Timeline for How to generate a random (Weyl) curvature operator ?
Current License: CC BY-SA 3.0
8 events
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| Jan 21, 2013 at 7:12 | comment | added | John Wiltshire-Gordon | @Thomas Richard All I mean is the following lemma: Suppose a compact group G acts on a probability space X (without preserving measures). Then we obtain a new measure on X which is G-invariant: to integrate a function on X, pull it back along the action map GxX-->X and integrate using the product measure. | |
| Jan 21, 2013 at 6:39 | comment | added | Thomas Richard | @Deane : My interest starts mainly in dimension >4, so I'd better be a bit careful with computational costs... @John : That's interesting... | |
| Jan 21, 2013 at 6:16 | comment | added | Deane Yang | To generate random vectors or tensors, you need to choose a probability measure on the vector- or tensor-space. For example, if you want to generate a random vector on $\mathbb{R}^n$, you can just use $n$ independent Gaussian random variables. This will be $O(n)$-invariant. Presumably, to generate a random 4th order tensor, you need to use $n^4$ independent random Gaussian variables. This would be costly if you want to do it in dimension 10 but presumably dimension 3 and 4 would be interesting enough? | |
| Jan 21, 2013 at 6:08 | comment | added | John Wiltshire-Gordon | Q1: Generate a random symmetric matrix any way you want and then hit it with a random element of O(n): en.wikipedia.org/wiki/Orthogonal_matrix#Randomization | |
| Jan 21, 2013 at 5:48 | comment | added | Thomas Richard | That might surely do the job. However, I'm worried about the computational cost and the $O(n,\mathbb{R})$ invariance. Maybe the best solution is to generate a random element $S^2(\Lambda^2\mathbb{R}^n)$ in an $O(\Lambda^2\mathbb{R}^n)$ invariant way, and then to project it to $S^2_B(\Lambda^2\mathbb{R}^n)$. But this needs an answer to Q1 ! | |
| Jan 21, 2013 at 5:33 | comment | added | Deane Yang | Instead of generating each irreducible piece separately, isn't it easier to just generate a random 4th order tensor and project it orthogonally into the space of curvature tensors? | |
| Jan 21, 2013 at 4:37 | history | edited | Thomas Richard | CC BY-SA 3.0 |
added 354 characters in body
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| Jan 21, 2013 at 4:28 | history | asked | Thomas Richard | CC BY-SA 3.0 |