Weighted Monte Carlo algorithms with branching corresponding member of the RAS (Q954187)

The present paper describes and studies algorithms for numerical statistical simulation with a trajectory branching when the current weighting factor exceeds unity. The weights of individual branches do not exceed unity and the variance of the estimate of the computed function is finite. In the introduction the integral equation \(\varphi(x) = \int_{X}k(x,x^{\prime})\varphi(x^{\prime}) d x^{\prime} + h(x),\) where \(X = {\mathbb R}^{n},\) \(h \in L_{\infty}(X)\) and the kernel \(k \in [L_{\infty} \to L_{\infty}]\) is considered. The functions \(k(x,x^{\prime}),\) \(h(x)\) and \(\varphi(x)\) are non-negative. To construct Monte Carlo estimates a homogeneous Markov chain with transition density is introduced. The so-called collision estimate of the Neumann series of \(\varphi\) is constructed. In part 2 the integer random variable \(\nu(x,x^{\prime})\) regarded as the number of branches is defined. The random variable \(\zeta_{x} = h(x) + \delta_{x}\sum_{k=1}^{\nu(x,x^{\prime})}\zeta_{x^{\prime}}^{(k)},\) where \(\delta_{x}\) is the indicator and \(\{\zeta_{x^{\prime}}^{(k)}\}\) are independent realization of \(\zeta_{x^{\prime}}\) is defined. In Theorem 1 an expression of the expectation \(E \zeta_{x_{0}}^{2}\) by a Neumann series is given. In part 3 the problem of computing the reproduction factor of particles by a medium contained in a convex domain, outside of which there is an absorbent is considered. In part 4 the boundary value problem \(\Delta u + c u = -g,\) \(u |_{\Gamma} = v\) in a domain \(D \in {\mathbb R}^{3}\) with boundary \(\Gamma\) is considered. The concept of the walk on spheres (WOS) is developed. The WOS collision estimate \(\xi_{\epsilon}\) is obtained. The mean number of branches for this algorithm is bounded by \({\mathcal O}(\epsilon^{-2}).\)