We design a new stochastic algorithm (called SALT) that sequentially approximates a given function in L2 w.r.t. a probability measure, using a finite sample of the distribution. By increasing the sets of approximating functions and the simulation effort, we compute a L2-approximation with higher and higher accuracy. The simulation effort is tuned in a robust way that ensures the convergence under rather general conditions. Then, we apply SALT to build efficient control variates for accurate numerical integration. Examples and numerical experiments support the mathematical analysis.
