In high dimensions (>20, say) the best known numerical integration method still usually seems to be the monte-carlo method "VEGAS" by G.Peter Lepage. This is monte carlo, enhanced with a crude (but often quite effective) attempt to add "importance-based sampling biases." (The effectiveness of that depends very heavily on the structure of the function being integrated.) G. Peter Lepage: A New Algorithm for Adaptive Multidimensional Integration, J. Computational Physics 27,2 (1978) 192-203. http://www-zeus.desy.de/~tassi/Lepage-slac-pub-1839.pdf Here's a new idea which might be better. View multi-D integration as 1D integration along a uniform space filling curve filling the unit D-dimensional hypercube (assuming that is the region of integration). Accomplish that 1D integration, by employing a crude differential-equation solver to solve for F(1)-F(0) in the equation F'(t) = G(t) where G is the integrand and t is the space filling curve parameter, 0<t<1. Make this solver employ "adaptive step sizes," i.e. if the integrand within the current step t-interval has too great a mean-square deviation from linearity, then redo the step with smaller size; if too small a variance, then increase step size for the next step. Versus VEGAS, this scheme seems on the surface to enjoy several advantages. 1. very simple algorithm [once you know how to convert t to locations (x,y,z...) within the cube] 2. essentially zero memory required 3. low "waste" i.e. function evaluations that are not directly used 4. seems to be more flexible and fine grained in its "adaption" i.e. importance sampling, unlike VEGAS which has a very procrustean importance sampling notion; also the error estimate which the integrator could output, also could be very fine grained. 5. random-point monte-carlo is bad in the sense that points which happen to be close together, are partially "wasted." So-called "quasi monte carlo" methods use "low discrepancy point sets" which avoid such defects and in practice, as well as theoretically asymptotically, perform better than plain monte carlo (i.e. random uniform points). Points equispaced along space filling curve would (I would expect) be somewhat "better than random" points, thus gaining the advantages of quasi monte carlo to some extent. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)