On stochastic and quasistochastic methods for solving equations

A modification of the quasi-Monte Carlo method previously suggested by the authors is discussed. This modification is successfully applied to solve integral equations of the second kind thanks to a number of advantages over the quasi-Monte Carlo method. Thus, solving these equations by the quasi-Monte Carlo method requires estimating the sum of a series whose terms are integrals with infinitely increasing constructive dimension. As is known, this is one of the main factors hindering the application of the quasi-Monte Carlo method. Another difficulty is the dominated convergence condition, which ensures the absolute convergence of the series whose sum is to be estimated.The modification under consideration makes it possible to overcome the first difficulty and relax the dominated convergence condition.

Two families of estimates are suggested, which can be applied in the modified algorithm of the quasi-Monte Carlo method. The integral equation

$phi (x) = int {k(x,y)phi (y)mu (dy) + f(x)(bmod mu ),} $

is considered, where x ∈ D ? ? ^s and f and k are given functions defined on the supports of the measures μ and μ ? μ. The values of ? _n (x) = ∫k(x, y)?_{n - 1}(y)μ(dy) + f(x)are estimated step by step. The first group of estimates serves to evaluate ? _n at a fixed point x’:

$xi _1 (x') = frac{1}{N}sumlimits_{j = 1}^N {xi _1^n (y_j )} ,wherexi _1^n (y) = frac{{k(x',y)hat phi _{n - 1} (y)}}{{p_{n - 1} (y)}} + f(x'),$

here, the y _j are distributed with density p _n?1 and (hat phi _{n - 1} (y)) is the estimate obtained at the preceding step. The other group of estimates makes it possible to evaluate ? _n at random points.

Optimal parameters of the estimates are found and the corresponding theorems are proved.The theory has been verified on the example of a difference analogue of the Navier-Stokes equation; experimental results are presented.