Line sampling

<h2 id="mathematical-approach">Mathematical approach</h2>
Firstly the importance direction must be determined. This can be achieved by finding the design point, or the gradient of the limit state function.
A set of samples is generated using <a href="/facts/Monte_Carlo_method/AkHjY7jc">Monte Carlo simulation</a> in the standard normal space. For each sample 
 
 
 
 
 x
 
 
 
 {\displaystyle {\boldsymbol {x}}}
 
, the probability of failure in the line parallel to the important direction is defined as:

p
          
            f
          
        
        (
        
          x
        
        )
        =
        
          ∫
          
            −
            ∞
          
          
            +
            ∞
          
        
        I
        (
        
          x
        
        +
        β
        ⋅
        
          α
        
        )
        φ
        (
        β
        )
        
        d
        β
      
    
    {\displaystyle p_{f}({\boldsymbol {x}})=\int _{-\infty }^{+\infty }I({\boldsymbol {x}}+\beta \cdot {\boldsymbol {\alpha }})\varphi (\beta )\,d\beta }

where 
 
 
 
 I
 (
 ⋅
 )
 
 
 {\displaystyle I(\cdot )}
 
  is equal to one for samples contributing to failure, and is zero otherwise:

I
          
            f
          
        
        (
        
          x
        
        )
        =
        
          
            {
            
              
                
                  1
                
                
                  
                    if 
                  
                  
                    x
                  
                  ∈
                  
                    Ω
                    
                      f
                    
                  
                
              
              
                
                  0
                
                
                  
                    else
                  
                
              
            
            
          
        
      
    
    {\displaystyle I_{f}({\boldsymbol {x}})={\begin{cases}1&{\text{if }}{\boldsymbol {x}}\in \Omega _{f}\\0&{\text{else}}\end{cases}}}

α
 
 
 
 {\displaystyle {\boldsymbol {\alpha }}}
 
  is the important direction, 
 
 
 
 φ
 
 
 {\displaystyle \varphi }
 
  is the probability density function of a <a href="/facts/Normal_distribution/UapjjPyQ">Gaussian distribution</a> (and 
 
 
 
 β
 
 
 {\displaystyle \beta }
 
  is a real number). In practice the roots of a nonlinear function must be found to estimate the partial probabilities of failure along each line. This is either done by interpolation of a few samples along the line, or by using the <a href="/facts/Newton%2527s_method/VlRhI2FL">Newton–Raphson method</a>.
The global probability of failure is the mean of the probability of failure on the lines:

p
                ~
              
            
          
          
            f
          
        
        =
        
          
            1
            
              N
              
                L
              
            
          
        
        
          ∑
          
            i
            =
            1
          
          
            
              N
              
                L
              
            
          
        
        
          p
          
            f
          
          
            (
            i
            )
          
        
      
    
    {\displaystyle {\tilde {p}}_{f}={\frac {1}{N_{L}}}\sum _{i=1}^{N_{L}}p_{f}^{(i)}}

where 
 
 
 
 
 N
 
 L
 
 
 
 
 {\displaystyle N_{L}}
 
  is the total number of lines used in the analysis and the 
 
 
 
 
 p
 
 f
 
 
 (
 i
 )
 
 
 
 
 {\displaystyle p_{f}^{(i)}}
 
  are the partial probabilities of failure estimated along all the lines. 
For problems in which the dependence of the performance function is only moderately non-linear with respect to the parameters modeled as random variables, setting the importance direction as the gradient vector of the performance function in the underlying standard normal space leads to highly efficient Line Sampling. In general it can be shown that the variance obtained by line sampling is always smaller than that obtained by conventional Monte Carlo simulation, and hence the line sampling algorithm converges more quickly.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> The rate of convergence is made quicker still by recent advancements which allow the importance direction to be repeatedly updated throughout the simulation, and this is known as adaptive line sampling.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

<h2 id="industrial-application">Industrial application</h2>
The algorithm is particularly useful for performing reliability analysis on computationally expensive industrial black box models, since the limit state function can be non-linear and the number of samples required is lower than for other reliability analysis techniques such as <a href="/facts/Subset_simulation/a6cHdlcd">subset simulation</a>.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> The algorithm can also be used to efficiently propagate <a href="/facts/Uncertainty_quantification/YzIJjhEE">epistemic uncertainty</a> in the form of <a href="/facts/Probability_box/MoQRSlGW">probability boxes</a>, or <a href="/facts/Random_compact_set/K7UJWzqR">random sets</a>.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a><a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> A numerical implementation of the method is available in the open source software OpenCOSSAN.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Rare_event_sampling/aQ9IarMf">Rare event sampling</a></li>
<li><a href="/facts/Curse_of_dimensionality/dk7SnY6L">Curse of dimensionality</a></li>
<li><a href="/facts/Risk_assessment/vlG9Uf0h">Quantitative risk assessment</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Schueller, G. I.; Pradlwarter, H. J.; Koutsourelakis, P. (2004). "A critical appraisal of reliability estimation procedures for high dimensions". Probabilistic Engineering Mechanics. 19 (4): 463–474. doi:10.1016/j.probengmech.2004.05.004. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Schueller, G. I.; Pradlwarter, H. J.; Koutsourelakis, P. (2004). "A critical appraisal of reliability estimation procedures for high dimensions". Probabilistic Engineering Mechanics. 19 (4): 463–474. doi:10.1016/j.probengmech.2004.05.004. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">de Angelis, Marco; Patelli, Edoardo; Beer, Michael (2015). "Advanced Line Sampling for efficient robust reliability analysis". Structural Safety. 52: 170–182. doi:10.1016/j.strusafe.2014.10.002. ISSN 0167-4730. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Zio, E; Pedroni, N (2009). "Subset simulation and line sampling for advanced Monte Carlo reliability analysis". Reliability, Risk, and Safety. doi:10.1201/9780203859759.ch94 (inactive 2024-11-12). ISBN 978-0-415-55509-8.{{cite book}}: CS1 maint: DOI inactive as of November 2024 (link) <a href="978-0-415-55509-8" target="_blank">978-0-415-55509-8</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">De Angelis, Marco (2015). Efficient Random Set Uncertainty Quantification by means of Advanced Sampling Techniques (Ph.D.). University of Liverpool. <a href="https://livrepository.liverpool.ac.uk/2038039/" target="_blank">https://livrepository.liverpool.ac.uk/2038039/</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Patelli, E; de Angelis, M (2015). "Line sampling approach for extreme case analysis in presence of aleatory and epistemic uncertainties". Safety and Reliability of Complex Engineered Systems. pp. 2585–2593. doi:10.1201/b19094-339 (inactive 2024-11-12). ISBN 978-1-138-02879-1.{{cite book}}: CS1 maint: DOI inactive as of November 2024 (link) <a href="978-1-138-02879-1" target="_blank">978-1-138-02879-1</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Patelli, Edoardo (2016). "COSSAN: A Multidisciplinary Software Suite for Uncertainty Quantification and Risk Management". Handbook of Uncertainty Quantification. pp. 1–69. doi:10.1007/978-3-319-11259-6_59-1. ISBN 978-3-319-11259-6. <a href="978-3-319-11259-6" target="_blank">978-3-319-11259-6</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
</ol>

Line sampling open-in-new

Line sampling