The point‐estimate method with large numbers of variables

Abstract

SUMMARYRosenbleuth's point-estimate method has become widely used in geotechnical practice for reliability calculations. Although the point-estimate method is a powerful and simple method for evaluating the moments of functions of random variables, it is limited by the need to make 2 n evaluations when there are n random variables. Modifications of the method reduce this to 2n evaluations by using points on the diameters of a hypersphere instead of at the corners of the inscribed hypercube. However, these techniques force the co-ordinates of the evaluation points farther from the means of the variables; for a bounded variable, the points may easily fall outside the domain of definition of the variable. The problem can be avoided by using other techniques for some special cases or by reducing the number of random variables that must be considered. Copyright # 2002 John Wiley & Sons, Ltd. THE BASIC METHODThe point-estimate method, originally proposed by Rosenblueth [1,2], is a simple but powerful technique for evaluating the moments of functions of random variables, and has been adopted in many geotechnical reliability analyses [3]. Miller and Rice [4] and Christian and Baecher [5] have shown that the point-estimate method is a form of Gaussian quadrature. Despite its simplicity, it can be accurate in many practical situations [3][4][5]. In this paper the authors describe current methods for dealing with the computational burdens that arise when the number of variables becomes large and show that these methods are themselves problematical. Examples from geotechnical reliability practice are used to illustrate both difficulties and alternative approaches.As in any Gaussian quadrature scheme, including more points in the calculation increases the order of polynomial functions that are integrated exactly, but the most widely used form of the method employs two points per variable. When there is one independent variable X ; the two values of X where calculations are made are chosen such that

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