Monte Carlo Integration

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Easy

Test: Easy: compile and run volesti. Read the CRAN package documentation, generate a random H-polytope and compute its volume.

• Compiled and ran volesti tests in C++ interface using cmake, make and ctest. Also installed all package dependencies like Rcpp, RcppEigen, BH etc and used devtools to install volesti Rcpp Package
• Generated a random H-polytope and computed its volume

Code

library(volesti)
P = gen_rand_hpoly(10, 50, generator = list(constants = "sphere"))
volumes <- list()
for (i in 1:10) {
volumes[[i]] <- volume(P) # default parameters
}
options(digits=10)
summary(as.numeric(volumes))

Medium

Test: Use the R package cubature to compute the integral of f(x) = exp^{-a||x||^2} over the cube [-1,1]^n, for various values of a and dimension n until it crashes.

• Installed R package cubeture and used cubeture library to compute the integral of f(x) = exp^{-a||x||^2} over the cube [-1,1]^n.
• I tried varios values of a=[0,1,2,3] and the values of n incrementing by 1 stating from the number 1.
• At the value of n=24 the program got crashed due to the higher dimention used showing the result 0.
• Ovserving this we can say this package is way more less efficient than volesti package. using volesti approximation we can compute up to a few hundreds of dimensions.

Code

library(cubature) # load the package "cubature"
testFnWeb <- function(x) {
  exp(-a * sum(x^2))
}
hcubature(testFnWeb, rep(-1,n), rep(1,n), tol=1e-4)

Hard

Test: Use volesti to approximate the same integrals as in previous test by simple Monte Carlo based on uniform sampling and by Importance Sampling using multivariate spherical Gaussian. Comment on the accuracy and run-time.

• Used library volesti to compute integral f(x) = exp^{-a||x||^2} till 100th dimension using uniform and gaussian sampling.

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Observation of f(x) = exp^{-a||x||^2}

• Accuracy : cubature is more accurate for small dimentations and it breaks easily for higher dimentions (n>=24). On otherside volesti perfectly works with a higher dimention with very less posibility of error.
• Runtime : Comparing upto 23rd dimention the runtime taken by volesti was definitely less then cubature.

Code

#installing volesti package
install.packages("volesti")
library(volesti)

myfunction <- function(x) {
  y = exp(-a *sum(x^2))

}

## SIMPLE MONTE CARLO USING UNIFORM DISTRIBUTION

P = gen_cube(d, 'H')
points = sample_points(P1, random_walk = list("walk" = "Baw"), n = 10000,
                        seed = 5)
plot(points[1,],points[2,])
                       

## SIMPLE MONTE CARLO USING GAUSSIAN DISTRIBUTION

P <- gen_cube(d,"H")

points = sample_points(P, n=10000 , distribution = list("density" = "gaussian", "variance" = 1)

plot(points[1,],points[2,])