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.