Scarce SampleBased Reliability Estimation and Optimization Using Importance Sampling
Abstract
:1. Introduction
2. Reliability Estimation Using Importance Sampling for Separable Limit States
3. Identifying Parameters of Gaussian ISD
Algorithm 1 Finding ${\mu}_{h}$. 

4. Estimation of Reliability and Its Confidence Bounds
Algorithm 2 Confidence bounds using bootstrap. 

TailIndex Estimation
5. Reliability Estimation Examples
5.1. Example 1: Concave Limit State 1
5.2. Example 2: Concave Limit State 2
5.3. Example 3: Roof Truss Example
5.4. Example 4: Propped Cantilever Beam Example
6. Application to RBDO Examples
Algorithm 3 RBDO using proposed importance sampling approach. 

6.1. Cantilever Beam Example
6.2. Bracket Structure Example
 (i)
 Maximum bending stress of beam CD at point B $\left({\sigma}_{B}\right)$ does not exceed its yield strength $\left({f}_{y}\right)$,
 (ii)
 Maximum axial load on beam AB $\left({F}_{AB}\right)$ does not exceed the Euler critical buckling load $\left({F}_{buckling}\right)$.
6.3. Torque Arm Example
6.4. Car SideImpact Problem—A MultiObjective ReliabilityBased Design Optimization (MORBDO) Example
7. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Appendix A. Kernel Density Estimation (KDE)
Appendix B. ThirdOrder Polynomial Normal Transformation Technique (TPNT)
Appendix C. Car SideImpact Problem
References
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Response Tail  ${\mathit{\xi}}_{\mathit{r}},{\mathit{\theta}}_{\mathit{r}},{\mathit{\sigma}}_{\mathit{r}}$  Capacity Tail  ${\mathit{\xi}}_{\mathit{c}},{\mathit{\theta}}_{\mathit{c}},{\mathit{\sigma}}_{\mathit{c}}$ 

Heavy  ($1.8,1,33.5$)  
Heavy  ($0.2,1,0$)  Medium  ($1,1,26.5$) 
Light  ($0.52,1,26.2$)  
Heavy  ($1.8,1,30$)  
Medium  ($0,1,0$)  Medium  ($1,1,10.7$) 
Light  ($0.52,1,9.5$)  
Heavy  ($1.8,1,30$)  
Light  ($0.12,1,0$)  Medium  ($1,1,9.4$) 
Light  ($0.52,1,6.4$) 
Heavy C  Medium C  Light C  

Percentile  Original Sample  Bootstrap Mean (Std)  Original Sample  Bootstrap Mean (Std)  Original Sample  Bootstrap Mean (Std)  
Heavy R  $25\mathrm{th}$  0.96  1.10 (0.26)  1.21  1.42 (0.29)  1.79  2.08 (0.61) 
$50\mathrm{th}$  1.06  1.17 (0.29)  1.50  1.54 (0.32)  2.13  2.46 (0.89)  
$75\mathrm{th}$  1.22  1.36 (0.31)  1.76  1.77 (0.33)  2.59  3.08 (1.31)  
Medium R  $25\mathrm{th}$  0.93  1.03 (0.20)  0.85  0.95 (0.18)  1.01  1.09 (0.21) 
$50\mathrm{th}$  1.02  1.09 (0.21)  0.97  1.01 (0.19)  1.13  1.20 (0.25)  
$75\mathrm{th}$  1.18  1.25 (0.23)  1.08  1.14 (0.22)  1.30  1.40 (0.32)  
Light R  $25\mathrm{th}$  0.91  1.04 (0.23)  0.78  0.85 (0.14)  0.85  0.91 (0.12) 
$50\mathrm{th}$  1.01  1.10 (0.25)  0.88  0.91 (0.14)  0.94  0.98 (0.13)  
$75\mathrm{th}$  1.18  1.28 (0.26)  0.99  1.04 (0.17)  1.03  1.09 (0.18) 
Heavy C  Medium C  Light C  

Percentile  Original Sample  Bootstrap Mean (Std)  Original Sample  Bootstrap Mean (Std)  Original Sample  Bootstrap Mean (Std)  
Heavy R  $25\mathrm{th}$  1.02  1.02 ($9\times {10}^{3}$)  1.67  1.71 (0.15)  3.16  3.63 (0.66) 
$50\mathrm{th}$  1.03  1.03 ($7\times {10}^{3}$)  1.78  1.72 (0.15)  3.64  3.67 (0.67)  
$75\mathrm{th}$  1.04  1.04 ($5\times {10}^{3}$)  1.81  1.79 (0.06)  4.00  4.09 (0.30)  
Medium R  $25\mathrm{th}$  0.99  0.99 ($2\times {10}^{3}$)  1.02  1.03 (0.06)  1.25  1.32 (0.20) 
$50\mathrm{th}$  1.00  1.00 ($2\times {10}^{3}$)  1.05  1.04 (0.05)  1.37  1.34 (0.20)  
$75\mathrm{th}$  1.00  1.00 ($2\times {10}^{3}$)  1.08  1.07 (0.03)  1.45  1.51 (0.14)  
Light R  $25\mathrm{th}$  0.99  0.99 ($2\times {10}^{3}$)  0.99  0.99 (0.02)  0.98  1.03 (0.10) 
$50\mathrm{th}$  1.00  1.00 ($2\times {10}^{3}$)  1.00  1.00 (0.02)  1.07  1.06 (0.10)  
$75\mathrm{th}$  1.00  1.00 ($2\times {10}^{3}$)  1.02  1.01 (0.02)  1.12  1.13 (0.07) 
Heavy C  Medium C  Light C  

Percentile  Original Sample  Bootstrap Mean (Std)  Original Sample  Bootstrap Mean (Std)  Original Sample  Bootstrap Mean (Std)  
Heavy R  $25\mathrm{th}$  0.96  0.98 (0.10)  0.99  0.99 ($6\times {10}^{3}$)  1.00  1.00 ($2\times {10}^{3}$) 
$50\mathrm{th}$  1.03  1.00 (0.08)  1.00  1.00 ($3\times {10}^{3}$)  1.00  1.00 ($2\times {10}^{3}$)  
$75\mathrm{th}$  1.06  1.04 (0.05)  1.00  1.00 ($2\times {10}^{3}$)  1.00  1.00 ($2\times {10}^{3}$)  
Medium R  $25\mathrm{th}$  0.93  1.13 (0.31)  0.87  0.92 (0.12)  0.96  0.96 (0.05) 
$50\mathrm{th}$  1.12  1.24 (0.35)  0.96  0.96 (0.11)  0.99  0.98 (0.04)  
$75\mathrm{th}$  1.42  1.52 (0.33)  1.03  1.02 (0.07)  1.00  1.00 (0.02)  
Light R  $25\mathrm{th}$  0.91  1.05 (0.23)  0.82  0.90 (0.13)  0.87  0.89 (0.09) 
$50\mathrm{th}$  1.03  1.12 (0.25)  0.93  0.95 (0.14)  0.94  0.94 (0.09)  
$75\mathrm{th}$  1.20  1.29 (0.29)  1.03  1.06 (0.14)  1.00  0.99 (0.08) 
Heavy C  Medium C  Light C  

Percentile  Original Sample  Bootstrap Mean (Std)  Original Sample  Bootstrap Mean (Std)  Original Sample  Bootstrap Mean (Std)  
Heavy R  $25\mathrm{th}$  0.95  0.96 (0.09)  0.95  1.14 (0.29)  0.92  1.13 (0.31) 
$50\mathrm{th}$  1.02  0.99 (0.07)  1.13  1.22 (0.31)  1.11  1.23 (0.35)  
$75\mathrm{th}$  1.03  1.03 (0.02)  1.43  1.42 (0.30)  1.38  1.51 (0.41)  
Medium R  $25\mathrm{th}$  0.99  0.99 (0.02)  0.84  0.89 (0.12)  0.77  0.89 (0.18) 
$50\mathrm{th}$  0.99  0.99 (0.01)  0.94  0.94 (0.11)  0.93  0.96 (0.20)  
$75\mathrm{th}$  1.00  1.00 ($2\times {10}^{3}$)  1.02  1.01 (0.09)  1.08  1.11 (0.21)  
Light R  $25\mathrm{th}$  0.99  0.99 ($2\times {10}^{3}$)  0.91  0.93 (0.07)  0.79  0.84 (0.12) 
$50\mathrm{th}$  1.00  1.00 ($1\times {10}^{3}$)  0.97  0.96 (0.06)  0.87  0.90 (0.12)  
$75\mathrm{th}$  1.00  1.00 ($1\times {10}^{3}$)  1.00  0.99 (0.04)  0.99  0.98 (0.12) 
Percentile  Original Sample  Bootstrap Mean (Std) 

25th  0.93  1.19 (0.39) 
50th  1.12  1.32 (0.53) 
75th  1.39  1.70 (1.02) 
Percentile  Original Sample  Bootstrap Mean (Std) 

25th  0.84  0.96 (0.20) 
50th  0.96  1.03 (0.20) 
75th  1.12  1.20 (0.22) 
Percentile  Original Sample  Bootstrap Mean (Std) 

25th  0.82  0.89 (0.16) 
50th  0.90  0.95 (0.16) 
75th  1.02  1.08 (0.19) 
Random Variable  Mean (SD) 

q (N/m)  20,000 (1600) 
l (m)  12 (0.24) 
${A}_{s}\phantom{\rule{0.166667em}{0ex}}\left({\mathrm{m}}^{2}\right)$  $9.82\times {10}^{4}(5.89\times {10}^{5})$ 
${A}_{c}\phantom{\rule{0.166667em}{0ex}}\left({\mathrm{m}}^{2}\right)$  0.04 (0.008) 
${E}_{s}\phantom{\rule{0.166667em}{0ex}}\left({\mathrm{N}/\mathrm{m}}^{2}\right)$  $1.2\times {10}^{11}(8.4\times {10}^{9})$ 
${E}_{c}\phantom{\rule{0.166667em}{0ex}}\left({\mathrm{N}/\mathrm{m}}^{2}\right)$  $3\times {10}^{10}(2.4\times {10}^{9})$ 
Percentile  Original Sample  Bootstrap Mean (Std) 

25th  0.84  0.94 (0.17) 
50th  0.94  1.02 (0.20) 
75th  1.07  1.18 (0.29) 
Random Variable  Mean (SD) 

${q}_{0}$ (kN/m)  20 (2) 
L (m)  6 (0.3) 
E (GPa)  210 (10) 
d (cm)  25 (0.5) 
${b}_{f}$ (cm)  25 (0.5) 
${t}_{w}$ (cm)  2 (0.2) 
${t}_{f}$ (cm)  2 (0.2) 
${\mathit{\nu}}_{\mathit{crit}}=4.0\phantom{\rule{0.166667em}{0ex}}\mathbf{mm}\phantom{\rule{0.166667em}{0ex}},{\mathit{\beta}}_{\mathit{t}}=2.98$  ${\mathit{\nu}}_{\mathit{crit}}=4.5\phantom{\rule{0.166667em}{0ex}}\mathbf{mm}\phantom{\rule{0.166667em}{0ex}},{\mathit{\beta}}_{\mathit{t}}=3.50$  ${\mathit{\nu}}_{\mathit{crit}}=5.0\phantom{\rule{0.166667em}{0ex}}\mathbf{mm}\phantom{\rule{0.166667em}{0ex}},{\mathit{\beta}}_{\mathit{t}}=3.97$  

Percentile  Original  Bootstrap Mean (Std)  Original  Bootstrap Mean (Std)  Original  Bootstrap Mean (Std) 
25th  0.84  1.03 (0.30)  0.82  0.99 (0.27)  0.81  0.99 (0.30) 
50th  0.99  1.13 (0.36)  0.99  1.09 (0.32)  0.96  1.12 (0.39) 
75th  1.29  1.31 (0.42)  1.24  1.29 (0.44)  1.26  1.35 (0.52) 
Random Variable  Mean (SD) 

$X\phantom{\rule{0.166667em}{0ex}}\left(lb\right)$  500 (100) 
$Y\phantom{\rule{0.166667em}{0ex}}\left(lb\right)$  ${10}^{3}$ (100) 
${\sigma}_{y}\phantom{\rule{0.166667em}{0ex}}\left(psi\right)$  $4\times {10}^{4}$ ($2\times {10}^{3}$) 
$E\phantom{\rule{0.166667em}{0ex}}\left(psi\right)$  $2.9\times {10}^{7}$ ($1.45\times {10}^{6}$) 
Surrogate Model for $\mathit{\beta}$ Constraints  Reliability Estimation  Optima $\left({\mathit{d}}^{*}\right)$  Objective Function Value  $\widehat{\mathit{\beta}}$ at ${\mathit{d}}^{*}$  ${\mathit{\beta}}_{\mathit{MCS}}$ at ${\mathit{d}}^{*}$  

$\mathit{w}$ (in)  $\mathit{t}$ (in)  $\mathit{A}$ (in${}^{\mathbf{2}}$)  ${\mathit{g}}_{\mathbf{1}}$  ${\mathit{g}}_{\mathbf{2}}$  ${\mathit{g}}_{\mathbf{1}}$  ${\mathit{g}}_{\mathbf{2}}$  
WAS  IS  2.59  3.74  9.69  3.00  3.64  3.25  3.69 
MCS  2.59  3.66  9.50  3.00  3.44  2.95  3.39 
Type  Variable  Distribution  Mean  C.o.V 

Random  P (kN)  Gumbel  100  15% 
E (GPa)  Gumbel  200  8%  
${f}_{y}$ (MPa)  Lognormal  225  8%  
$\rho (\mathrm{kg}\xb7{\mathrm{m}}^{3})$  Weibull  7860  10%  
L (m)  Gaussian  5  5%  
Design  ${w}_{AB}$ (mm)  Gaussian  ${\mu}_{{w}_{AB}}$  5% 
${w}_{CD}$ (mm)  Gaussian  ${\mu}_{{w}_{CD}}$  5%  
t (mm)  Gaussian  ${\mu}_{t}$  5% 
Surrogate Model for $\mathit{\beta}$ Constraints  Reliability Estimation  Optima $\left({\mathit{d}}^{*}\right)$  Objective Function Value  $\widehat{\mathit{\beta}}$ at ${\mathit{d}}^{*}$  ${\mathit{\beta}}_{\mathit{MCS}}$ at ${\mathit{d}}^{*}$  

${\mathit{w}}_{\mathit{AB}}\phantom{\rule{3.33333pt}{0ex}}\left(\mathrm{mm}\right)$  ${\mathit{w}}_{\mathit{CD}}\phantom{\rule{3.33333pt}{0ex}}\left(\mathrm{mm}\right)$  $\mathit{t}\phantom{\rule{3.33333pt}{0ex}}\left(\mathrm{mm}\right)$  $\mathit{Weight}$ (kg)  ${\mathit{g}}_{\mathbf{1}}$  ${\mathit{g}}_{\mathbf{2}}$  ${\mathit{g}}_{\mathbf{1}}$  ${\mathit{g}}_{\mathbf{2}}$  
WAS  IS  58  89  300  1576  2.00  2.00  2.59  2.87 
MCS  62  77  300  1474  2.00  2.00  2.02  3.55 
Random Variable  Distribution Type  Mean; SD 

${F}_{x}$ (N)  Normal  $2789;278.9$ 
${F}_{y}$ (N)  Normal  $5066;506.6$ 
${\sigma}_{all}$ (MPa)  Lognormal  $800;80$ 
DV  ${\mathit{d}}_{\mathit{L}}$  ${\mathit{d}}_{\mathit{U}}$  ${\mathit{d}}^{*}$ (Optimum) 

${d}_{1}$  1.80  3.20  2.15 
${d}_{2}$  1.25  1.60  1.28 
${d}_{3}$  1.20  4.60  1.59 
${d}_{4}$  −0.10  0.40  −0.09 
${d}_{5}$  −0.30  0.30  0.30 
${d}_{6}$  −0.90  0.80  0.30 
${d}_{7}$  0.40  1.80  0.54 
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Pannerselvam, K.; Yadav, D.; Ramu, P. Scarce SampleBased Reliability Estimation and Optimization Using Importance Sampling. Math. Comput. Appl. 2022, 27, 99. https://doi.org/10.3390/mca27060099
Pannerselvam K, Yadav D, Ramu P. Scarce SampleBased Reliability Estimation and Optimization Using Importance Sampling. Mathematical and Computational Applications. 2022; 27(6):99. https://doi.org/10.3390/mca27060099
Chicago/Turabian StylePannerselvam, Kiran, Deepanshu Yadav, and Palaniappan Ramu. 2022. "Scarce SampleBased Reliability Estimation and Optimization Using Importance Sampling" Mathematical and Computational Applications 27, no. 6: 99. https://doi.org/10.3390/mca27060099