Monte Carlo Integration

This package provides multidimensional integration
algorithms based on monte carlo methods. The biggest
advantage of using monte carlo methods is that their
convergence rate is independent of the dimension of
the integral.

Currently, this package only provides a routine
called VEGAS:

vegas(f, st, en, kwargs...)

VEGAS is a Monte Carlo algorithm for
multidimensional integration based on
adaptive importance sampling. It divides
each dimension into bins and adaptively adjusts
bin widths so points are sampled from the
region where the function has highest magnitude.

Arguments:

• st: Array of starting values in each dimension.
  Defaults to zeros(2)
• end: Array of ending values in each dimension.
  Defaults to ones(2)

Kwargs:

• nbins: Number of bins in each dimension.
  Defaults to 100.
• ncalls: Number of function calls per iteration.
  Defaults to 1000.
• maxiter: Maximum number of iterations.
  Defaults to 100.
• rtol: Relative tolerance required.
  Defaults to 1e-4.
• atol: Absolute tolerance required.
  Defaults to 1e-2.
• debug: Prints abs(sd/I) every 100 iterations.
  Defaults to false.
• batch: Whether f returns batches of function
  evaluations. f is assumed to take one argument
  pts, an ncalls × ndims matrix. Each row
  is a unique point and returns an ncalls length
  vector of function evals. This argument defaults
  to false.

Output:

• Estimate for the integral
• Standard deviation
• χ^2 / (numiter - 1): should be less than 1
  otherwise integral estimate should not be trusted.

References:

• Lepage, G. Peter. "A new algorithm for adaptive
  multidimensional integration." Journal of
  Computational Physics 27.2 (1978): 192-203.

Batch interface

Most of the computation time in an integration
algorithm is usually spent in function evaluations.
The batch inteface allows users to provide
batches of function evaluations, instead of supplying
a function directly to be integrated. Users can now
evaluate a number of points in parallel.

Roadmap

• Supporting vector valued functions
• Other integration algorithms