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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. 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.
80:. Once the importance direction has been set to point towards the failure region, samples are randomly generated from the standard normal space and lines are drawn parallel to the importance direction in order to compute the distance to the limit state function, which enables the probability of failure to be estimated for each sample. These failure probabilities can then be averaged to obtain an improved estimate.
72: in the input parameter space, which points towards the region which most strongly contributes to the overall failure probability. The importance direction can be closely related to the center of mass of the failure region, or to the failure point with the highest probability density, which often falls at the closest point to the origin of the limit state function, when the
616:
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
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360:
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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
430: 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
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function. Conceptually, this is achieved by averaging the result of different FORM simulations. In practice, this is made possible by identifying the importance direction
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443:
222:{\displaystyle p_{f}({\boldsymbol {x}})=\int _{-\infty }^{+\infty }I({\boldsymbol {x}}+\beta \cdot {\boldsymbol {\alpha }})\varphi (\beta )\,d\beta }
88:
Firstly the importance direction must be determined. This can be achieved by finding the design point, or the gradient of the limit state function.
27:, in which the performance function exhibits moderate non-linearity with respect to the uncertain parameters The method is suitable for analyzing
692:
Schueller, G. I.; Pradlwarter, H. J.; Koutsourelakis, P. (2004). "A critical appraisal of reliability estimation procedures for high dimensions".
639:
808:
Patelli, E; de
Angelis, M (2015). "Line sampling approach for extreme case analysis in presence of aleatory and epistemic uncertainties".
355:{\displaystyle I_{f}({\boldsymbol {x}})={\begin{cases}1&{\text{if }}{\boldsymbol {x}}\in \Omega _{f}\\0&{\text{else}}\end{cases}}}
23:
to compute small (i.e., rare event) failure probabilities encountered in engineering systems. The method is particularly suitable for
719:
de
Angelis, Marco; Patelli, Edoardo; Beer, Michael (2015). "Advanced Line Sampling for efficient robust reliability analysis".
858:
825:
775:
841:
Patelli, Edoardo (2016). "COSSAN: A Multidisciplinary
Software Suite for Uncertainty Quantification and Risk Management".
624:
An illustration of the line sampling algorithm. Two line samples are shown approaching the limit state surface.
43:
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20:
650:. A numerical implementation of the method is available in the open source software OpenCOSSAN.
235:
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117:, the probability of failure in the line parallel to the important direction is defined as:
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The global probability of failure is the mean of the probability of failure on the lines:
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794:
Efficient Random Set
Uncertainty Quantification by means of Advanced Sampling Techniques
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756:"Subset simulation and line sampling for advanced Monte Carlo reliability analysis"
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538:{\displaystyle {\tilde {p}}_{f}={\frac {1}{N_{L}}}\sum _{i=1}^{N_{L}}p_{f}^{(i)}}
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613: are the partial probabilities of failure estimated along all the lines.
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The basic idea behind line sampling is to refine estimates obtained from the
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575: is the total number of lines used in the analysis and the
46:(FORM), which may be incorrect due to the non-linearity of the
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638:. The algorithm can also be used to efficiently propagate
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of the problem have been transformed into the standard
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379:{\displaystyle {\boldsymbol {\alpha }}}
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65:{\displaystyle {\boldsymbol {\alpha }}}
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843:Handbook of Uncertainty Quantification
25:high-dimensional reliability problems
91:A set of samples is generated using
694:Probabilistic Engineering Mechanics
386: is the important direction,
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797:(Ph.D.). University of Liverpool.
706:10.1016/j.probengmech.2004.05.004
110:{\displaystyle {\boldsymbol {x}}}
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760:Reliability, Risk, and Safety
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754:Zio, E; Pedroni, N (2009).
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606:{\displaystyle p_{f}^{(i)}}
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791:De Angelis, Marco (2015).
768:10.1201/9780203859759.ch94
254:{\displaystyle I(\cdot )}
399:{\displaystyle \varphi }
31:systems, and unlike the
665:Curse of dimensionality
21:reliability engineering
812:. pp. 2585–2593.
629:Industrial application
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423:{\displaystyle \beta }
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727:: 170–182.
648:random sets
48:limit state
875:Categories
676:References
35:method of
741:0167-4730
488:∑
455:~
418:β
394:φ
373:α
323:Ω
319:∈
246:⋅
217:β
207:β
201:φ
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190:⋅
187:β
168:∞
160:∞
157:−
153:∫
59:α
29:black box
654:See also
310:if
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232:where
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