An efficient interval moment method for uncertainty propagation analysis with non-parameterized probability-box

Abstract

This paper proposes an efficient interval moment method (IMM) for uncertainty propagation analysis with non-parameterized probability-box (p-box), where the bounds of statistical moments and cumulative distribution function (CDF) of output response can be simultaneously obtained. Firstly, two output response bounds are defined based on the equivalent probability transformation, which converts the original imprecise uncertainty propagation problem into two precise uncertainty propagation problems. Then, sparse grid numerical integration (SGNI) is employed to estimate the statistical moments of output response bounds. To improve computational efficiency, a multi-interval efficient global optimization (MI-EGO) algorithm is developed to capture the minimum and maximum responses on all collocation intervals of SGNI. By reconstructing the distributions of output response bounds using the maximum entropy method, the CDF bounds of output response can be acquired accordingly. Furthermore, by reusing the previous functional evaluations in the interval multiplication, the bounds of the statistical moments of output response can be estimated by SGNI again. Three numerical examples are investigated to verify the accuracy and efficiency of the proposed method.

Appendix: Interval Monte Carlo simulation for estimating the bounds of statistical moments and CDF

The IMCS method [22, 23] is substantially a double-loop strategy, and its detailed procedures are as follows:

• Step 1: Generate N_IMCS standard Uniform distribution sample points r_i, i = 1, 2, …, N_IMCS, in the outer loop.

• Step 2: For each sample point r_i, generate an interval sample point x_i, whose lower and upper bounds $\underline{{\mathbf{x}}}_{i}$ and $\overline{{\mathbf{x}}}_{i}$ are obtained by the following inverse probability transformation operators:

  $$ \underline{{{\mathbf{x}}_{i} }} = \overline{F}_{{\mathbf{X}}}^{ - 1} ({\mathbf{r}}_{i} ), $$

  (61)
  $$ \overline{{{\mathbf{x}}_{i} }} = \underline{F}_{{\mathbf{X}}}^{ - 1} ({\mathbf{r}}_{i} ). $$

  (62)

  where $\overline{F}_{{\mathbf{X}}}^{ - 1} ( \cdot )$ and $\underline{F}_{{\mathbf{X}}}^{ - 1} ( \cdot )$ are the inverse functions of $\overline{F}_{{\mathbf{X}}} ( \cdot )$ and $\underline{F}_{{\mathbf{X}}} ( \cdot )$, respectively.

• Step 3: For each interval sample point x_i, compute the minimum and maximum of G(x_i) in the inner loop:

  $$ \underline{G} ({\mathbf{x}}_{i} ) = \mathop {\min }\limits_{{{\mathbf{x}}_{i} \in [\underline{{{\mathbf{x}}_{i} }} ,\overline{{{\mathbf{x}}_{i} }} ]}} G({\mathbf{x}}_{i} ), $$

  (63)
  $$ \overline{G} ({\mathbf{x}}_{i} ) = \mathop {\max }\limits_{{{\mathbf{x}}_{i} \in [\underline{{{\mathbf{x}}_{i} }} ,\overline{{{\mathbf{x}}_{i} }} ]}} G({\mathbf{x}}_{i} ). $$

  (64)

• Step 4: Estimate two bounds of F_Y(y) by:

  $$ \underline{F}_{Y} (y) = \frac{1}{{N_{IMCS} }}\sum\limits_{i = 1}^{{N_{IMCS} }} {I[\overline{G} ({\mathbf{x}}_{i} ) \le y]} , $$

  (65)
  $$ \overline{F}_{Y} (y) = \frac{1}{{N_{IMCS} }}\sum\limits_{i = 1}^{{N_{IMCS} }} {I[\underline{G} ({\mathbf{x}}_{i} ) \le y]} . $$

  (66)

  where I[·] is the indicator function, whose value is equal to one when the event in brackets occurs.

• Step 5: Estimate two bounds of v_kY by:

  $$ \underline{v}_{kY} = \frac{1}{{N_{IMCS} }}\sum\limits_{i = 1}^{{N_{IMCS} }} {\underline{{G({\mathbf{x}}_{i} )^{k} }} } , $$

  (67)
  $$ \overline{v}_{kY} = \frac{1}{{N_{IMCS} }}\sum\limits_{i = 1}^{{N_{IMCS} }} {\overline{{G({\mathbf{x}}_{i} )^{k} }} } . $$

  (68)

  where $\underline{{G({\mathbf{x}}_{i} )^{k} }}$ and $\overline{{G({\mathbf{x}}_{i} )^{k} }}$ for each specific interval sample point x_i are obtained by the interval multiplication proposed in Sect. 3.4 such that the evaluations of $\underline{G} ({\mathbf{x}}_{i} )$ and $\overline{G} ({\mathbf{x}}_{i} )$ can be reused.