A Robust and Accurate Quasi-Monte Carlo Algorithm for Estimating Eigenvalue of Homogeneous Integral Equations

We present an efficient numerical algorithm for computing the eigenvalue of the linear homogeneous integral equations. The proposed algorithm is based on antithetic Monte Carlo algorithm and a low-discrepancy sequence, namely, Faure sequence. To reduce the computational time we reduce the variance by using the antithetic variance reduction procedure. Numerical results show that our scheme is robust and accurate. 1. Introduction In Monte Carlo (MC) methods the random variables are simulated by computer generated pseudorandom sequences, and the numerical solution is performed by averaging over a large number of simulations. Pseudorandom numbers mimic the realizations of independent identically distributed random variables. Due to a fundamental result in probability theory, Monte Carlo methods based on such pseudorandom numbers can converge only proportionally to the square root of the number of simulations. It was noticed that the randomness of the simulating sequence is not essential for numerical integration. Deterministic sequences which fill the space uniformly can also be used. One of the most powerful methods for improving Monte Carlo simulation is to use low-discrepancy numbers [1]. Therefore, instead of using random numbers, we can employ a deterministic sequence of numbers. Such a sequence is called a low-discrepancy sequence. The great advantage of low-discrepancy sequences is that their rate of convergence is rather than , which greatly increases the competitiveness of Monte Carlo simulations. Using low-discrepancy sequences to carry out Monte Carlo algorithm is sometimes called quasi-Monte Carlo (QMC) algorithm [1]. The Faure sequence is one of the well-known quasirandom sequences used in quasi-Monte Carlo applications. In [2, 3] details of Faure sequence and its various scrambling methods are described. The problem of using Monte Carlo and quasi-Monte Carlo methods for finding an eigenpair of matrices has been extensively studied [4]. In [5] a numerical method for finding the eigenvalue of the linear homogenous integral equation is proposed by applying antithetic Monte Carlo method to obtain a reliable scheme. In this paper we present a new numerical method based on antithetic variance reduction method and quasi-Monte Carlo algorithm on scrambled Faure sequence for estimating the eigenvalue of the following linear homogenous integral equation: It can be written in the form of If , then and are called the eigenvalue and eigenfunction of the above equations, respectively. Also, is called the kernel of this integral equation. Throughout this