Timeline for Computing the volume of intersection between a ball and a box
Current License: CC BY-SA 4.0
11 events
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| Feb 15, 2020 at 19:35 | history | edited | Christian Chapman | CC BY-SA 4.0 |
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| Feb 15, 2020 at 19:30 | history | edited | Christian Chapman | CC BY-SA 4.0 |
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| May 30, 2018 at 0:40 | comment | added | fedja | he obvious idea would be to approximate the characteristic function of $[e^{-1},1]$ by a polynomial $\sum c_k x^k$ and say that the characteristic function of the unit ball is just $\sum c_k e^{-k|x|^2}$. The integral of a Gaussian over a cuboid is a piece of cake and the approximation is not hard but keep in mind that the coefficients are huge (exponential in $d$), so you need to use arbitrary precision arithmetic and to find a way to compute $Erf()$ to arbitrary precision in decent time near infinity (otherwise you'll be outputting total garbage). Does anybody know how to do the latter? | |
| May 30, 2018 at 0:05 | history | edited | Christian Chapman | CC BY-SA 4.0 |
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| May 30, 2018 at 0:04 | comment | added | Christian Chapman | That is only one possibility for $C$. $C$ isn't necessarily the hypercube with vertices +/-1, it is instead the hyper-rectangle bounded by the vectors $\underline{c}, \overline{c}\in \mathbb{R}^d$, each component of the two vectors in the range $[-1,1]$. | |
| May 30, 2018 at 0:01 | comment | added | Mark L. Stone | I don't follow. If, as suggested by the question title, $C$ is the hypercube with vertices +/- 1 and B(0,1) is the unit ball $\|x\|_2 \le 1$, then their intersection is just the unit ball. So what is wrong with my interpretation? Is $C$ wholly contained within the hypercube, but not equal to it, in which case, what can you tell us about $C$? | |
| May 29, 2018 at 23:59 | history | edited | Christian Chapman | CC BY-SA 4.0 |
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| May 29, 2018 at 23:58 | comment | added | Christian Chapman | Yes this is correct. | |
| May 29, 2018 at 23:57 | comment | added | SBK | I interpret this as: if you have e.g. $\bar{c} = (c_1,\dots,c_n)$ , then the region $C$ would have $x \in C\ \Rightarrow\ x_i \leq c_i$. Similarly for being lower bounded by the components of $\underline{c}$. Is that right? | |
| May 29, 2018 at 22:40 | history | edited | Christian Chapman | CC BY-SA 4.0 |
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| May 29, 2018 at 22:27 | history | asked | Christian Chapman | CC BY-SA 4.0 |