Monte Carlo method via a numerical algorithm to solve a parabolic problem

Markov chain Monte Carlo method to solve Fredholm integral equations

“…From the numerical examples it can be seen that the proposed Monte Carlo method is efficient and accurate to estimate the solution of Fredholm integral eqs. (1).…”

“…The Monte Carlo method is preferable for solving both high-dimensional multiple integrals and large sparse systems of linear algebraic equations [1][2][3]. Recently, Farnoosh and Ebrahimi [4] employed Monte Carlo method based on the simulation of a continuous Markov chain for solving Fredholm integral equations of the second kind.…”

Combined probabilistic algorithm for solving high dimensional problems

“…; n 0 ; g [ ð0; 1; n 0 ¼ ð2 n mÞ 2 2 1. It is called the Jacobi over-relaxation method with relaxation parameter g [11,12]. Consider…”

“…In this case for the large sparse LSAEs, they are more efficient than iterative or direct methods. For example the time of the algorithm (or computational complexity) of Gaussian elimination and Gauss -Jordan methods for solving LSAEs (1) is Oðn 3 Þ [12]. For iterative methods such as Jacobi, Gauss -Sidel and relaxation methods, the computational complexity is of order Oðn 2 kÞ in which k is the number of iterations [11].…”

“…One of the well-known stochastic algorithms which is preferable for solving LSAEs, such as those obtained from discretization of differential equations, is the MC method [12]. This method is organized based on simulation of Markov chains.…”