[Numpy-discussion] Numeric integration of higher order integrals

[Anne Archibald]

When the dimensionality gets high, grid methods like you're describing
start to be a problem ("the curse of dimensionality"). The standard
approaches are simple Monte Carlo integration or its refinements
(Metropolis-Hasings, for example). These converge somewhat slowly, but
are not much affected by the dimensionality.

Anne

On 1 June 2011 05:44, Mario Bettenbuehl <bettenbu at uni-potsdam.de> wrote:
> Hello everyone,
>
> I am currently tackling the issue to numerically solve an integral of higher
> dimensions numerically. I am comparing models
> and their dimension increase with 2^n order.
> Taking a closer look to its projections along the axes, down to a two
> dimensions picture, the projections are of Gaussian nature, thus
> they show a Gaussian bump.
>
> I already used to approaches:
>    1. brute force:        Process the values at discrete grid points and
> calculate the area of the obtained rectangle, cube, ... with a grid of
> 5x5x5x5 for a 4th order equation.
>    2. Gaussian quad:    Cascading Gaussian quadrature given from numpy/
> scipy with a grid size of 100x100x...
>
> The problem I have:
> For 1:    How reliable are the results and does anyone have experience with
> equations whose projections are Gaussian like and solved these with the
> straight-forward-method? But how large should the grid be.
> For 2:    How large do I need to choose the grid to still obtain reliable
> results? Is a grid of 10x10 sufficiently large?
>
> Thanks in advance for any reply. If needed, I'll directly provide further
> informations about the problem.
>
> Greetings,
> Mario
>
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