Integration by a Monte Carlo Algorithm

Implement a Monte Carlo algorithm for multidimensional numerical integration. This algorithm uses importance sampling as a variance-reduction technique. Vegas iteratively builds up a piecewise constant weight function, represented on a rectangular grid. Each iteration consists of a sampling step followed by a refinement of the grid.

Usage

vegas(
  f,
  nComp = 1L,
  lowerLimit,
  upperLimit,
  ...,
  relTol = 1e-05,
  absTol = 1e-12,
  minEval = 0L,
  maxEval = 10^6,
  flags = list(verbose = 0L, final = 1L, smooth = 0L, keep_state = 0L, load_state = 0L,
    level = 0L),
  rngSeed = 12345L,
  nVec = 1L,
  nStart = 1000L,
  nIncrease = 500L,
  nBatch = 1000L,
  gridNo = 0L,
  stateFile = NULL
)

Arguments

f

The function (integrand) to be integrated as in cuhre(). Optionally, the function can take two additional arguments in addition to the variable being integrated: - cuba_weight which is the weight of the point being sampled, - cuba_iter the current iteration number. The function author may choose to use these in any appropriate way or ignore them altogether.

nComp

The number of components of f, default 1, bears no relation to the dimension of the hypercube over which integration is performed.

lowerLimit

The lower limit of integration, a vector for hypercubes.

upperLimit

The upper limit of integration, a vector for hypercubes.

...

All other arguments passed to the function f.

relTol

The maximum tolerance, default 1e-5.

absTol

the absolute tolerance, default 1e-12.

minEval

the minimum number of function evaluations required

maxEval

The maximum number of function evaluations needed, default 10^6. Note that the actual number of function evaluations performed is only approximately guaranteed not to exceed this number.

flags

flags governing the integration. The list here is exhaustive to keep the documentation and invocation uniform, but not all flags may be used for a particular method as noted below. List components:

verbose

encodes the verbosity level, from 0 (default) to 3. Level 0 does not print any output, level 1 prints reasonable information on the progress of the integration, level 2 also echoes the input parameters, and level 3 further prints the subregion results.

final

when 0, all sets of samples collected on a subregion during the various iterations or phases contribute to the final result. When 1, only the last (largest) set of samples is used in the final result.

smooth

Applies to Suave and Vegas only. When 0, apply additional smoothing to the importance function, this moderately improves convergence for many integrands. When 1, use the importance function without smoothing, this should be chosen if the integrand has sharp edges.

keep_state

when nonzero, retain state file if argument stateFile is non-null, else delete stateFile if specified.

load_state

Applies to Vegas only. Reset the integrator state even if a state file is present, i.e. keep only the grid. Together with keep_state this allows a grid adapted by one integration to be used for another integrand.

level

applies only to Divonne, Suave and Vegas. When 0, Mersenne Twister random numbers are used. When nonzero Ranlux random numbers are used, except when rngSeed is zero which forces use of Sobol quasi-random numbers. Ranlux implements Marsaglia and Zaman's 24-bit RCARRY algorithm with generation period \(p\), i.e. for every 24 generated numbers used, another \(p-24\) are skipped. The luxury level for the Ranlux generator may be encoded in level as follows:

Level 1 (p = 48): gives very long period, passes the gap test but fails spectral test
Level 2 (p = 97): passes all known tests, but theoretically still defective
Level 3 (p = 223): any theoretically possible correlations have very small chance of being observed
Level 4 (p = 389): highest possible luxury, all 24 bits chaotic
Levels 5-23: default to 3, values above 24 directly specify the period p. Note that Ranlux's original level 0, (mis)used for selecting Mersenne Twister in Cuba, is equivalent to level = 24

rngSeed

seed, default 0, for the random number generator. Note the articulation with level settings for flag

nVec

the number of vectorization points, default 1, but can be set to an integer > 1 for vectorization, for example, 1024 and the function f above needs to handle the vector of points appropriately. See vignette examples.

nStart

the number of integrand evaluations per iteration to start with.

nIncrease

the increase in the number of integrand evaluations per iteration. The j-th iteration evaluates the integrand at nStart+(j-1)*nincrease points.

nBatch

Vegas samples points not all at once, but in batches of a predetermined size, to avoid excessive memory consumption. nbatch is the number of points sampled in each batch. Tuning this number should usually not be necessary as performance is affected significantly only as far as the batch of samples fits into the CPU cache.

gridNo

an integer. Vegas may accelerate convergence to keep the grid accumulated during one integration for the next one, if the integrands are reasonably similar to each other. Vegas maintains an internal table with space for ten grids for this purpose. If gridno is a number between 1 and 10, the grid is not discarded at the end of the integration, but stored in the respective slot of the table for a future invocation. The grid is only re-used if the dimension of the subsequent integration is the same as the one it originates from. In repeated invocations it may become necessary to flush a slot in memory. In this case the negative of the grid number should be set. Vegas will then start with a new grid and also restore the grid number to its positive value, such that at the end of the integration the grid is again stored in the indicated slot.

stateFile

the name of an external file. Vegas can store its entire internal state (i.e. all the information to resume an interrupted integration) in an external file. The state file is updated after every iteration. If, on a subsequent invocation, Vegas finds a file of the specified name, it loads the internal state and continues from the point it left off. Needless to say, using an existing state file with a different integrand generally leads to wrong results. Once the integration finishes successfully, i.e. the prescribed accuracy is attained, the state file is removed. This feature is useful mainly to define ‘check-points’ in long-running integrations from which the calculation can be restarted.

Value

A list with components:

neval: the actual number of integrand evaluations needed
returnCode: if zero, the desired accuracy was reached, if -1, dimension out of range, if 1, the accuracy goal was not met within the allowed maximum number of integrand evaluations.
integral: vector of length nComp; the integral of integrand over the hypercube
error: vector of length nComp; the presumed absolute error of integral
prob: vector of length nComp; the \(\chi^2\)-probability (not the \(\chi^2\)-value itself!) that error is not a reliable estimate of the true integration error.

Details

See details in the documentation.

References

G. P. Lepage (1978) A new algorithm for adaptive multidimensional integration. J. Comput. Phys., 27, 192-210.

G. P. Lepage (1980) VEGAS - An adaptive multi-dimensional integration program. Research Report CLNS-80/447. Cornell University, Ithaca, N.-Y.

T. Hahn (2005) CUBA-a library for multidimensional numerical integration. Computer Physics Communications, 168, 78-95.

Examples


integrand <- function(arg, weight) {
  x <- arg[1]
  y <- arg[2]
  z <- arg[3]
  ff <- sin(x)*cos(y)*exp(z);
return(ff)
} # end integrand
vegas(integrand, lowerLimit = rep(0, 3), upperLimit = rep(1, 3),
             relTol=1e-3,  absTol=1e-12,
             flags=list(verbose=2, final=0))
#> $integral
#> [1] 0.6652311
#> 
#> $error
#> [1] 0.0006170268
#> 
#> $neval
#> [1] 13500
#> 
#> $prob
#> [1] 0.6053001
#> 
#> $returnCode
#> [1] 0
#>