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

Abstract

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.

Simulating diffusions with piecewise constant coefficients using a kinetic approximation

“…With Remark 2, it can provide the law of the exit time from the domain, which is crucial in the computation of the principal eigenelements of the operator by means of Monte Carlo methods [28,29]. In a recent work [5] it has also been used successfully for solving the Poisson-Boltzmann equation of molecular electrostatics for which the finite differences method was originally designed.…”

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

“…A first idea is to estimate F N ðtÞ at two times t 0 and t 1 , and use the difference of the values at these times to estimate k 1 [28,32]. Another possible approach, developed in [25], is to find a time window ½t 0 ; t 1 , in which F N ðtÞ is a good approximation of FðtÞ. A last approach is to note that for t > t 0 and c ¼ 0, the exit time, s, from D is distributed like an exponential random variable with parameter Àk 1 .…”

“…The purpose of this article is to improve the results of [32] and [25]. To achieve this improvement, we propose a variance reduction scheme for the empirical approximation of FðtÞ which is very easy to implement.…”

“…We fix T > 0 and we get compute the estimator b p T of p T using a Monte Carlo method, then we simulate the first exit time, s, from D for the process X with b p T as initial distribution and we compute k 1 using the methods previously discussed in [32,25].…”

Simulation of exit times and positions for Brownian motions and Diffusions

“…This technique could be applied to solve many linear problems arising in mathematical modelling, such as solving Poisson equations or bi-harmonic equations, evaluating effective coefficients in geophysics [6], computing the first eigenvalue of the Laplace operator [7,8], computing barrier options in finance etc.…”