divonne: Integration by Stratified Sampling for Variance Reduction

In R2Cuba: Multidimensional Numerical Integration

Description

Divonne works by stratified sampling, where the partioning of the integration region is aided by methods from numerical optimization.

Usage

divonne(ndim, ncomp, integrand, ...,
        lower=rep(0,ndim), upper=rep(1,ndim), 
        rel.tol= 0.001,   abs.tol = 0,
        flags=list(verbose=1, final=1, pseudo.random=0, mersenne.seed=NULL),
        min.eval=0,  max.eval=50000,
        key1=47,  key2=1, key3=1, 
        max.pass=5, border=0, max.chisq=10,
        min.deviation=0.25,
        xgiven=NULL,  nextra=0, peakfinder=NULL)

Arguments

ndim	same as cuhre
ncomp	same as cuhre
integrand	same as cuhre. But, here, the input argument phw indicates the integration phase: 0: sampling of the points in xgiven, 1: partitioning phase, 2: final integration phase, 3: refinement phase. This information might be useful if the integrand takes long to compute and a sufficiently accurate approximation of the integrand is available. The actual value of the integrand is only of minor importance in the partitioning phase, which is instead much more dependent on the peak structure of the integrand to find an appropriate tessellation. An approximation which reproduces the peak structure while leaving out the fine details might hence be a perfectly viable and much faster substitute when phw < 2. In all other instances, phw can be ignored.
...	same as cuhre
lower	same as cuhre
upper	same as cuhre
rel.tol	same as cuhre
abs.tol	same as cuhre
flags	same as cuhre pseudo.random and mersenne.seed are only taken into account when the argument key1 is negative.
min.eval	same as cuhre
max.eval	same as cuhre
key1	integer that determines sampling in the partitioning phase: key1 = 7, 9, 11, 13 selects the cubature rule of degree key1. Note that the degree-11 rule is available only in 3 dimensions, the degree-13 rule only in 2 dimensions. For other values of key1, a quasi-random sample of \code{n=|key1|} points is used, where the sign of key1 determines the type of sample, key1 = 0, use the default rule. key1 > 0, use a Korobov quasi-random sample, key1 < 0, use a “standard” sample (a Mersenne Twister pseudo-random sample if flags$pseudo.random=1, otherwise a Sobol quasi-random sample).
key2	integer that determines sampling in the final integration phase: same as key1, but here \code{n = |key2|} determines the number of points, \code{n > 39}, sample n points, \code{n < 40}, sample \code{n} nneed points, where nneed is the number of points needed to reach the prescribed accuracy, as estimated by Divonne from the results of the partitioning phase.
key3	integer that sets the strategy for the refinement phase: key3 = 0, do not treat the subregion any further. key3 = 1, split the subregion up once more. Otherwise, the subregion is sampled a third time with key3 specifying the sampling parameters exactly as key2 above.
max.pass	integer that controls the thoroughness of the partitioning phase: The partitioning phase terminates when the estimated total number of integrand evaluations (partitioning plus final integration) does not decrease for max.pass successive iterations. A decrease in points generally indicates that Divonne discovered new structures of the integrand and was able to find a more effective partitioning. max.pass can be understood as the number of “safety” iterations that are performed before the partition is accepted as final and counting consequently restarts at zero whenever new structures are found.
border	the relative width of the border of the integration region. Points falling into the border region will not be sampled directly, but will be extrapolated from two samples from the interior. Use a non-zero border if the integrand subroutine cannot produce values directly on the integration boundary. The relative width of the border is identical in all the dimensions. For example, set border=0.1 for a border of width equal to 10% of the width of the integration region.
max.chisq	the maximum Chi2 value a single subregion is allowed to have in the final integration phase. Regions which fail this Chi2 test and whose sample averages differ by more than min.deviation move on to the refinement phase.
min.deviation	a bound, given as the fraction of the requested error of the entire integral, which determines whether it is worthwhile further examining a region that failed the Chi2 test. Only if the two sampling averages obtained for the region differ by more than this bound is the region further treated.
xgiven	a matrix ( ndim, ngiven). A list of ngiven points where the integrand might have peaks. Divonne will consider these points when partitioning the integration region. The idea here is to help the integrator find the extrema of the integrand in the presence of very narrow peaks. Even if only the approximate location of such peaks is known, this can considerably speed up convergence.
nextra	the maximum number of extra points the peak-finder subroutine will return. If nextra is zero, peakfinder is not called and an arbitrary object may be passed in its place, e.g. just 0.
peakfinder	the peak-finder subroutine. This R function is called whenever a region is up for subdivision and is supposed to point out possible peaks lying in the region, thus acting as the dynamic counterpart of the static list of points supplied in xgiven. It is expected to be declared as peakfinder <- function(bounds) where bounds is a matrix of dimension (ndim, 2) which contains the upper and lower bounds of the subregion. The names of the columns are c("lower", "upper"). The returned value should be a matrix (ndim, nx) where nx is the actual number of points (should be less or equal to nextra).

Details

Divonne uses stratified sampling for variance reduction, that is, it partitions the integration region such that all subregions have an approximately equal value of a quantity called the spread (volume times half-range).

See details in the documentation.

Value

Idem as cuhre. Here ifail may be >1 when the accuracy goal was not met within the allowed maximum number of integrand evaluations. Divonne can estimate the number of points by which maxeval needs to be increased to reach the desired accuracy and returns this value.

References

J. H. Friedman, M. H. Wright (1981) A nested partitioning procedure for numerical multiple integration. ACM Trans. Math. Software, 7(1), 76-92.

J. H. Friedman, M. H. Wright (1981) User's guide for DIVONNE. SLAC Report CGTM-193-REV, CGTM-193, Stanford University.

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

See Also

cuhre, suave, vegas

Examples

NDIM <- 3
NCOMP <- 1
integrand <- function(arg, phase) {
  x <- arg[1]
  y <- arg[2]
  z <- arg[3]
  ff <- sin(x)*cos(y)*exp(z);
return(ff)
}
divonne(NDIM, NCOMP, integrand, rel.tol=1e-3,  abs.tol=1e-12,
        flags=list(verbose=2),  key1= 47)

# Example with a peak-finder function
NMAX <- 4

peakf <- function(bounds) {
#  print(bounds) # matrix (ndim,2)
  x <- matrix(0, ncol=NMAX, nrow=NDIM)
   pas <- 1/(NMAX-1)
   # 1ier point
   x[,1] <- rep(0, NDIM)
   # Les autres points
   for (i in 2:NMAX) {
      x[,i] <- x[,(i-1)] + pas
    }
  return(x)
} #end peakf

divonne(NDIM, NCOMP, integrand,
               flags=list(verbose=0) ,
                peakfinder=peakf, nextra=NMAX)

Example output

Divonne input parameters:
  ndim 3
  ncomp 1
  rel.tol 0.001
  abs.tol 1e-12
  pseudo.random  0
  final 0
  verbose 2
  min.eval 0
  max.eval 50000
  key1 47
  key2 1
  key3 1
  max.pass 5
  border 0
  max.chisq 10
  min.deviation 0.25
  ngiven 0
  nextra 0
Partitioning phase:
Iteration 1 (pass 0):  8 regions
    836 integrand evaluations so far,
    406 in optimizing regions,
     70 in finding cuts
[1] 0.665011 +- 0.00470198
Iteration 2 (pass 0):  9 regions
    966 integrand evaluations so far,
    478 in optimizing regions,
     80 in finding cuts
[1] 0.664964 +- 0.00429467
Iteration 3 (pass 1):  10 regions
   1096 integrand evaluations so far,
    550 in optimizing regions,
     90 in finding cuts
[1] 0.664949 +- 0.00388393
Iteration 4 (pass 2):  11 regions
   1194 integrand evaluations so far,
    590 in optimizing regions,
    100 in finding cuts
[1] 0.664887 +- 0.0035842
Iteration 5 (pass 3):  12 regions
   1308 integrand evaluations so far,
    646 in optimizing regions,
    110 in finding cuts
[1] 0.664853 +- 0.0033385
Iteration 6 (pass 4):  13 regions
   1438 integrand evaluations so far,
    718 in optimizing regions,
    120 in finding cuts
[1] 0.664825 +- 0.0031276
Iteration 7 (pass 5):  14 regions
   1568 integrand evaluations so far,
    790 in optimizing regions,
    130 in finding cuts
[1] 0.664816 +- 0.00292074

Main integration on 14 regions with 211 samples per region.integral: 0.6646195 (+-0.00064)
nregions: 14; number of evaluations:  3052; probability:  1.110223e-16 
integral: 0.6646195 (+-0.00064)
nregions: 14; number of evaluations:  3136; probability:  1.110223e-16 

R2Cuba documentation built on May 29, 2017, 7:53 p.m.