Recursive Estimation of a Failure Probability for a Lipschitz Function

Let g:Ω=[0,1] d →ℝ denote a Lipschitz function that can be evaluated at each point, but the price of heavy computational time. X stand for random variable with values in Ω such one is able to simulate, least approximately, according restriction law any subset Ω. For example, thanks Markov chain Monte Carlo techniques, this always possible when admits density known up normalizing constant. In context, given deterministic threshold T failure probability p:=ℙ(g(X)>T) may very low, our goal estimate latter minimal number calls g. aim, building on Cohen et al. [9], we propose recursive and (in certain sens) optimal algorithm selects fly areas interest estimates their respective probabilities.