performance of adaptIntegrate vs. integrate

Question

I'd like to perform a numerical integration in one dimension, where the integrand is vector-valued. integrate() only allows scalar integrands, thus I would need to call it several times. The cubature package seems well suited, but it seems to perform quite poorly for 1D integrals. Consider the following example (scalar-valued integrand and 1D integration),

library(cubature)
integrand <- function(x, a=0.01) exp(-x^2/a^2)*cos(x)
Nmax <- 1e3
tolerance <- 1e-4

# using cubature's adaptIntegrate
time1 <- system.time(replicate(1e3, {
  a <<- adaptIntegrate(integrand, -1, 1, tol=tolerance, fDim=1, maxEval=Nmax)
}) )

# using integrate
time2 <- system.time(replicate(1e3, {
  b <<- integrate(integrand, -1, 1, rel.tol=tolerance, subdivisions=Nmax)
}) )

time1
user  system elapsed 
  2.398   0.004   2.403 
time2
user  system elapsed 
  0.204   0.004   0.208 

a$integral
> [1] 0.0177241
b$value
> [1] 0.0177241

a$functionEvaluations
> [1] 345
b$subdivisions
> [1] 10

Somehow, adaptIntegrate seems to be using many more function evaluations for a similar precision. Both methods apparently use Gauss-Kronrod quadrature (1D case: 15-point Gaussian quadrature rule), though ?integrate adds a "Wynn's Epsilon algorithm". Would that explain the large timing difference?

I'm open to suggestions of alternative ways of dealing with vector-valued integrands such as

integrand <- function(x, a = 0.01) c(exp(-x^2/a^2), cos(x))
adaptIntegrate(integrand, -1, 1, tol=tolerance, fDim=2, maxEval=Nmax)
$integral
[1] 0.01772454 1.68294197

$error
[1] 2.034608e-08 1.868441e-14

$functionEvaluations
[1] 345

Thanks.

Answer 1

There is also R2Cuba package in CRAN which implements several multidimensional integration algorithms:

I tried to test this with your example function, and in such a simple case I couldn't get all the algorithms to work (although I didn't try really hard), and few methods I did get to work were considerably slower than adaptIntegrate with default setting, but perhaps in your true application this package could be worth of trying.