Sylvain Maire

Reducing variance using iterated control variates

In this paper we describe a new variance reduction method for Monte Carlo integration based on an iterated computation of L^{2} approximations using control variates. This computation leads to non linear unbiased estimates for each of the coefficients of the truncated L^{2} expansion. We give estimations of the variance of these estimates without further hypotheses on the approximation basis. We study especially the convergence of our algorithm in the case of a polynomial decay of these coefficients. As an application, regular monodimensional functions will be approximated using a Fourier basis on periodised functions, Legendre and Tchebychef polynomial L^2 approximations. The order of our method will appear to be almost optimal in this case. Numerical examples will be given as a comparison with standard Monte Carlo estimates.

An iterative computation of approximations on Korobov-like spaces

This paper treats the multidimensional application of a previous iterative Monte Carlo algorithm that enables the computation of approximations in L2. The case of regular functions is studied using a Fourier basis on periodised functions, Legendre and Tchebychef polynomial bases. The dimensional effect is reduced by computing these approximations on Korobov-like spaces. Numerical results show the efficiency of the algorithm for both approximation and numerical integration.

Computing the principal eigenvalue of the Laplace operator by a stochastic method

We describe a Monte Carlo method for the numerical computation of the principal eigenvalue of the Laplace operator in a bounded domain with Dirichlet conditions. It is based on the estimation of the speed of absorption of the Brownian motion by the boundary of the domain. Various tools of statistical estimation and different simulation schemes are developed to optimize the method. Numerical examples are studied to check the accuracy and the robustness of our approach.

New Monte Carlo schemes for simulating diffusions in discontinuous media

We introduce new Monte Carlo simulation schemes for diffusions in a discontinuous media divided in subdomains with piecewise constant diffusivity. These schemes are higher order extensions of the usual schemes and take into account the two dimensional aspects of the diffusion at the interface between subdomains. This is achieved using either stochastic process techniques or an approach based on finite differences. Numerical tests on elliptic, parabolic and eigenvalue problems involving an operator in divergence form show the efficiency of these new schemes.

Some new simulations schemes for the evaluation of Feynman-Kac representations

Abstract We describe new variants of the Euler scheme and of the walk on spheres method for the Monte Carlo computation of Feynman–Kac representations. We optimize these variants using quantization for both source and boundary terms. Numerical tests are given on basic examples and on Monte Carlo versions of spectral methods for the Poisson equation. We especially introduce a new stochastic spectral formulation with very good properties in terms of conditioning.

A spectral Monte Carlo method for the Poisson equation

Using a sequential Monte Carlo algorithm, we compute a spectral approximation of the solution of the Poisson equation in dimension 1 and 2. The Feyman-Kac computation of the pointwise solution is achieved using either an integral representation or a modified walk on spheres method. The variances decrease geometrically with the number of steps. A global solution is obtained, accurate up to the interpolation error. Surprisingly, the accuracy depends very little on the absorption layer thickness of the walk on spheres.

Computing the principal eigenelements of some linear operators using a branching Monte Carlo method

In earlier work, we developed a Monte Carlo method to compute the principal eigenvalue of linear operators, which was based on the simulation of exit times. In this paper, we generalize this approach by showing how to use a branching method to improve the efficacy of simulating large exit times for the purpose of computing eigenvalues. Furthermore, we show that this new method provides a natural estimation of the first eigenfunction of the adjoint operator. Numerical examples of this method are given for the Laplace operator and an homogeneous neutron transport operator.

Polynomial approximations of multivariate smooth functions from quasi-random data

We improve a Monte Carlo algorithm which computes accurate approximations of smooth functions on multidimensional Tchebychef polynomials by using quasi-random sequences. We first show that the convergence of the previous algorithm is twice faster using these sequences. Then, we slightly modify this algorithm to make it work from a single set of random or quasi-random points. This especially leads to a Quasi-Monte Carlo method with an increased rate of convergence for numerical integration.

Adaptive integration and approximation over hyper-rectangular regions with applications to basket option pricing

Abstract We describe an adaptive algorithm to compute piecewise sparse polynomial approximations and the integral of a multivariate function over hyper-rectangular regions in medium dimensions. The key ingredient is a quasi-Monte Carlo quadrature rule which can handle the numerical integration of both very regular and less regular functions. Numerical tests are performed on functions taken from Genz package in dimensions up to 5 and on basket options pricing.

Stochastic Spectral Formulations for Elliptic Problems

We describe new stochastic spectral formulations with very good properties in terms of conditioning. These formulations are built by combining Monte Carlo approximations of the Feynman-Kac formula and standard deterministic approximations on basis functions. We give error bounds on the solutions obtained using these formulations in the case of linear approximations. Some numerical tests are made on an anisotropic diffusion equation using a tensor product Tchebychef polynomial basis and one random point schemes quantized or not.