mcODE (version 1.1)

ODE.MVT.agg: Monte Carlo ODE Solver Based on Mean Value Theorem - Replicated

Description

Given g' = G(x, g) and g(x0) = g0 satisfying conditions for existence and uniqueness of solution, a Monte Carlo approximation to the solution is found. The method is essentially a Monte Carlo version of Euler and Runge-Kutta type methods. Repeated calls are made to the ODE.MVT function to obtain replicated estimates of the solution.

Usage

ODE.MVT.agg(G, initvalue, endpoint, initpoint = 0, Niter = 2, npoints = 1000, nsims = 10)

Value

A list consisting of

x

a vector of length npoints, consisting of the x-coordinate of the solution.

y

a vector of length npoints, consisting of the average of replicated y-coordinates of the solution, i.e. g(x).

stdev

a vector of length npoints, consisting of the standard deviation estimates of the solution estimate at each point.

Arguments

G

a function having two numeric vector arguments.

initvalue

a numeric initial value

endpoint

a numeric interval endpoint value.

initpoint

a numeric interval starting point value.

Niter

an integer value specifying the number of iterations at each step.

npoints

an integer value specifying the number of subintervals to build the approximation on.

nsims

an integer value specifying the number of replicated solutions required.

Author

W.J. Braun

Examples

G <- function(x, g) {
    -1000*g + sin(x)
}
g0 <- -1/1000001; x0 <- 0 # initial condition
xF <- 1 # interval endpoint
nMVT <- 1000 # number of subintervals used
# 10 reps of Monte Carlo solution based on two iterations:
ghat2 <- ODE.MVT.agg(G, initvalue = g0, endpoint = xF, initpoint = x0, 
    Niter = 2, npoints = nMVT, nsims = 10)
# 10 reps of Monte Carlo solution based on three iterations:
ghat3 <- ODE.MVT.agg(G, initvalue = g0, endpoint = xF, initpoint = x0, 
    Niter = 3, npoints = nMVT, nsims = 10)
gTrue <- function(x) (1000*sin(x) - cos(x))/1000001 # true solution
oldpar <- par(mfrow=c(1,2))
curve(gTrue(x), from = 0.3, to = 0.3015) # graph of solution on small interval
lines(ghat2, col = 4, lty = 4, ylab="g(x)", lwd = 2)  # estimated MC solution - 2 iterations
lines(ghat2$y - 1.96*ghat2$stdev ~ ghat2$x, lty = 3, col = 4) # lower conf. limits
lines(ghat2$y + 1.96*ghat2$stdev ~ ghat2$x, lty = 3, col = 4) # upper conf. limits
curve(gTrue(x), from = 0.3, to = 0.3015) # graph of solution on small interval
lines(ghat3, col = 4, lty = 4, ylab="g(x)", lwd = 2)  # estimated MC solution - 3 iterations
lines(ghat3$y - 1.96*ghat3$stdev ~ ghat3$x, lty = 3, col = 4) # lower conf. limits
lines(ghat3$y + 1.96*ghat3$stdev ~ ghat3$x, lty = 3, col = 4) # upper conf. limits
par(oldpar)