Vegas algorithm

function montecarlo(config::Configuration{Ni,V,P,O,T}, integrand::Function, neval,
    print=0, save=0, timer=[], debug=false;
    measure::Union{Nothing,Function}=nothing, measurefreq=1, kwargs...) where {Ni,V,P,O,T}

This algorithm implements the classic Vegas algorithm.

The main idea of the algorithm can be found in this link.

The algorithm uses a simple important sampling scheme. Consider an one-dimensional integral with the integrand $f(x)$, the algorithm will try to learn an optimized distribution $\rho(x)>0$ which mimic the integrand as good as possible (a.k.a, the Vegas map trick, see Dist.Continuous) for more detail.

One then sample the variable $x$ with the distribution $\rho(x)$, and estimate the integral by averging the estimator $f(x)/\rho(x)$.

NOTE: If there are more than one integrals, then all integrals are sampled and measured at each Monte Carlo step.

This algorithm is very efficient for low-dimensional integrations, but can be less efficient and less robust than the Markov-chain Monte Carlo solvers for high-dimensional integrations.

Arguments

• integrand : User-defined function with the following signature:

function integrand(var, config)
    # calculate your integrand values
    # return integrand1, integrand2, ...
end

The first parameter var is either a Variable struct if there is only one type of variable, or a tuple of Varibles if there are more than one types of variables. The second parameter passes the MC Configuration struct to the integrand, so that user has access to userdata, etc.

• measure : User-defined function with the following signature:

function measure(var, obs, weights, config)
    # accumulates the weight into the observable
    # For example,
    # obs[1] = weights[1] # integral 1
    # obs[2] = weights[2] # integral 2
    # ...
end

The first argument var is either a Variable struct if there is only one type of variable, or a tuple of Varibles if there are more than one types of variables. The second argument passes the user-defined observable to the function, it should be a vector with the length same as the integral number. The third argument is the integrand weights to be accumulated to the observable, it is a vector with the length same as the integral number. The last argument passes the MC Configuration struct to the integrand, so that user has access to userdata, etc.

Examples

The following command calls the Vegas solver,

julia> integrate((x, c)->(x[1]^2+x[2]^2); var = Continuous(0.0, 1.0), dof = 2, print=-1, solver=:vegas)
Integral 1 = 0.667203631824444 ± 0.0005046485925614018   (chi2/dof = 1.46)