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###   Meta Model I18N string                       #
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dispname=Importance Sampling
desc=Importance Sampling
longdesc=\
    <b>Importance Sampling Reliability Method</b>\
	<br><br> \
	<i>Importance sampling</i> is a type of Monte Carlo \
	reliability method where a distribution other than the \
	original distribution is used to compute the probability \
	of failure more efficiently. Given one or more identified \
	random variables, the focus in structural reliability analysis \
	is to assess the <i>probability of failure</i> - the \
	probability of violating a constraint - of a structural component \
	or system, resulting from performance (output) variation caused \
	by the variation of uncertain, random (input) variables. The \
	structural reliability is then defined as the probability of \
	satisfying a constraint, and is equal to 1-probability of failure. \
	<br><br> \
	Importance sampling, like First Order Reliability Method (FORM), \
	computes the Most Probable Point, i.e. the point on the constraint \
	that is closest to the design point in the standard normal space. \
	A transformation is used to map the original random vector \
	<b>X</b> (in X-space) to the standard, uncorrelated normal vector \
	using <b>U</b>=T(<b>X</b>). \
	<br><br> \
	Importance sampling then generates a set of designs around the MPP, \
	unlike the traditional Monte Carlo method that uses the \
	Mean Value Point (MVP), to estimate the probability of failure. \
	In other words, the mean value of this new sampling density \
	function is the MPP, whereas the variance is the same as the \
	original density function. If \u03c6(\u03bc, \u03c3, x) is the \
	standard normal probability density function (pdf) where \u03bc \
	is the mean and \u03c3<sup>2</sup> is the variance, then pdf of the sample \
	set is \u03c6(MPP, \u03c3, x) while that of the original problem is \
	\u03c6(MVP, \u03c3, x).\
	<br><br> \
	Simple importance sampling estimates the probability of failure as follows: \
	<br> \
	P<sub>f</sub> = (1/N) \u03a3<sup>N</sup><sub>i=1</sub> \
	I(x<sub>i</sub>) \u03c6(MVP,\u03c3, x<sub>i</sub>) = \
	(1/N) \u03a3<sup>N</sup><sub>i=1</sub> \
	I(y<sub>i</sub>) \u03c6(MVP,\u03c3, y<sub>i</sub>) / \
	\u03c6(MPP,\u03c3, y<sub>i</sub>) \
	 <br>\
	 where x<sub>i</sub> are N samples drawn around the MVP, \
	 y<sub>i</sub> are N samples drawn around the MPP and I(x or y) \
	 is an indicator function that is 1 when g(x or y) \u003c 0 and 0 otherwise. \
	 <br><br> \
	 FORM uses a linear approximation of the constraint surface to \
	 compute the probability of failure as:\
	 <br>\
	 P<sub>f, FORM</sub> = \u03A6(-\u03B2) \
	 <br>\
	 where \u03A6 is the standard normal distribution function and \u03B2 \
	 is the distance to the MPP. If the failure function is nonlinear, \
	 or the random variables are not normally distributed, the FORM \
	 estimate can be corrected using importance sampling as follows: \
	 <br>\
	 P<sub>f</sub> = P<sub>f, FORM</sub> + (1/N) \
	 \u03a3<sup>N</sup><sub>i=1</sub>( I(y<sub>i</sub>)- \
	 I<sub>FORM</sub>(y<sub>i</sub>)) \u03c6(MVP,\u03c3, y<sub>i</sub>) / \
	 \u03c6(MPP,\u03c3, y<sub>i</sub>) \
	 <br> \
	 where  I<sub>FORM</sub>(y) is the indicator function of the linear \
	 approximation used by FORM. \
	 <br><br> \
	 Adaptive Importance Sampling samples mainly in the failure region to \
	 compute the probability of failure. The failure region is approximated \
	 by a second order polynomial (like SORM) using principle curvatures. \
	 The curvature values are updated iteratively to cover the failure region.

prop.dispname.importancesamplingmethod=Importance Sampling Method
prop.dispname.numberofsamplepoints=Number of Sample Points
prop.dispname.correctformestimate=Correct FORM Estimate
prop.dispname.aismaxiterations=AIS Maximum Iterations
prop.dispname.mppmaxiterations=MPP Maximum Iterations
prop.dispname.finitedifferencestepsize=Finite Difference Step Size
prop.dispname.minimumfinitedifferencestep=Minimum Finite Difference Step
prop.dispname.mppabsconvergence=MPP Absolute Convergence
prop.dispname.mpprelconvergence=MPP Relative Convergence
prop.dispname.fixedrandomseed=Use fixed random seed
prop.dispname.randomseed=Random seed value
