Monte Carlo integration with adaptive variance reduction: an asymptotic analysis
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The crude Monte Carlo approximates the integral S(f)=∫_a^b f(x) dx with expected error (deviation) σ(f)N^-1/2, where σ(f)^2 is the variance of f and N is the number of random samples. If f∈ C^r then special variance reduction techniques can lower this error to the level N^-(r+1/2). In this paper, we consider methods of the form M_N,r(f)=S(L_m,rf)+M_n(f-L_m,rf), where L_m,r is the piecewise polynomial interpolation of f of degree r-1 using a partition of the interval [a,b] into m subintervals, M_n is a Monte Carlo approximation using n samples of f, and N is the total number of function evaluations used. We derive asymptotic error formulas for the methods M_N,r that use nonadaptive as well as adaptive partitions. Although the convergence rate N^-(r+1/2) cannot be beaten, the asymptotic constants make a huge difference. For example, for ∫_0^1(x+d)^-1dx and r=4 the best adaptive methods overcome the nonadaptive ones roughly 10^12 times if d=10^-4, and 10^29 times if d=10^-8. In addition, the proposed adaptive methods are easily implementable and can be well used for automatic integration. We believe that the obtained results can be generalized to multivariate integration.
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