Numeric integration of higher order integrals
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
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
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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participants (2)
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Anne Archibald
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Mario Bettenbuehl