01436922

j

01436922 Arsenjev, D.G. Ivanov, V.M. Kul'chitsky, O.Yu. Adaptive control of stochastic calculating processes. J. Stat. Plann. Inference 85, No.1-2, 213-226 (2000). 2000 Elsevier Science B.V. (North-Holland), Amsterdam EN adaptive control stochastic calculating process Monte-Carlo procedure adaptive stochastic learning multidimensional integration

doi:10.1016/S0378-3758(99)00082-8

The authors propose adaptive control methods for Monte-Carlo procedures of multidimensional integration and numerical solution of Fredholm-type integral equations. Let, for example, the $n$-dimensional integral $ J=\int_D f(x) dx=E\{f(\xi)/p(\xi)\} $ over a bounded closed set $D\subset\bbfR^n$ be evaluated. Here $\xi$ is a random variable with values in $D$ distributed with a density $p=p(x)>0$, $x\in D$. Suppose that the estimate $\widehat J_N$ of $J$ is represented in the form $\widehat J_N=\widehat\theta^T_N\int_D\psi(x)p(x) dx$, where the integral $\int_D\psi(x)p(x) dx$ is considered to be known. The authors use a recurrent system of equations satisfied by $\widehat\theta_N$ as a stochastic object of control with the control action represented by a function of unit distribution density of a random grid of integration. The criterion of evaluation accuracy is chosen to be the criterion of optimal functioning. Vigirdas Mackevi\v{c}ius (Vilnius)