Based Design Optimization (RBDO)

Reliability-based design optimization (RBDO) methods are used to perform design optimization accounting for reliability metrics. The reliability analysis capabilities described in Section Local Reliability Methods provide a substantial foundation for exploring a variety of gradient-based RBDO formulations. [EAP+07] investigated bi-level, fully-analytic bi-level, and first-order sequential RBDO approaches employing underlying first-order reliability assessments. [EB06] investigated fully-analytic bi-level and second-order sequential RBDO approaches employing underlying second-order reliability assessments. These methods are overviewed in the following sections.

Bi-level RBDO

The simplest and most direct RBDO approach is the bi-level approach in which a full reliability analysis is performed for every optimization function evaluation. This involves a nesting of two distinct levels of optimization within each other, one at the design level and one at the MPP search level.

Since an RBDO problem will typically specify both the $\overline{z}$ level and the $\overline{p}/\overline{\beta }$ level, one can use either the RIA or the PMA formulation for the UQ portion and then constrain the result in the design optimization portion. In particular, RIA reliability analysis maps $\overline{z}$ to $p/\beta$, so RIA RBDO constrains $p/\beta$:

(265)$$\begin{array}{r}\begin{array}{rl}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}& f\\ \mathrm{s}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{t}\mathrm{o}& \beta \ge \overline{\beta }\\ \mathrm{o}\mathrm{r}& p\le \overline{p}\end{array}\end{array}$$

And PMA reliability analysis maps $\overline{p}/\overline{\beta }$ to $z$, so PMA RBDO constrains $z$:

(266)$$\begin{array}{r}\begin{array}{rl}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}& f\\ \mathrm{s}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{t}\mathrm{o}& z\ge \overline{z}\end{array}\end{array}$$

where $z\ge \overline{z}$ is used as the RBDO constraint for a cumulative failure probability (failure defined as $z\le \overline{z}$) but $z\le \overline{z}$ would be used as the RBDO constraint for a complementary cumulative failure probability (failure defined as $z\ge \overline{z}$). It is worth noting that Dakota is not limited to these types of inequality-constrained RBDO formulations; rather, they are convenient examples. Dakota supports general optimization under uncertainty mappings [EGWojtkiewiczJrT02] which allow flexible use of statistics within multiple objectives, inequality constraints, and equality constraints.

An important performance enhancement for bi-level methods is the use of sensitivity analysis to analytically compute the design gradients of probability, reliability, and response levels. When design variables are separate from the uncertain variables (i.e., they are not distribution parameters), then the following first-order expressions may be used [AM04, HR86, KC92]:

(267)$$\begin{array}{r}\begin{array}{rlr}{\mathrm{\nabla }}_{\mathbf{d}}z& =& {\mathrm{\nabla }}_{\mathbf{d}}g\end{array}\end{array}$$

(268)$$\begin{array}{r}\begin{array}{rlr}{\mathrm{\nabla }}_{\mathbf{d}}{\beta }_{cdf}& =& \frac{1}{\parallel {\mathrm{\nabla }}_{\mathbf{u}}G\parallel }{\mathrm{\nabla }}_{\mathbf{d}}g\end{array}\end{array}$$

(269)$$\begin{array}{rlr}{\mathrm{\nabla }}_{\mathbf{d}}{p}_{cdf}& =& -\varphi (-{\beta }_{cdf}){\mathrm{\nabla }}_{\mathbf{d}}{\beta }_{cdf}\end{array}$$

where it is evident from Eqs. (57) that ${\mathrm{\nabla }}_{\mathbf{d}}{\beta }_{ccdf}=-{\mathrm{\nabla }}_{\mathbf{d}}{\beta }_{cdf}$ and ${\mathrm{\nabla }}_{\mathbf{d}}{p}_{ccdf}=-{\mathrm{\nabla }}_{\mathbf{d}}{p}_{cdf}$. In the case of second-order integrations, Eq. (267) must be expanded to include the curvature correction. For Breitung’s correction (Eq. (73) ),

(270)$${\mathrm{\nabla }}_{\mathbf{d}}{p}_{cdf}=[\mathrm{\Phi }(-{\beta }_p)\sum _{i=1}^{n-1}\left(\frac{-{\kappa }_i}{2(1+{\beta }_p{\kappa }_i{)}^{\frac{3}{2}}}\prod _{\stackrel{j=1}{j\ne i}}^{n-1}\frac{1}{\sqrt{1+{\beta }_p{\kappa }_j}}\right)-\varphi (-{\beta }_p)\prod _{i=1}^{n-1}\frac{1}{\sqrt{1+{\beta }_p{\kappa }_i}}]{\mathrm{\nabla }}_{\mathbf{d}}{\beta }_{cdf}$$

where ${\mathrm{\nabla }}_{\mathbf{d}}{\kappa }_i$ has been neglected and ${\beta }_p\ge 0$ (see Section Local Reliability Integration. Other approaches assume the curvature correction is nearly independent of the design variables [Rac02], which is equivalent to neglecting the first term in Eq. (270) .

To capture second-order probability estimates within an RIA RBDO formulation using well-behaved $\beta$ constraints, a generalized reliability index can be introduced where, similar to Eq. eq:beta_cdf ,

(271)$${\beta }_{cdf}^{\ast }=-{\mathrm{\Phi }}^{-1}(p_{cdf})$$

for second-order $p_{cdf}$. This reliability index is no longer equivalent to the magnitude of $\mathbf{u}$, but rather is a convenience metric for capturing the effect of more accurate probability estimates. The corresponding generalized reliability index sensitivity, similar to Eq. (267), is

(272)$${\mathrm{\nabla }}_{\mathbf{d}}{\beta }_{cdf}^{\ast }=-\frac{1}{\varphi (-{\beta }_{cdf}^{\ast })}{\mathrm{\nabla }}_{\mathbf{d}}{p}_{cdf}$$

where ${\mathrm{\nabla }}_{\mathbf{d}}{p}_{cdf}$ is defined from Eq. (270). Even when ${\mathrm{\nabla }}_{\mathbf{d}}g$ is estimated numerically, Eqs. (267) - (272) can be used to avoid numerical differencing across full reliability analyses.

When the design variables are distribution parameters of the uncertain variables, ${\mathrm{\nabla }}_{\mathbf{d}}g$ is expanded with the chain rule and Eqs.  (267) and  (268) (2) become

(273)$$\begin{array}{r}\begin{array}{rlr}{\mathrm{\nabla }}_{\mathbf{d}}z& =& {\mathrm{\nabla }}_{\mathbf{d}}\mathbf{x}{\mathrm{\nabla }}_{\mathbf{x}}g\end{array}\end{array}$$

(274)$$\begin{array}{rlr}{\mathrm{\nabla }}_{\mathbf{d}}{\beta }_{cdf}& =& \frac{1}{\parallel {\mathrm{\nabla }}_{\mathbf{u}}G\parallel }{\mathrm{\nabla }}_{\mathbf{d}}\mathbf{x}{\mathrm{\nabla }}_{\mathbf{x}}g\end{array}$$

where the design Jacobian of the transformation (${\mathrm{\nabla }}_{\mathbf{d}}\mathbf{x}$) may be obtained analytically for uncorrelated $\mathbf{x}$ or semi-analytically for correlated $\mathbf{x}$ (${\mathrm{\nabla }}_{\mathbf{d}}\mathbf{L}$ is evaluated numerically) by differentiating Eqs. eq:trans_zx and  eq:trans_zu _ with respect to the distribution parameters. Eqs. (267) - (272) remain the same as before. For this design variable case, all required information for the sensitivities is available from the MPP search.

Since Eqs. (267) - (274) are derived using the Karush-Kuhn-Tucker optimality conditions for a converged MPP, they are appropriate for RBDO using AMV+, AMV${}^2$+, TANA, FORM, and SORM, but not for RBDO using MVFOSM, MVSOSM, AMV, or AMV${}^2$.

Sequential/Surrogate-based RBDO

An alternative RBDO approach is the sequential approach, in which additional efficiency is sought through breaking the nested relationship of the MPP and design searches. The general concept is to iterate between optimization and uncertainty quantification, updating the optimization goals based on the most recent probabilistic assessment results. This update may be based on safety factors [WSSC01] or other approximations [DC04].

A particularly effective approach for updating the optimization goals is to use the $p/\beta /z$ sensitivity analysis of Eqs.  (267) - (274) in combination with local surrogate models [ZMR04]. In [EAP+07] and [EB06], first-order and second-order Taylor series approximations were employed within a trust-region model management framework [GE00] in order to adaptively manage the extent of the approximations and ensure convergence of the RBDO process. Surrogate models were used for both the objective function and the constraints, although the use of constraint surrogates alone is sufficient to remove the nesting.

In particular, RIA trust-region surrogate-based RBDO employs surrogate models of $f$ and $p/\beta$ within a trust region ${\mathrm{\Delta }}^k$ centered at ${\mathbf{d}}_c$. For first-order surrogates:

(275)$$\begin{array}{r}\begin{array}{rl}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}& f({\mathbf{d}}_c)+{\mathrm{\nabla }}_{d}f({\mathbf{d}}_c{)}^T(\mathbf{d}-{\mathbf{d}}_c)\\ \mathrm{s}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{t}\mathrm{o}& \beta ({\mathbf{d}}_c)+{\mathrm{\nabla }}_d\beta ({\mathbf{d}}_c{)}^T(\mathbf{d}-{\mathbf{d}}_c)\ge \overline{\beta }\\ \mathrm{o}\mathrm{r}& p({\mathbf{d}}_c)+{\mathrm{\nabla }}_{d}p({\mathbf{d}}_c{)}^T(\mathbf{d}-{\mathbf{d}}_c)\le \overline{p}\\ & {\parallel \mathbf{d}-{\mathbf{d}}_c\parallel }_{\mathrm{\infty }}\le {\mathrm{\Delta }}^k\end{array}\end{array}$$

and for second-order surrogates:

(276)$$\begin{array}{r}\begin{array}{rl}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}& f({\mathbf{d}}_c)+{\mathrm{\nabla }}_{\mathbf{d}}f({\mathbf{d}}_c{)}^T(\mathbf{d}-{\mathbf{d}}_c)+\frac{1}{2}(\mathbf{d}-{\mathbf{d}}_c{)}^T{\mathrm{\nabla }}_{\mathbf{d}}^{2}f({\mathbf{d}}_c)(\mathbf{d}-{\mathbf{d}}_c)\\ \mathrm{s}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{t}\mathrm{o}& \beta ({\mathbf{d}}_c)+{\mathrm{\nabla }}_{\mathbf{d}}\beta ({\mathbf{d}}_c{)}^T(\mathbf{d}-{\mathbf{d}}_c)+\frac{1}{2}(\mathbf{d}-{\mathbf{d}}_c{)}^T{\mathrm{\nabla }}_{\mathbf{d}}^2\beta ({\mathbf{d}}_c)(\mathbf{d}-{\mathbf{d}}_c)\ge \overline{\beta }\\ \mathrm{o}\mathrm{r}& p({\mathbf{d}}_c)+{\mathrm{\nabla }}_{\mathbf{d}}p({\mathbf{d}}_c{)}^T(\mathbf{d}-{\mathbf{d}}_c)+\frac{1}{2}(\mathbf{d}-{\mathbf{d}}_c{)}^T{\mathrm{\nabla }}_{\mathbf{d}}^{2}p({\mathbf{d}}_c)(\mathbf{d}-{\mathbf{d}}_c)\le \overline{p}\\ & {\parallel \mathbf{d}-{\mathbf{d}}_c\parallel }_{\mathrm{\infty }}\le {\mathrm{\Delta }}^k\end{array}\end{array}$$

For PMA trust-region surrogate-based RBDO, surrogate models of $f$ and $z$ are employed within a trust region ${\mathrm{\Delta }}^k$ centered at ${\mathbf{d}}_c$. For first-order surrogates:

(277)$$\begin{array}{r}\begin{array}{rl}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}& f({\mathbf{d}}_c)+{\mathrm{\nabla }}_{d}f({\mathbf{d}}_c{)}^T(\mathbf{d}-{\mathbf{d}}_c)\\ \mathrm{s}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{t}\mathrm{o}& z({\mathbf{d}}_c)+{\mathrm{\nabla }}_{d}z({\mathbf{d}}_c{)}^T(\mathbf{d}-{\mathbf{d}}_c)\ge \overline{z}\\ & {\parallel \mathbf{d}-{\mathbf{d}}_c\parallel }_{\mathrm{\infty }}\le {\mathrm{\Delta }}^k\end{array}\end{array}$$

and for second-order surrogates:

(278)$$\begin{array}{r}\begin{array}{rl}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}& f({\mathbf{d}}_c)+{\mathrm{\nabla }}_{\mathbf{d}}f({\mathbf{d}}_c{)}^T(\mathbf{d}-{\mathbf{d}}_c)+\frac{1}{2}(\mathbf{d}-{\mathbf{d}}_c{)}^T{\mathrm{\nabla }}_{\mathbf{d}}^{2}f({\mathbf{d}}_c)(\mathbf{d}-{\mathbf{d}}_c)\\ \mathrm{s}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{t}\mathrm{o}& z({\mathbf{d}}_c)+{\mathrm{\nabla }}_{\mathbf{d}}z({\mathbf{d}}_c{)}^T(\mathbf{d}-{\mathbf{d}}_c)+\frac{1}{2}(\mathbf{d}-{\mathbf{d}}_c{)}^T{\mathrm{\nabla }}_{\mathbf{d}}^{2}z({\mathbf{d}}_c)(\mathbf{d}-{\mathbf{d}}_c)\ge \overline{z}\\ & {\parallel \mathbf{d}-{\mathbf{d}}_c\parallel }_{\mathrm{\infty }}\le {\mathrm{\Delta }}^k\end{array}\end{array}$$

where the sense of the $z$ constraint may vary as described previously. The second-order information in Eqs. (276) and (278) will typically be approximated with quasi-Newton updates.

Stochastic Expansion-Based Design Optimization (SEBDO)

Stochastic Sensitivity Analysis

Section Expansion RVSA describes sensitivity analysis of the polynomial chaos expansion with respect to the expansion variables. Here we extend this analysis to include sensitivity analysis of probabilistic moments with respect to nonprobabilistic (i.e., design or epistemic uncertain) variables.

Local sensitivity analysis: first-order probabilistic expansions

With the introduction of nonprobabilistic variables $\mathit{s}$ (for example, design variables or epistemic uncertain variables), a polynomial chaos expansion only over the probabilistic variables $\mathit{\xi }$ has the functional relationship:

(279)$$R(\mathit{\xi },\mathit{s})\cong \sum _{j=0}^P{\alpha }_j(\mathit{s}){\mathrm{\Psi }}_j(\mathit{\xi })$$

For computing sensitivities of response mean and variance, the $ij$ indices may be dropped from Eqs. eq:mean_pce and eq:covar_pce , simplifying to

(280)$$\mu (s)={\alpha }_0(s),{\sigma }^2(s)=\sum _{k=1}^P{\alpha }_k^2(s)⟨{\mathrm{\Psi }}_k^2⟩$$

Sensitivities of Eq. (280) with respect to the nonprobabilistic variables are as follows, where independence of $\mathit{s}$ and $\mathit{\xi }$ is assumed:

(281)$$\begin{array}{r}\begin{array}{rlr}\frac{d\mu }{ds}& =& \frac{d{\alpha }_0}{ds}=⟨\frac{dR}{ds}⟩\end{array}\end{array}$$

(282)$$\begin{array}{rlr}\frac{d{\sigma }^2}{ds}& =& \sum _{k=1}^P⟨{\mathrm{\Psi }}_k^2⟩\frac{d{\alpha }_k^2}{ds}=2\sum _{k=1}^P{\alpha }_k⟨\frac{dR}{ds},{\mathrm{\Psi }}_k⟩\end{array}$$

where

(283)$$\frac{d{\alpha }_k}{ds}=\frac{⟨\frac{dR}{ds},{\mathrm{\Psi }}_k⟩}{⟨{\mathrm{\Psi }}_k^2⟩}$$

has been used. Due to independence, the coefficients calculated in Eq. (283) may be interpreted as either the derivatives of the expectations or the expectations of the derivatives, or more precisely, the nonprobabilistic sensitivities of the chaos coefficients for the response expansion or the chaos coefficients of an expansion for the nonprobabilistic sensitivities of the response. The evaluation of integrals involving $\frac{dR}{ds}$ extends the data requirements for the PCE approach to include response sensitivities at each of the sampled points. The resulting expansions are valid only for a particular set of nonprobabilistic variables and must be recalculated each time the nonprobabilistic variables are modified.

Similarly for stochastic collocation,

(284)$$R(\mathit{\xi },\mathit{s})\cong \sum _{k=1}^{N_p}{r}_k(\mathit{s}){\mathit{L}}_k(\mathit{\xi })$$

leads to

(285)$$\begin{array}{r}\begin{array}{rlr}\mu (s)& =& \sum _{k=1}^{N_p}{r}_k(s)w_k,{\sigma }^2(s)=\sum _{k=1}^{N_p}{r}_k^2(s)w_k-{\mu }^2(s)\end{array}\end{array}$$

(286)$$\begin{array}{r}\begin{array}{rlr}\frac{d\mu }{ds}& =& \sum _{k=1}^{N_p}{w}_k\frac{d{r}_k}{ds}\end{array}\end{array}$$

(287)$$\begin{array}{rlr}\frac{d{\sigma }^2}{ds}& =& \sum _{k=1}^{N_p}2w_k{r}_k\frac{d{r}_k}{ds}-2\mu \frac{d\mu }{ds}=\sum _{k=1}^{N_p}2w_k(r_k-\mu )\frac{d{r}_k}{ds}\end{array}$$

Local sensitivity analysis: zeroth-order combined expansions

Alternatively, a stochastic expansion can be formed over both $\mathit{\xi }$ and $\mathit{s}$. Assuming a bounded design domain ${\mathit{s}}_L\le \mathit{s}\le {\mathit{s}}_U$ (with no implied probability content), a Legendre chaos basis would be appropriate for each of the dimensions in $\mathit{s}$ within a polynomial chaos expansion.

(288)$$R(\mathit{\xi },\mathit{s})\cong \sum _{j=0}^P{\alpha }_j{\mathrm{\Psi }}_j(\mathit{\xi },\mathit{s})$$

In this case, design sensitivities for the mean and variance do not require response sensitivity data, but this comes at the cost of forming the PCE over additional dimensions. For this combined variable expansion, the mean and variance are evaluated by performing the expectations over only the probabilistic expansion variables, which eliminates the polynomial dependence on $\mathit{\xi }$, leaving behind the desired polynomial dependence of the moments on $\mathit{s}$:

(289)$$\begin{array}{r}\begin{array}{rlr}{\mu }_R(\mathit{s})& =& \sum _{j=0}^P{\alpha }_j⟨{\mathrm{\Psi }}_j(\mathit{\xi },\mathit{s}){⟩}_{\mathit{\xi }}\end{array}\end{array}$$

(290)$$\begin{array}{rlr}{\sigma }_R^2(\mathit{s})& =& \sum _{j=0}^P\sum _{k=0}^P{\alpha }_j{\alpha }_k⟨{\mathrm{\Psi }}_j(\mathit{\xi },\mathit{s}){\mathrm{\Psi }}_k(\mathit{\xi },\mathit{s}){⟩}_{\mathit{\xi }}-{\mu }_R^2(\mathit{s})\end{array}$$

The remaining polynomials may then be differentiated with respect to $\mathit{s}$. In this approach, the combined PCE is valid for the full design variable range (${\mathit{s}}_L\le \mathit{s}\le {\mathit{s}}_U$) and does not need to be updated for each change in nonprobabilistic variables, although adaptive localization techniques (i.e., trust region model management approaches) can be employed when improved local accuracy of the sensitivities is required.

Similarly for stochastic collocation,

(291)$$R(\mathit{\xi },\mathit{s})\cong \sum _{j=1}^{N_p}{r}_j{\mathit{L}}_j(\mathit{\xi },\mathit{s})$$

leads to

(292)$$\begin{array}{rlr}{\mu }_R(\mathit{s})& =& \sum _{j=1}^{N_p}{r}_j⟨{\mathit{L}}_j(\mathit{\xi },\mathit{s}){⟩}_{\mathit{\xi }}\end{array}$$

(293)$$\begin{array}{rlr}{\sigma }_R^2(\mathit{s})& =& \sum _{j=1}^{N_p}\sum _{k=1}^{N_p}{r}_j{r}_k⟨{\mathit{L}}_j(\mathit{\xi },\mathit{s}){\mathit{L}}_k(\mathit{\xi },\mathit{s}){⟩}_{\mathit{\xi }}-{\mu }_R^2(\mathit{s})\end{array}$$

where the remaining polynomials not eliminated by the expectation over $\mathit{\xi }$ are again differentiated with respect to $\mathit{s}$.

Inputs and outputs

There are two types of nonprobabilistic variables for which sensitivities must be calculated: “augmented,” where the nonprobabilistic variables are separate from and augment the probabilistic variables, and “inserted,” where the nonprobabilistic variables define distribution parameters for the probabilistic variables. Any inserted nonprobabilistic variable sensitivities must be handled using Eqs. (281) and Eqs. (285) where $\frac{dR}{ds}$ is calculated as $\frac{dR}{dx}\frac{dx}{ds}$ and $\frac{dx}{ds}$ is the Jacobian of the variable transformation $\mathbf{x}=T^{-1}(\mathit{\xi })$ with respect to the inserted nonprobabilistic variables. In addition, parameterized polynomials (generalized Gauss-Laguerre, Jacobi, and numerically-generated polynomials) may introduce a $\frac{d\mathrm{\Psi }}{ds}$ or $\frac{d\mathit{L}}{ds}$ dependence for inserted $s$ that will introduce additional terms in the sensitivity expressions.

While moment sensitivities directly enable robust design optimization and interval estimation formulations which seek to control or bound response variance, control or bounding of reliability requires sensitivities of tail statistics. In this work, the sensitivity of simple moment-based approximations to cumulative distribution function (CDF) and complementary cumulative distribution function (CCDF) mappings (Eqs. eq:mv_ria – eq:mv_pma ) are employed for this purpose, such that it is straightforward to form approximate design sensitivities of reliability index $\beta$ (forward reliability mapping $\overline{z}\to \beta$) or response level $z$ (inverse reliability mapping $\overline{\beta }\to z$) from the moment design sensitivities and the specified levels $\overline{\beta }$ or $\overline{z}$.

Optimization Formulations

Given the capability to compute analytic statistics of the response along with design sensitivities of these statistics, Dakota supports bi-level, sequential, and multifidelity approaches for optimization under uncertainty (OUU). The latter two approaches apply surrogate modeling approaches (data fits and multifidelity modeling) to the uncertainty analysis and then apply trust region model management to the optimization process.

Bi-level SEBDO

The simplest and most direct approach is to employ these analytic statistics and their design derivatives from Section SEBDO SSA directly within an optimization loop. This approach is known as bi-level OUU, since there is an inner level uncertainty analysis nested within an outer level optimization.

Consider the common reliability-based design example of a deterministic objective function with a reliability constraint:

(294)$$\begin{array}{r}\begin{array}{rl}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}& f\\ \mathrm{s}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{t}\mathrm{o}& \beta \ge \overline{\beta }\end{array}\end{array}$$

where $\beta$ is computed relative to a prescribed threshold response value $\overline{z}$ (e.g., a failure threshold) and is constrained by a prescribed reliability level $\overline{\beta }$ (minimum allowable reliability in the design), and is either a CDF or CCDF index depending on the definition of the failure domain (i.e., defined from whether the associated failure probability is cumulative, $p(g\le \overline{z})$, or complementary cumulative, $p(g>\overline{z})$).

Another common example is robust design in which the constraint enforcing a reliability lower-bound has been replaced with a constraint enforcing a variance upper-bound ${\overline{\sigma }}^2$ (maximum allowable variance in the design):

(295)$$\begin{array}{r}\begin{array}{rl}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}& f\\ \mathrm{s}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{t}\mathrm{o}& {\sigma }^2\le {\overline{\sigma }}^2\end{array}\end{array}$$

Solving these problems using a bi-level approach involves computing $\beta$ and $\frac{d\beta }{d\mathit{s}}$ for Eq. (294) or ${\sigma }^2$ and $\frac{d{\sigma }^2}{d\mathit{s}}$ for Eq. (295) for each set of design variables $\mathit{s}$ passed from the optimizer. This approach is supported for both probabilistic and combined expansions using PCE and SC.

Sequential/Surrogate-Based SEBDO

An alternative OUU approach is the sequential approach, in which additional efficiency is sought through breaking the nested relationship of the UQ and optimization loops. The general concept is to iterate between optimization and uncertainty quantification, updating the optimization goals based on the most recent uncertainty assessment results. This approach is common with the reliability methods community, for which the updating strategy may be based on safety factors [WSSC01] or other approximations [DC04].

A particularly effective approach for updating the optimization goals is to use data fit surrogate models, and in particular, local Taylor series models allow direct insertion of stochastic sensitivity analysis capabilities. In Ref. [EAP+07], first-order Taylor series approximations were explored, and in Ref. [EB06], second-order Taylor series approximations are investigated. In both cases, a trust-region model management framework [ED06] is used to adaptively manage the extent of the approximations and ensure convergence of the OUU process. Surrogate models are used for both the objective and the constraint functions, although the use of surrogates is only required for the functions containing statistical results; deterministic functions may remain explicit is desired.

In particular, trust-region surrogate-based optimization for reliability-based design employs surrogate models of $f$ and $\beta$ within a trust region ${\mathrm{\Delta }}^k$ centered at ${\mathbf{s}}_c$:

(296)$$\begin{array}{r}\begin{array}{rl}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}& f({\mathbf{s}}_c)+{\mathrm{\nabla }}_{s}f({\mathbf{s}}_c{)}^T(\mathbf{s}-{\mathbf{s}}_c)\\ \mathrm{s}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{t}\mathrm{o}& \beta ({\mathbf{s}}_c)+{\mathrm{\nabla }}_s\beta ({\mathbf{s}}_c{)}^T(\mathbf{s}-{\mathbf{s}}_c)\ge \overline{\beta }\\ & {\parallel \mathbf{s}-{\mathbf{s}}_c\parallel }_{\mathrm{\infty }}\le {\mathrm{\Delta }}^k\end{array}\end{array}$$

and trust-region surrogate-based optimization for robust design employs surrogate models of $f$ and ${\sigma }^2$ within a trust region ${\mathrm{\Delta }}^k$ centered at ${\mathbf{s}}_c$:

(297)$$\begin{array}{r}\begin{array}{rl}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}& f({\mathbf{s}}_c)+{\mathrm{\nabla }}_{s}f({\mathbf{s}}_c{)}^T(\mathbf{s}-{\mathbf{s}}_c)\\ \mathrm{s}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{t}\mathrm{o}& {\sigma }^2({\mathbf{s}}_c)+{\mathrm{\nabla }}_s{\sigma }^2({\mathbf{s}}_c{)}^T(\mathbf{s}-{\mathbf{s}}_c)\le {\overline{\sigma }}^2\\ & {\parallel \mathbf{s}-{\mathbf{s}}_c\parallel }_{\mathrm{\infty }}\le {\mathrm{\Delta }}^k\end{array}\end{array}$$

Second-order local surrogates may also be employed, where the Hessians are typically approximated from an accumulation of curvature information using quasi-Newton updates [NJ99] such as Broyden-Fletcher-Goldfarb-Shanno (BFGS, Eq. (75) or symmetric rank one (SR1, Eq. eq:sr1 . The sequential approach is available for probabilistic expansions using PCE and SC.

Multifidelity SEBDO

The multifidelity OUU approach is another trust-region surrogate-based approach. Instead of the surrogate UQ model being a simple data fit (in particular, first-/second-order Taylor series model) of the truth UQ model results, distinct UQ models of differing fidelity are now employed. This differing UQ fidelity could stem from the fidelity of the underlying simulation model, the fidelity of the UQ algorithm, or both. In this section, we focus on the fidelity of the UQ algorithm. For reliability methods, this could entail varying fidelity in approximating assumptions (e.g., Mean Value for low fidelity, SORM for high fidelity), and for stochastic expansion methods, it could involve differences in selected levels of $p$ and $h$ refinement.

Here, we define UQ fidelity as point-wise accuracy in the design space and take the high fidelity truth model to be the probabilistic expansion PCE/SC model, with validity only at a single design point. The low fidelity model, whose validity over the design space will be adaptively controlled, will be either the combined expansion PCE/SC model, with validity over a range of design parameters, or the MVFOSM reliability method, with validity only at a single design point. The combined expansion low fidelity approach will span the current trust region of the design space and will be reconstructed for each new trust region. Trust region adaptation will ensure that the combined expansion approach remains sufficiently accurate for design purposes. By taking advantage of the design space spanning, one can eliminate the cost of multiple low fidelity UQ analyses within the trust region, with fallback to the greater accuracy and higher expense of the probabilistic expansion approach when needed. The MVFOSM low fidelity approximation must be reformed for each change in design variables, but it only requires a single evaluation of a response function and its derivative to approximate the response mean and variance from the input mean and covariance (Eqs. eq:mv_mean1  – eq:mv_std_dev from which forward/inverse CDF/CCDF reliability mappings can be generated using Eqs. eq:mv_ria – eq:mv_pma . This is the least expensive UQ option, but its limited accuracy [1] may dictate the use of small trust regions, resulting in greater iterations to convergence. The expense of optimizing a combined expansion, on the other hand, is not significantly less than that of optimizing the high fidelity UQ model, but its representation of global trends should allow the use of larger trust regions, resulting in reduced iterations to convergence. The design derivatives of each of the PCE/SC expansion models provide the necessary data to correct the low fidelity model to first-order consistency with the high fidelity model at the center of each trust region, ensuring convergence of the multifidelity optimization process to the high fidelity optimum. Design derivatives of the MVFOSM statistics are currently evaluated numerically using forward finite differences.

Multifidelity optimization for reliability-based design can be formulated as:

(298)$$\begin{array}{r}\begin{array}{rl}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}& f(\mathbf{s})\\ \mathrm{s}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{t}\mathrm{o}& \widehat{{\beta }_{hi}}(\mathbf{s})\ge \overline{\beta }\\ & {\parallel \mathbf{s}-{\mathbf{s}}_c\parallel }_{\mathrm{\infty }}\le {\mathrm{\Delta }}^k\end{array}\end{array}$$

and multifidelity optimization for robust design can be formulated as:

(299)$$\begin{array}{r}\begin{array}{rl}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}& f(\mathbf{s})\\ \mathrm{s}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{t}\mathrm{o}& {\widehat{{\sigma }_{hi}}}^2(\mathbf{s})\le {\overline{\sigma }}^2\\ & {\parallel \mathbf{s}-{\mathbf{s}}_c\parallel }_{\mathrm{\infty }}\le {\mathrm{\Delta }}^k\end{array}\end{array}$$

where the deterministic objective function is not approximated and $\widehat{{\beta }_{hi}}$ and ${\widehat{{\sigma }_{hi}}}^2$ are the approximated high-fidelity UQ results resulting from correction of the low-fidelity UQ results. In the case of an additive correction function:

(300)$$\begin{array}{r}\begin{array}{rlr}\widehat{{\beta }_{hi}}(\mathbf{s})& =& {\beta }_{lo}(\mathbf{s})+{\alpha }_{\beta }(\mathbf{s})\end{array}\end{array}$$

(301)$$\begin{array}{rlr}{\widehat{{\sigma }_{hi}}}^2(\mathbf{s})& =& {\sigma }_{lo}^2(\mathbf{s})+{\alpha }_{{\sigma }^2}(\mathbf{s})\end{array}$$

where correction functions $\alpha (\mathbf{s})$ enforcing first-order consistency [EGC04] are typically employed. Quasi-second-order correction functions [EGC04] can also be employed, but care must be taken due to the different rates of curvature accumulation between the low and high fidelity models. In particular, since the low fidelity model is evaluated more frequently than the high fidelity model, it accumulates curvature information more quickly, such that enforcing quasi-second-order consistency with the high fidelity model can be detrimental in the initial iterations of the algorithm [2]. Instead, this consistency should only be enforced when sufficient high fidelity curvature information has been accumulated (e.g., after $n$ rank one updates).

Sampling-based OUU

Gradient-based OUU can also be performed using random sampling methods. In this case, the sample-average approximation to the design derivative of the mean and standard deviation are:

(302)$$\begin{array}{r}\begin{array}{rlr}\frac{d\mu }{ds}& =& \frac{1}{N}\sum _{i=1}^N\frac{dQ}{ds}\\ \frac{d\sigma }{ds}& =& [\sum _{i=1}^N(Q\frac{dQ}{ds})-N\mu \frac{d\mu }{ds}]/(\sigma (N-1))\end{array}\end{array}$$

This enables design sensitivities for mean, standard deviation or variance (based on final_moments type), and forward/inverse reliability index mappings ($\overline{z}\to \beta$, $\overline{\beta }\to z$).

[1]

MVFOSM is exact for linear functions with Gaussian inputs, but quickly degrades for nonlinear and/or non-Gaussian.

[2]

Analytic and numerical Hessians, when available, are instantaneous with no accumulation rate concerns.