Proposed interval importance sampling method

5.2.2.1 Constructing the importance sampling function

The approach of constructing the importance sampling function is mainly based on the search-based approach (Melchers, 1990), in which the sampling function h_v(.) approaches the optimal point during the simulation.

Let (x₁, . . . , x_s) be the basic random variables, the form of h_v(.) is taken as:

h_v(.) =

s

Y

i=1

h_vi(.), (5.9)

where h_vi is assumed to be the same as the distribution function of f_xi. The mean and standard deviation of hvi are changed to the optimal point through the simulation.

The first sampling point is randomly simulated from h_v⁽¹⁾ with its mean equal to the mean of random variables µX and its standard deviation is taken as:

σ_vj =

in which j is the order of simulation and n is the total number of simulation. In the first simulation; j = 1, then σ_v1 = 2σ_X.

The function of the standard deviation is proposed in Eq. (5.10) has two main properties. The first is the range of the function varies between σ_X and 2σ_X. Secondly, the function can only be reduced or stay the same during the simulation.

These properties are consistent with recommendations byMelchers (1990). When all the values of v₁ can be generated, then the value of the limit state function g (v₁) and the value of f_X(v₁) at this point can also be calculated. These values are recorded as the minimum value of limit state function and maximum value of joint PDF.

In the next step, the second sampling point v₂ is also randomly generated from h_v⁽¹⁾ with its mean and standard deviation equal to the first point v₁. Similarly, the values of g (v₂) and the value of f_X(v₂) are calculated. If this second limit state function g (v₂) is lower than the first limit state function g (v₁) and the second value of f_X(v₂) is greater than f_X(v₁), then the importance sampling function h_v⁽¹⁾ is changed to the new sampling function h_v⁽²⁾ with its mean of µ_v2 and its standard deviation of σ_v2. Also, the minimum value of limit state function is changed to g (v₁) and maximum value of joint PDF f_X(v₁) is changed to g (v₂) and f_X(v₂) respectively. Otherwise, the importance sampling is h_v⁽¹⁾. Also, the minimum value of limit state function and maximum value of joint PDF equal to the previous point which mean equal to g (v₁) and f_X(v₁) respectively.

Repeat this process n times. The sampling function approaches the optimal point when n is approximate 20 − 100 (Melchers, 1990).

5.2.2.2 Determine the bounds of a set of PDFs

In case of independent p-boxes, the bounds of PDF can be obtained numerically.

For some commonly distribution types used in reliability analysis, analytical so-lution can be obtained for a PDF with interval parameters.Zhang(2012) derived the analytical solutions in the case of normal distribution and log-normal distri-bution with an interval mean µ, µ and a deterministic standard deviation of σ.

in which φ (.) is the PDF for standard normal distribution.

• In the log-normal case:

The lower (or upper) bound of PDF is not a member PDF because the member PDF normally cross over each other. Thus, the bound is not even a valid density

function, and integration of f_X(.) (or f_X(.) ) is not equal unity. Consequently, the bounds calculated byEq. (5.7)are always conservative. In a case of wide intervals between the lower and upper bound, this calculation becomes over excessive.

Zhang(2012) proposed the adaptive partition method to reduce the conservatism.

In case of dependent p-boxes, the proposed method uses Latin Hypercube Sam-pling (LHS) to obtain the bounds of PDF numerically. In this study, the proposed algorithm is shown for the case of p-boxes with interval parameters such as mean and copula parameter. The standard deviation of each p-box is assumed to be constant. The case of the p-box that has many different distribution types is excluded from the study.

When all random variables are independent, the joint PDF is given as:

f_X(.) =

s

Y

i=1

f_xi(.) . (5.18)

When random variables are dependent with copula C, the joint PDF at a sampling vector v is given as:

in which fX(.) is the probability density function of random variables X. c [.] is the copula density function, and u = F_v(v) where F_v is the CDF of the sampling vector v.

Based on the method of distribution with interval parameters which was intro-duced in Section 4.2.3, the bounds of f_X(.) are derived as:

fX(.) =

It is noted that the bounds of f

xi(.) and f_xi(.) are only dependent on the statisti-cal parameters of each random variable x_i, such as mean and standard deviation.

They are independent of the copula parameter θ. The copula density functions of c [.] and c [.] in Eq. (5.20)are only dependent on θ.

Let [µ

X, µ_X] denote the interval mean of random variables X, and [θ, θ] denote the interval of a copula parameter θ.

By Latin Hypercube method, [µ

Similarly, the bounds of the copula c [.] and c [.] are defined from k of the copula parameter as following:

c [.] = min (c[θ₁], . . . , c[θ_k]) ,

c [.] = max (c[θ₁], . . . , c[θ_k]) . (5.24)

It can be seen that the computation of P_f and P_f requires double-loop simulation.

For every simulation of v_j in the outer loop, k simulations in the inner loop are required to calculate the bounds of the bounds of f_X(v_j). If n is the number of simulations is required for the outer loop, the total required simulation is n × k. The accuracy and efficiency of the proposed method are demonstrated in numerical examples in Section 5.3.1.

5.2.2.3 The algorithm of the proposed interval importance sampling method

The proposed algorithm can be summarised in the following steps:

1. Form the limit state function g (.).

2. Establish the importance sampling h_v⁽¹⁾, as suggested in Section5.2.2.1, in which h_v⁽¹⁾ is assumed to be independent. In the case of random variables involving copula dependence, h_v⁽¹⁾ can still be chosen to have independent components (Melchers,1999b). When random variables have interval mean, the mid point value is assumed to be its mean value.

3. Generate random variables v₁ from h_v⁽¹⁾.

4. Calculate the limit state function g (v₁) and consider g (v₁) as the minimum limit state function.

5. Determine the bounds of f

X(v₁) and f_X(v₁), as proposed in Section5.2.2.2.

Consider them as the maximum values of joint PDF of f

X(v₁) and f_X(v₁).

6. If the limit state function g (v₁) is positive. Repeat step 3 to step 5, in which random variables v2 are generating from the same importance sampling h_v⁽¹⁾.

7. If the limit station function g (v1) is negative, calculate the bounds of Pf and P_f using Eq. (5.7). Record the difference between P_f and P_f as wid(P_f,1).

8. If the limit state function is lower than the minimum value of limit state function and the value of joint PDF is greater than the maximum value of joint PDF, then the importance sampling function h_v⁽¹⁾ is changed to the new sampling function as suggested in Section5.2.2.1. Also, the new values of minimum limit state function and maximum joint PDF are recorded.

Otherwise, the importance sampling function is not changed.

9. Repeat Step 3 to Step 8, in which new random variables v_i are generat-ing from the importance samplgenerat-ing function which is obtained in Step 8.

Record the value of wid(P_f,i). The loop will end when the result satisfies the following criterion:

wid (P_f,i) − wid (P_f,i+1)

wid (Pf,i+1) ≤ ε, (5.25)

in which, the threshold ε can be chosen between 1% and 10%.

Note that the method to find the bounds of the bounds of f

X(v1) and f_X(v1) in Step 5 does not increase the required number of finite element analysis in the whole process. The determination of limit station function g (vi) and importance sampling function h_vi are independent of bounds of f

X(v_i) and f_X(v_i). Thus g (v_i) and hvi are retained and used in calculating the bounds of Pf and Pf. In other words, the additional computing cost of using LHS may not be significant.

5.3 Numerical examples

5.3.1 Example 1: a fundamental case of structural