The Latin hypercube method was described in 79' by researchers in scientific computing and is quite popular in data science. The objective of the Latin hypercube method is to integrate a function on the hypercube [0,1]^m where the integration points are drawn at random, but on a cartesian grid. It is an evolution to the Monte-Carlo method and belongs to the general family of Lattice rules (Korobov 59') and quasi Monte-Carlo methods.
I will show a non probabilistic proof of the efficiency of the Latin hypercube method. The final formula is obtained by exact calculations classical in numerical analysis, without any need of random variable of any kind. The final formula shows convergence independantely of the dimension m. Even if this convergence can be interpreted in the mean so in the sense of probability, it is nevertheless only discrete probability like throwing a dice.
Application to simple problems an to the design of new finite volume solvers obtained by training will serve as as illustration.