A Study of Biased and Unbiased Stochastic Algorithms for Solving Integral Equations

In this paper we propose and analyse a new unbiased stochastic method for solving a class of integral equations, namely the second kind Fredholm integral equations. We study and compare three possible approaches to compute linear functionals of the integral under consideration: i) biased Monte Carlo method based on evaluation of truncated Liouville-Neumann series, ii) transformation of this problem into the problem of computing a finite number of integrals, and iii) unbiased stochastic approach. Five Monte Carlo algorithms for numerical integration have been applied for approach (ii). Error balancing of both stochastic and systematic errors has been discussed and applied during the numerical implementation of the biased algorithms. Extensive numerical experiments have been performed to support the theoretical studies regarding the convergence rate of Monte Carlo methods for numerical integration done in our previous studies. We compare the results obtained by some of the best biased stochastic approaches with the results obtained by the proposed unbiased approach. Conclusions about the applicability and efficiency of the algorithms have been drawn

A New Walk on Equations Monte Carlo Method for Linear Algebraic Problems

International audienceA new Walk on Equations (WE) Monte Carlo algorithm for Linear Algebra (LA) problem is proposed and studied. This algorithm relies on a non-discounted sum of an absorbed random walk. It can be applied for either real or complex matrices. Several techniques like simultaneous scoring or the sequential Monte Carlo method are applied to improve the basic algorithm. Numerical tests are performed on examples with matrices of different size and on systems coming from various applications. Comparisons with standard deterministic or Monte Carlo algorithms are also done

Sexual maturation in adolescents from the region of Eastern Rodopi in connection with their social environment

The purpose of this paper is to characterize the influence of socio-economic environment on the processes of sexual maturation and the age of adolescent's menarche from the Eastern Rhodopi. Subjects of study are 1378 children and adolescents from local schools in Ivaylovgrad, Svilengrad and Lyubimets. These are 730 girls and 648 boys aged 8 to 17 years. In connection with studying the influence of environmental factors on the sexual maturation processes, interviews with parents were conducted . The survey contains 22 questions with parents' data, which were used to clarify the social, economic and professional status, living conditions of family life, etc. The results show that although genetically determinated, the processes of sexual maturation are significantly influenced by exogenous factors, and in a peculiar way they reflect the specific social and economic situation in a given geographical area at a randomly selected time period