Introduction to Monte-Carlo Methods
نویسنده
چکیده
converges to E(f(U)) almost surely when N tends to infinity. This suggests a very simple algorithm to approximate I: call a random number generator N times and compute the average (??). Observe that the method converges for any integrable function on [0, 1] : f is not necessarily a smooth function. In order to efficiently use the above Monte-Carlo method, we need to know its rate of convergence and to determine when it is more efficient than deterministic algorithms. The Central Limit Theorem provides the asymptotic distribution of √ N(SN − I) when N tends to +∞. Various refinements of the Central Limit Theorem, such as Berry-Essen and Bikelis theorems, provide non asymptotic estimates. The preceding consideration shows that the convergence rate of a Monte Carlo method is rather slow (1/ √ N). Moreover, the approximation error is random and may take large values even if N is large (however, the probability of such an event tends to 0 when N tends to infinity). Nevertheless, the Monte-Carlo methods are useful in practice. For instance, consider an integral in a hypercube [0, 1], with d large (d = 40, e.g.). It is clear that the quadrature methods require too many points (the number of points increases exponentially with the dimension of the space). Low discrepancy sequences are efficient for moderate value of d but this efficiency decreases drastically when d becomes large (the discrepancy behaves like C(d) log (N) N where the constant C(d) may be extremely large.). A Monte-Carlo method does not have such disadvantages : it requires the simulation of independent random vectors (X1, . . . , Xd), whose coordinates are independent. Thus, compared to the computation of the one-dimensional situation, the number of trials is multiplied by d only and therefore the method remains tractable even when d is large. In addition, another advantage of the Monte-Carlo methods is their parallel nature: each processor of a parallel computer can be assigned the task of making a random trial.
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