Basics

Phase spacing

  1. One- and two-particle phase space
  2. Three and more particles
  3. MC Integration: Sampling
  4. Smarter sampling methods

    VEGAS IN GENERAL

    When information about the function to be integrated is not available, the preference clearly is on adaptive techniques, i.e. on algorithms that learn about the function when integrating them. Vegas, which will be discussed here is a prime example for such an adaptive Monte Carlo algorithm; it automatically concentrates on those regions of the integrand where it is largest.
    Vegas starts by dividing the n-dimensional hypercube [0,1]n into smaller ones - usually of identical size - and performs a sampling in each of them. The results are then used to refine the integration grid for the next iteration of the sampling procedure. The refinement is done by approximating the optimal p.d.f. p of the function f,
    by a step function in each bin. The trick in Vegas is to factorise the p.d.f. p in n dimensions through a product of n p.d.f.'s pi in each dimension, i.e.
    which clearly saves memory and obviously limits the usability of Vegas to those integrands which have factorising structure. The latter requirement is often met in particle physics, a fact that explains why Vegas is widely used in this field.
    However, the nice thing about Vegas is that the evaluation of an integral can be divided into two phases: Firstly, the grid is initialised and optimised with many iterations and comparably few sampling points, and only then the "true" sampling leading to a result starts with comparably many sampling points distributed over the now frozen grid. In each iteration j, the estimator and the variance are calculated afresh, based on the sampling points of this iteration only,
    These individual results can be combined to yield a global estimator,
    where each estimator contributes weighted by the number of sampling points used for it and by the variance σ attached to it. The reliablitiy of the total procedure is then judged according to the χ² per degree of freedom,
    which checks whether the individual results are consistent with each other. This is signalled by values around 1 for the χ² per d.o.f..

    WEIGHTS IN VEGAS: THE P.D.F.'s AND THEIR EVOLUTION

    In fact, a closer look into the algorithm reveals that instead of reweighting the bins in each iteration step, their boundaries are shifted. This is done in the following way (for simplicity, scetched for one dimension only): The first iteration starts out with a fixed number Nbins of bins, each of which having the same size Δxi= 1/Nbins.
    After the first iteration, each bin i is subdivided into mi bins according to its contribution <Ii> to the total estimator,
    where K is a constant steering the granularity of the subdivision step. Of course, each of the individual estimators is given by
    (Obviously the total number of sampling points cancels out here.) Assuming that each sub-bin contributes in due proportion to the corresponding bin, the sub-bins are regrouped into Nbins bins again, such that the new anticipated contributions of each new bin to the total estimator is the same. In practise this translates into few sub-bins to form a new bin in those regions where the integrand is largest in magnitude and many sub-bins to form a new bin in those region where the function is smallest. Of course, in further iterations these newly found bins can be refined even more.
    In so doing, however, the binning and rebinning may lead to oscillations in the grid which destabilise the convergence. to avoid them, a damping term may be included into the evaluation of the mi, for example
    this is in fact the rebinning distribution implemented in Vegas, with the convergence parameter α typically in the interval [0.2,2].

  5. Selection according to a distribution: Unweighting
  6. Unweighting: Hit-or-miss
  7. Problems