Variance Reduction Methods and Multilevel Monte Carlo Strategy for Estimating Densities of Solutions to Random Second-Order Linear Differential Equations

Abstract

This paper concerns the estimation of the density function of the solution to a random non-autonomous second-order linear differential equation with analytic data processes. In a recent contribution, we proposed to express the density function as an expectation, and we used a standard Monte Carlo algorithm to approximate the expectation. Although the algorithms worked satisfactorily for most test-problems, some numerical challenges emerged for others, due to large statistical errors. In these situations, the convergence of the Monte Carlo simulation slows down severely, and noisy features plague the estimates. In this paper, we focus on computational aspects and propose several variance reduction methods to remedy these issues and speed up the convergence. First, we introduce a path-wise selection of the approximating processes which aims at controlling the variance of the estimator. Second, we propose a hybrid method, combining Monte Carlo and deterministic quadrature rules, to estimate the expectation. Third, we exploit the series expansions of the solutions to design a multilevel Monte Carlo estimator. The proposed methods are implemented and tested on several numerical examples to highlight the theoretical discussions and demonstrate the significant improvements achieved.

Extending the applicability of the RVT technique for the randomized radioactive decay chain model

“…Finally, notice that, for a fixed M , the kernel density estimation and the parametric estimation have practically the same cost, since both rely on M realizations of Z 1 and Z 2 (the kernel density estimation also needs realizations of A, evaluates the kernel function and considers a bandwidth, while the parametric estimation evaluates f A ). Despite all the favorable properties of the parametric estimation method, there are some situations, which were not analyzed in [12], in which it may present slow convergence: since (2) involves random denominators, the variance of the random quantity inside the expectation may be high for some z, which produces "noise" that plagues the PDF estimate [17,18]. This issue is not observed for kernel density estimation.…”