The variance of the estimator for the probability of failure can be reduced by in troducing analytical partial solutions. This is done in the Directional Sampling procedure (Deák 1980), (Bjerager 1988). The original random variables are trans- formed into the space of standard Gaussian variables. A random vector in standard Gaussian space is represented in polar coordinates
 | (5–24) |
with A being a unit vector, and R the Euclidian norm of U. With this, Equation 5–4 can be re-written as
 | (5–25) |
Due to the transformation of the standard Gaussian variables from Cartesian into polar coordinates, the unit vectors A are uniformly distributed over the surface of a hypersphere with radius 1, centered at the origin. The radii R are independent of A and follow a X 2-distribution with k degrees of freedom, k being the dimension.
The procedure is to generate unit vectors a
i, for each of which the distance of the failure surface to the
origin 
is found by a suitable iteration. Then the conditional probability of
failure given a direction ai is computed as
 | (5–26) |
and the total failure probability is the mean of the conditional failure probabilities with respect to the random vectors A.
 | (5–27) |
 | (5–28) |
A basic prerequisite for Directional Sampling is, that the mean vector of the original
random variables 
has to lie within the safe domain, and the limit surface in each
direction shall be unique. Failure should be formulated in a way that the iteration is
able to operate in the failure domain (no premature exit from the solver). However,
because of the monotonic decrease with increasing distance to the mean vector of the
joint density function, a non-unique failure surface may be acceptable, as long as the
closest root of 
is found.
The Directional Sampling method is sensitive towards dimension. It does not put any requirements on smoothness nor continuity of the limit state function. Moreover, the implementation in optiSLang is able to handle a moderate number of unsuccessful solver calls.
Figure 5.5: Parabola Example: Anthill Plot of Directional Sampling Procedure shows the Directional Sampling results for the parabola example introduced in Adaptive Sampling (ADSAP). A bisection algorithm is used to iterate the failure surface. The iteration does not exceed a distance to the mean vector of eight standard deviations.