Subset inverse sampling on an R-S model

This example demonstrates SubsetInverseSampling on a simple Resistance-Stress (R-S) model.

The limit state function is:

where represents the resistance and represents the stress. Failure occurs when .

The subset inverse simulation algorithm estimates the threshold value such that:

where is a target probability. This is the inverse problem: instead of computing the probability for a given threshold, we find the threshold for a given probability.

import openturns as ot
import otrobopt

ot.RandomGenerator.SetSeed(0)

Define the limit state function and the probabilistic model.

limitState = ot.SymbolicFunction(['u1', 'u2'], ['u1 - u2'])
dim = limitState.getInputDimension()

mean = [7.0, 2.0]
sigma = [1.0, 1.0]
R = ot.IdentityMatrix(dim)
distribution = ot.Normal(mean, sigma, R)

# Build the event
vect = ot.RandomVector(distribution)
output = ot.CompositeRandomVector(limitState, vect)
event = ot.ThresholdEvent(output, ot.Less(), 0.0)

Run the subset inverse sampling algorithm for a target probability .

targetProbability = 0.0002

algo = otrobopt.SubsetInverseSampling(event, targetProbability)
algo.setMaximumOuterSampling(10000)
algo.run()

Retrieve and display the results.

result = algo.getResult()
pf = result.getProbabilityEstimate()
threshold = algo.getThresholdPerStep()[-1]
threshold_cl = algo.getThresholdConfidenceLength(0.90)

print('Target probability =', targetProbability)
print('Estimated probability =', pf)
print('Threshold at target =', threshold)
print('Threshold confidence length (90%) =', threshold_cl)
print('Number of steps =', algo.getNumberOfSteps())
print('Coefficient of variation =', result.getCoefficientOfVariation())

Target probability = 0.0002
Estimated probability = 0.00020000000000000025
Threshold at target = 0.020271550198725088
Threshold confidence length (90%) = 0.02133651921248595
Number of steps = 4
Coefficient of variation = 0.08323641030222287