Variance reduction for numerical integration and for critical neutron transport calculations.

Summary This thesis is devoted to Monte-Carlo methods and more particularly to variance reduction. In the first part, we study a probabilistic algorithm, based on an iterative use of the method of control variables, allowing the calculation of quadratic approximations. Its use in dimension one for regular functions using the Fourier basis after periodization, the bases of orthogonal Legendre and Tchebychef polynomials, provides estimators with an increased order of convergence for Monte-Carlo integration. It is then extended to the multidimensional setting by a judicious choice of basis functions, allowing to attenuate the dimensional effect. Numerical validation is performed on many examples and applications. The second part is devoted to the study of the critical regime in neutron transport. The method developed consists in numerically calculating the principal eigenvalue of the neutron transport operator by combining the asymptotic development of the solution of the associated evolution problem with the calculation of its probabilistic interpretation by a Monte-Carlo method. Different techniques of variance reduction are implemented in the study of many homogeneous and inhomogeneous models. A probabilistic interpretation of the principal eigenvalue is given for a particular homogeneous model.