Metodo de Monte Carlo y su aplicación a ecuaciones diferenciales parciales elípticas (2017)
The Monte Carlo Method is a numerical method for solving physical and mathematical problems through the simulation of random variables. The Monte Carlo method was named for its analogy with roulette games in casinos, the most famous of which is the Monte Carlo casino whose built in 1856 by Prince Charles III of Monaco, 1861. The great importance of the Monte Carlo Method lies in the fact that it can attack problems very complicated that are linked to random processes or that can be associated with a probabilistic artificial structure (solving integrals of several variables, minimizing functions, etc.) with a relatively simple algorithm. Thanks to advances in computational constructions, with the Monte Carlo Method today you can solve problems that would have been unthinkable. In these methods, an approximate error of 1 P N Where N is the number of iterations and thus to gain an additional tenth of an approximation you need to increase by 100 times To N. The problems we will address here are the ordinary equations and elliptic equations, whose theories of existence and regularity have been studied extensively, see [5]. In this work we will study a Monte Carlo method for finding the numerical solution of a linear elliptic equation. This thesis Is structured as follows. In the first chapter we describe some elementary notions of probability that will be used in this work. In Chapter 2 we give a review of the finite differences and their relationship with Markov chains. In chapter 3, we give a simple example where the Monte Carlo method is used, one of them is the calculation of integrals, and another the resolution of a boundary problem for an ordinary differential equation (here we make use of the famous ruin game). In chapter 4 we apply the Monte Carlo Method to differential equations ellipticals. It should be noted that this paper shows results that although they are in the literature, have been worked independently by the author (unless stated otherwise), seeking.