Evaluation of moments of performance functions based on polynomial chaos expansions

Abstract

Simulating the higher-order statistical moments of response functions is a great challenge to research community in the filed of probabilistic analysis, and generally the polynomial surrogate model can be applied to evaluate these statistical moments. In this study, an analytical solution is developed for calculating the first four raw moments of response functions. Then, a novel criterion is proposed to choose the best analytical formulas that gives the most accurate values of response functions. The intrusive and non-intrusive PC expansion models are considered. Several applications are provided to demonstrate the accuracy and efficiency of the developed model, in which Monte Carlo methods are adopted as an exact criterion for comparison.

Appendices

Appendix 1: Equations for linearization coefficients

1.1 Jocobi polynomials

$$B_{n} ({\text{j}},{\text{k}}) = ( - 1)^{j + k + n} \frac{k!}{{(k + \beta )!}}\tilde{B}_{n} ({\text{j}},{\text{k}})$$

where the computational coefficients $\tilde{B}_{n} ({\text{j}},{\text{k}})$ starting from

$$\tilde{B}_{j - k - 1} ({\text{j}},{\text{k}}) = 0$$

$$\tilde{B}_{j - k} ({\text{j}},{\text{k}}) = \frac{{[2({\text{j}} - {\text{k}}) + \alpha + \beta + 1]!(2{\text{k}} + \alpha + \beta )!({\text{j}} + \alpha )!({\text{j}} + \beta )!}}{{(2{\text{j}} + \alpha + \beta + 1)!({\text{k}} + \alpha + \beta )!({\text{j}} - {\text{k}} + \alpha )!({\text{j}} - {\text{k}})!{\text{j}}!}}.$$

1.2 Hermite polynomials

$$B_{j + k - 2n} ({\text{j}},{\text{k}}) = \frac{{\gamma_{a} ({\text{j}})\gamma_{a} ({\text{k}})}}{{\gamma_{a} ({\text{j}} + {\text{k}} - 2{\text{n}})}}\sum\limits_{p = 0}^{j/2} {\sum\limits_{q = 0}^{k/2} {\frac{{\gamma_{a} ({\text{j}} + {\text{k}} - 2({\text{p}} + {\text{q}}))}}{{\gamma_{a} ({\text{j}} - 2{\text{p}})\gamma_{a} ({\text{k}} - 2{\text{q}}){\text{p}}!{\text{q}}!}}} } \frac{{( - {\text{n}})_{p + q} }}{n!}$$

$$B_{l} ({\text{j}},{\text{k}}) = \frac{{\sqrt \pi 2^{n} j!k!l!}}{(n - j)!(n - k)!(n - l)!}$$

1.3 Laguerre polynomials

$$B_{j + k - n} (j,k) = \frac{{( - 2)^{n} (j + k - {\text{n}})!}}{(j + k - n + \alpha )!(n)!} \times \sum\limits_{{l = max(0,{\text{n}} - {\text{j}},{\text{n}} - {\text{k}})}}^{n/2} {\left( { - \frac{n}{2}} \right)}_{l} \left( { - \frac{n - 1}{2}} \right)_{l} \frac{(j + k - n + \alpha + l)!}{{(j - n + l)!(k - n + l)!}}.$$

Appendix 2. Efficient Cubature Formula I–III

2.1 Cubature Formula I

$$\begin{aligned} I[{\text{r}}] & = a\left[ {r\left( {\sqrt 2 \eta ,\sqrt 2 \eta , \ldots \sqrt 2 \eta } \right) + r\left( { - \sqrt 2 \eta , - \sqrt 2 \eta , \ldots - \sqrt 2 \eta } \right)} \right] \\ & \quad + \;b\left\{ {\sum\limits_{Permution} {\left[ {{\text{r}}\left( {\sqrt 2 \lambda ,\sqrt 2 \xi , \ldots ,\sqrt 2 \xi } \right) + r\left( { - \sqrt 2 \lambda , - \sqrt 2 \xi , \ldots , - \sqrt 2 \xi } \right)} \right]} } \right\} \\ & \quad + \;c\left\{ {\sum\limits_{Permution} {\left[ {{\text{r}}\left( {\sqrt 2 \mu ,\sqrt 2 \mu ,\sqrt 2 \gamma \ldots ,\sqrt 2 \gamma } \right) + r\left( { - \sqrt 2 \mu , - \sqrt 2 \mu , - \sqrt 2 \gamma , \ldots , - \sqrt 2 \gamma } \right)} \right]} } \right\}. \\ \end{aligned}$$

2.2 Cubature Formula II

$$\begin{aligned} I[{\text{r}}] & = \frac{2}{d + 2}r(0) + \frac{{d^{2} (7 - {\text{d}})}}{{2({\text{d}} + 1)^{2} ({\text{d}} + 2)^{2} }}\sum\limits_{j = 1}^{d + 1} {\left[ {r\left( {\sqrt {d + 2} \times {\text{a}}^{{({\text{j}})}} } \right) + r\left( { - \sqrt {d + 2} \times {\text{a}}^{{({\text{j}})}} } \right)} \right]} \\ & \quad + \;\frac{{2({\text{d}} - 1)}}{{({\text{d}} + 1)^{2} ({\text{d}} + 2)^{2} }}\sum\limits_{j = 1}^{d(d + 2)/2} {\left[ {r\left( {\sqrt {{\text{d}} + 2} \times {\text{b}}^{{({\text{j}})}} } \right) + r\left( { - \sqrt {d + 2} \times {\text{b}}^{{({\text{j}})}} } \right)} \right]} \\ \end{aligned}$$

$$a^{{({\text{j}})}} = (a_{1}^{{({\text{j}})}} ,a_{2}^{{({\text{j}})}} , \ldots ,a_{d}^{{({\text{j}})}} ),{\text{j}} = 1,2, \ldots ,{\text{d}} + 1$$

$$b^{{({\text{j}})}} = \sqrt {\frac{d}{{2({\text{d}} - 1)}}} (a^{{({\text{k}})}} + a^{{({\text{l}})}} ),\;{\text{k}} < {\text{l}},{\text{l}} = 1,2 \ldots {\text{d}} + 1$$

$$a_{i}^{{({\text{j}})}} = \left\{ {\begin{array}{*{20}l} {\left[ {\frac{d + 1}{{d({\text{d}} - {\text{i}} + 2)({\text{d}} - {\text{i}} + 1)}}} \right]^{1/2} } \hfill & {for\;{\text{i}} < {\text{j}}} \hfill \\ {\left[ {\frac{d + 1}{{d({\text{d}} - {\text{i}} + 2)({\text{d}} - {\text{i}} + 1)}}} \right]^{1/2} } \hfill & {for\;{\text{i}} = {\text{j}}} \hfill \\ {0,} \hfill & {for\;{\text{i}} > {\text{j}}} \hfill \\ \end{array} } \right..$$

2.3 Cubature Formula III

$$I({\text{r}}) = \frac{2}{d + 2}r(0) + \frac{4 - d}{{2({\text{d}} + 2)^{2} }}\sum {r(\sqrt {d + 2} , \ldots 0)} + \frac{1}{{({\text{d}} + 2)^{2} }}\sum {r\left( {\sqrt {d/2 + 1} ,\sqrt {d/2 + 1} , \ldots 0} \right)} .$$