Adaptive multivariate integration over hypercubes

Description

The function performs adaptive multidimensional integration (cubature) of (possibly) vector-valued integrands over hypercubes.

Usage

adaptIntegrate(f, lowerLimit, upperLimit, ..., tol = 1e-05, fDim = 1,
               maxEval = 0, absError=0, doChecking=FALSE)

Arguments

Details

The function merely calls Johnson's C code and returns the results. The original C code by Johnson was modified for use with R memory allocation functions and a helper function does the callback.

One can specify a maximum number of function evaluations (default is 0 for no limit). Otherwise, the integration stops when the estimated error is less than the absolute error requested, or when the estimated error is less than tol times the integral, in absolute value.

Value

The returned value is a list of three items:

Author(s)

Balasubramanian Narasimhan

References

See http://ab-initio.mit.edu/wiki/index.php/Cubature.

Examples

## Test function 0
## Compare with original cubature result of
## ./cubature_test 2 1e-4 0 0
## 2-dim integral, tolerance = 0.0001
## integrand 0: integral = 0.708073, est err = 1.70943e-05, true err = 7.69005e-09
## #evals = 17

testFn0 <- function(x) {
  prod(cos(x))
}

adaptIntegrate(testFn0, rep(0,2), rep(1,2), tol=1e-4)

M_2_SQRTPI <- 2/sqrt(pi)

## Test function 1
## Compare with original cubature result of
## ./cubature_test 3 1e-4 1 0
## 3-dim integral, tolerance = 0.0001
## integrand 1: integral = 1.00001, est err = 9.67798e-05, true err = 9.76919e-06
## #evals = 5115

testFn1 <- function(x) {
  scale = 1.0
  val = 0
  dim = length(x)
  val = sum (((1-x) / x)^2)
  scale = prod(M_2_SQRTPI/x^2)
  exp(-val) * scale
}

adaptIntegrate(testFn1, rep(0, 3), rep(1, 3), tol=1e-4)

##
## Test function 2
## Compare with original cubature result of
## ./cubature_test 2 1e-4 2 0
## 2-dim integral, tolerance = 0.0001
## integrand 2: integral = 0.19728, est err = 1.97261e-05, true err = 4.58316e-05
## #evals = 166141

testFn2 <- function(x) {
  ## discontinuous objective: volume of hypersphere
  radius = as.double(0.50124145262344534123412)
  ifelse(sum(x*x) < radius*radius, 1, 0)
}

adaptIntegrate(testFn2, rep(0, 2), rep(1, 2), tol=1e-4)

##
## Test function 3
## Compare with original cubature result of
## ./cubature_test 3 1e-4 3 0
## 3-dim integral, tolerance = 0.0001
## integrand 3: integral = 1, est err = 0, true err = 2.22045e-16
## #evals = 33

testFn3 <- function(x) {
  prod(2*x)
}

adaptIntegrate(testFn3, rep(0,3), rep(1,3), tol=1e-4)

##
## Test function 4 (Gaussian centered at 1/2)
## Compare with original cubature result of
## ./cubature_test 2 1e-4 4 0
## 2-dim integral, tolerance = 0.0001
## integrand 4: integral = 1, est err = 9.84399e-05, true err = 2.78894e-06
## #evals = 1853

testFn4 <- function(x) {
  a = 0.1
  s = sum((x-0.5)^2)
  (M_2_SQRTPI / (2. * a))^length(x) * exp (-s / (a * a))
}

adaptIntegrate(testFn4, rep(0,2), rep(1,2), tol=1e-4)


##
## Test function 5 (double Gaussian)
## Compare with original cubature result of
## ./cubature_test 3 1e-4 5 0
## 3-dim integral, tolerance = 0.0001
## integrand 5: integral = 0.999994, est err = 9.98015e-05, true err = 6.33407e-06
## #evals = 59631

testFn5 <- function(x) {
  a = 0.1
  s1 = sum((x-1/3)^2)
  s2 = sum((x-2/3)^2)
  0.5 * (M_2_SQRTPI / (2. * a))^length(x) * (exp(-s1 / (a * a)) + exp(-s2 / (a * a)))
}

adaptIntegrate(testFn5, rep(0,3), rep(1,3), tol=1e-4)

##
## Test function 6 (Tsuda's example)
## Compare with original cubature result of
## ./cubature_test 4 1e-4 6 0
## 4-dim integral, tolerance = 0.0001
## integrand 6: integral = 0.999998, est err = 9.99685e-05, true err = 1.5717e-06
## #evals = 18753

testFn6 <- function(x) {
  a = (1+sqrt(10.0))/9.0
  prod(a/(a+1)*((a+1)/(a+x))^2)
}

adaptIntegrate(testFn6, rep(0,4), rep(1,4), tol=1e-4)


##
## Test function 7
##   test integrand from W. J. Morokoff and R. E. Caflisch, "Quasi=
##   Monte Carlo integration," J. Comput. Phys 122, 218-230 (1995).
##   Designed for integration on [0,1]^dim, integral = 1. */
## Compare with original cubature result of
## ./cubature_test 3 1e-4 7 0
## 3-dim integral, tolerance = 0.0001
## integrand 7: integral = 1.00001, est err = 9.96657e-05, true err = 1.15994e-05
## #evals = 7887

testFn7 <- function(x) {
  n <- length(x)
  p <- 1/n
  (1+p)^n * prod(x^p)
}

adaptIntegrate(testFn7, rep(0,3), rep(1,3), tol=1e-4)


## Example from web page
## http://ab-initio.mit.edu/wiki/index.php/Cubature
##
## f(x) = exp(-0.5(euclidean_norm(x)^2)) over the three-dimensional
## hyperbcube [-2, 2]^3
## Compare with original cubature result
testFnWeb <-  function(x) {
  exp(-0.5*sum(x^2))
}

adaptIntegrate(testFnWeb, rep(-2,3), rep(2,3), tol=1e-4)

## Test function I.1d from
## Numerical integration using Wang-Landau sampling
## Y. W. Li, T. Wust, D. P. Landau, H. Q. Lin
## Computer Physics Communications, 2007, 524-529
## Compare with exact answer: 1.63564436296
##
I.1d <- function(x) {
  sin(4*x) *
    x * ((x * ( x * (x*x-4) + 1) - 1))
}

adaptIntegrate(I.1d, -2, 2, tol=1e-7)

## Test function I.2d from
## Numerical integration using Wang-Landau sampling
## Y. W. Li, T. Wust, D. P. Landau, H. Q. Lin
## Computer Physics Communications, 2007, 524-529
## Compare with exact answer: -0.01797992646
##
## Test function I.2d from
## Numerical integration using Wang-Landau sampling
## Y.W. Li, T. Wust, D.P. Landau, H.Q. Lin
## Computer Physics Communications, 2007 524-529
## Compare with exact answer: -0.01797992646
##
I.2d <- function(x) {
  x1 = x[1]
  x2 = x[2]
  sin(4*x1+1) * cos(4*x2) * x1 * (x1*(x1*x1)^2 - x2*(x2*x2 - x1) +2)
}

adaptIntegrate(I.2d, rep(-1, 2), rep(1, 2), maxEval=10000)

##
## Example of multivariate normal integration borrowed from
## package mvtnorm (on CRAN) to check ... argument
## Compare with output of
## pmvnorm(lower=rep(-0.5, m), upper=c(1,4,2), mean=rep(0, m), corr=sigma, alg=Miwa())
##     0.3341125.  Blazing quick as well!  Ours is, not unexpectedly, much slower.
##
dmvnorm <- function (x, mean, sigma, log = FALSE) {
    if (is.vector(x)) {
        x <- matrix(x, ncol = length(x))
    }
    if (missing(mean)) {
        mean <- rep(0, length = ncol(x))
    }
    if (missing(sigma)) {
        sigma <- diag(ncol(x))
    }
    if (NCOL(x) != NCOL(sigma)) {
        stop("x and sigma have non-conforming size")
    }
    if (!isSymmetric(sigma, tol = sqrt(.Machine$double.eps),
        check.attributes = FALSE)) {
        stop("sigma must be a symmetric matrix")
    }
    if (length(mean) != NROW(sigma)) {
        stop("mean and sigma have non-conforming size")
    }
    distval <- mahalanobis(x, center = mean, cov = sigma)
    logdet <- sum(log(eigen(sigma, symmetric = TRUE, only.values = TRUE)$values))
    logretval <- -(ncol(x) * log(2 * pi) + logdet + distval)/2
    if (log)
        return(logretval)
    exp(logretval)
}

m <- 3
sigma <- diag(3)
sigma[2,1] <- sigma[1, 2] <- 3/5 ; sigma[3,1] <- sigma[1, 3] <- 1/3
sigma[3,2] <- sigma[2, 3] <- 11/15
adaptIntegrate(dmvnorm, lower=rep(-0.5, m), upper=c(1,4,2),
                        mean=rep(0, m), sigma=sigma, log=FALSE,
               maxEval=10000)

Results