Skip to main content
MathOverflow

Return to Question

simplified language
Source Link
user44143
user44143

My general question is how one mightto construct an isotropic random vector field $\vec f: \mathbb{R}^3 \to \mathbb{R}^3$ which haswith a given mean magnitude $\mathbb{E}[\|\vec f(\vec x)\|]=\mu$ such thatand with vector magnitudes and direction are correlated up to some length scales $l$ (beyond which the correlation goes to zero). Furthermore, we wish thatI prefer a construction satisfying $\nabla \cdot \vec f=0$, althoughbut a construction which does not satisfy thiswithout that condition is already very interesting.

More precisely, we are given a vector $\vec{\mu}$ in $\mathbb{R^3}$ and a matrix-valued correlation function $C(\vec x_1,\vec x_2): \mathbb{R^3} \times \mathbb{R^3} \to M_3(\mathbb{R})$$C: \mathbb{R^3} \times \mathbb{R^3} \to M_3(\mathbb{R})$ which is isotropic, i.e $C(\vec x_1,\vec x_2)=C(\|\vec x_1-\vec x_2\|)$$C(\vec v,\vec w)=C(\|\vec v-\vec w\|)$. One may define a Gaussian Process $f(\vec x) \sim GP(\vec{\mu},C(\vec x_1,\vec x_2))$$f(\vec x) \sim GP(\vec{\mu},C)$ such that $\mathbb{E}[\vec f(\vec x)]=\vec\mu$ and $Cov(\vec f(\vec x_1),\vec f(\vec x_1))=C(\vec x_1,\vec x_2)$$Cov(\vec f(v),\vec f(w))=C(\vec v,\vec w)$. I believe it is known how to generatethat constructions of such a random field $f$ are known. It seems theThe equivalent problem for a scalar field $f$ isseems well-studied, where: one can for examplefirst draw the field first in Fourier space by normalizing a white noise field with the appropriate power spectrum $P(k)$, and then transform back to real space. However it would be useful if someone could describe hereHere I am looking for a simple procedure for the case of $\mathbb{R}^3$, which; I think this is known but I haven't found a clear description anywhere.

Question: howHow can one generate such a random field $f$ if one imposeswith a zero mean vector of zero ($\vec{\mu}=\vec 0$), and in additiona mean magntitudemagnitude of $\mu$ ($\mathbb{E}[\|\vec f(\vec x)\|]=\mu$)? And nowwhat if we impose also impose $\nabla \cdot \vec f=0$?

My general question is how one might construct an isotropic random vector field $\vec f: \mathbb{R}^3 \to \mathbb{R}^3$ which has a given mean magnitude $\mathbb{E}[\|\vec f(\vec x)\|]=\mu$ such that vector magnitudes and direction are correlated up to some length scales $l$ (beyond which the correlation goes to zero). Furthermore, we wish that $\nabla \cdot \vec f=0$, although a construction which does not satisfy this condition is already very interesting.

More precisely, we are given a vector $\vec{\mu}$ in $\mathbb{R^3}$ and matrix-valued correlation function $C(\vec x_1,\vec x_2): \mathbb{R^3} \times \mathbb{R^3} \to M_3(\mathbb{R})$ which is isotropic, i.e $C(\vec x_1,\vec x_2)=C(\|\vec x_1-\vec x_2\|)$. One may define a Gaussian Process $f(\vec x) \sim GP(\vec{\mu},C(\vec x_1,\vec x_2))$ such that $\mathbb{E}[\vec f(\vec x)]=\vec\mu$ and $Cov(\vec f(\vec x_1),\vec f(\vec x_1))=C(\vec x_1,\vec x_2)$. I believe it is known how to generate such a random field $f$. It seems the equivalent problem for a scalar field $f$ is well-studied, where one can for example draw the field first in Fourier space by normalizing a white noise field with the appropriate power spectrum $P(k)$, and then transform back to real space. However it would be useful if someone could describe here a simple procedure for the case of $\mathbb{R}^3$, which I think is known but I haven't found a clear description anywhere.

Question: how can one generate such a random field $f$ if one imposes a zero mean vector $\vec{\mu}=\vec 0$, and in addition mean magntitude $\mathbb{E}[\|\vec f(\vec x)\|]=\mu$? And now if we impose also $\nabla \cdot \vec f=0$?

My general question is how to construct an isotropic random vector field $\vec f: \mathbb{R}^3 \to \mathbb{R}^3$ with a given mean magnitude $\mathbb{E}[\|\vec f(\vec x)\|]=\mu$ and with vector magnitudes and direction correlated up to some length scales $l$ (beyond which the correlation goes to zero). I prefer a construction satisfying $\nabla \cdot \vec f=0$, but a construction without that condition is already interesting.

More precisely, we are given a vector $\vec{\mu}$ in $\mathbb{R^3}$ and a matrix-valued correlation function $C: \mathbb{R^3} \times \mathbb{R^3} \to M_3(\mathbb{R})$ which is isotropic, i.e $C(\vec v,\vec w)=C(\|\vec v-\vec w\|)$. One may define a Gaussian Process $f(\vec x) \sim GP(\vec{\mu},C)$ such that $\mathbb{E}[\vec f(\vec x)]=\vec\mu$ and $Cov(\vec f(v),\vec f(w))=C(\vec v,\vec w)$. I believe that constructions of such a random field $f$ are known. The equivalent problem for a scalar field $f$ seems well-studied: one can first draw the field in Fourier space by normalizing a white noise field with the appropriate power spectrum $P(k)$, and then transform back to real space. Here I am looking for a simple procedure for $\mathbb{R}^3$; I think this is known but I haven't found a clear description anywhere.

Question: How can one generate such a random field $f$ with a mean vector of zero ($\vec{\mu}=\vec 0$), and a mean magnitude of $\mu$ ($\mathbb{E}[\|\vec f(\vec x)\|]=\mu$)? And what if we also impose $\nabla \cdot \vec f=0$?

Minor Math Jaxing (used $\|\cdot\|$ instead of $||\cdot||$)
Source Link
Daniele Tampieri
Daniele Tampieri
  • 6.8k
  • 10
  • 34
  • 49

My general question is how one might construct an isotropic random vector field $\vec f: \mathbb{R}^3 \to \mathbb{R}^3$ which has a given mean magnitude $\mathbb{E}[||\vec f(\vec x)||]=\mu$$\mathbb{E}[\|\vec f(\vec x)\|]=\mu$ such that vector magnitudes and direction are correlated up to some length scales $l$ (beyond which the correlation goes to zero). Furthermore, we wish that $\nabla \cdot \vec f=0$, although a construction which does not satisfy this condition is already very interesting.

More precisely, we are given a vector $\vec{\mu}$ in $\mathbb{R^3}$ and matrix-valued correlation function $C(\vec x_1,\vec x_2): \mathbb{R^3} \times \mathbb{R^3} \to M_3(\mathbb{R})$ which is isotropic, i.e $C(\vec x_1,\vec x_2)=C(||\vec x_1-\vec x_2||)$$C(\vec x_1,\vec x_2)=C(\|\vec x_1-\vec x_2\|)$. One may define a Gaussian Process $f(\vec x) \sim GP(\vec{\mu},C(\vec x_1,\vec x_2))$ such that $\mathbb{E}[\vec f(\vec x)]=\vec\mu$ and $Cov(\vec f(\vec x_1),\vec f(\vec x_1))=C(\vec x_1,\vec x_2)$. I believe it is known how to generate such a random field $f$. It seems the equivalent problem for a scalar field $f$ is well-studied, where one can for example draw the field first in Fourier space by normalizing a white noise field with the appropriate power spectrum $P(k)$, and then transform back to real space. However it would be useful if someone could describe here a simple procedure for the case of $\mathbb{R}^3$, which I think is known but I haven't found a clear description anywhere.

Question: how can one generate such a random field $f$ if one imposes a zero mean vector $\vec{\mu}=\vec 0$, and in addition mean magntitude $\mathbb{E}[||\vec f(\vec x)||]=\mu$$\mathbb{E}[\|\vec f(\vec x)\|]=\mu$? And now if we impose also $\nabla \cdot \vec f=0$?

My general question is how one might construct an isotropic random vector field $\vec f: \mathbb{R}^3 \to \mathbb{R}^3$ which has a given mean magnitude $\mathbb{E}[||\vec f(\vec x)||]=\mu$ such that vector magnitudes and direction are correlated up to some length scales $l$ (beyond which the correlation goes to zero). Furthermore, we wish that $\nabla \cdot \vec f=0$, although a construction which does not satisfy this condition is already very interesting.

More precisely, we are given a vector $\vec{\mu}$ in $\mathbb{R^3}$ and matrix-valued correlation function $C(\vec x_1,\vec x_2): \mathbb{R^3} \times \mathbb{R^3} \to M_3(\mathbb{R})$ which is isotropic, i.e $C(\vec x_1,\vec x_2)=C(||\vec x_1-\vec x_2||)$. One may define a Gaussian Process $f(\vec x) \sim GP(\vec{\mu},C(\vec x_1,\vec x_2))$ such that $\mathbb{E}[\vec f(\vec x)]=\vec\mu$ and $Cov(\vec f(\vec x_1),\vec f(\vec x_1))=C(\vec x_1,\vec x_2)$. I believe it is known how to generate such a random field $f$. It seems the equivalent problem for a scalar field $f$ is well-studied, where one can for example draw the field first in Fourier space by normalizing a white noise field with the appropriate power spectrum $P(k)$, and then transform back to real space. However it would be useful if someone could describe here a simple procedure for the case of $\mathbb{R}^3$, which I think is known but I haven't found a clear description anywhere.

Question: how can one generate such a random field $f$ if one imposes a zero mean vector $\vec{\mu}=\vec 0$, and in addition mean magntitude $\mathbb{E}[||\vec f(\vec x)||]=\mu$? And now if we impose also $\nabla \cdot \vec f=0$?

My general question is how one might construct an isotropic random vector field $\vec f: \mathbb{R}^3 \to \mathbb{R}^3$ which has a given mean magnitude $\mathbb{E}[\|\vec f(\vec x)\|]=\mu$ such that vector magnitudes and direction are correlated up to some length scales $l$ (beyond which the correlation goes to zero). Furthermore, we wish that $\nabla \cdot \vec f=0$, although a construction which does not satisfy this condition is already very interesting.

More precisely, we are given a vector $\vec{\mu}$ in $\mathbb{R^3}$ and matrix-valued correlation function $C(\vec x_1,\vec x_2): \mathbb{R^3} \times \mathbb{R^3} \to M_3(\mathbb{R})$ which is isotropic, i.e $C(\vec x_1,\vec x_2)=C(\|\vec x_1-\vec x_2\|)$. One may define a Gaussian Process $f(\vec x) \sim GP(\vec{\mu},C(\vec x_1,\vec x_2))$ such that $\mathbb{E}[\vec f(\vec x)]=\vec\mu$ and $Cov(\vec f(\vec x_1),\vec f(\vec x_1))=C(\vec x_1,\vec x_2)$. I believe it is known how to generate such a random field $f$. It seems the equivalent problem for a scalar field $f$ is well-studied, where one can for example draw the field first in Fourier space by normalizing a white noise field with the appropriate power spectrum $P(k)$, and then transform back to real space. However it would be useful if someone could describe here a simple procedure for the case of $\mathbb{R}^3$, which I think is known but I haven't found a clear description anywhere.

Question: how can one generate such a random field $f$ if one imposes a zero mean vector $\vec{\mu}=\vec 0$, and in addition mean magntitude $\mathbb{E}[\|\vec f(\vec x)\|]=\mu$? And now if we impose also $\nabla \cdot \vec f=0$?

Source Link
math_lover
math_lover
  • 147
  • 1
  • 8

Divergence-free Gaussian vector field with given mean magnitude and correlation function

My general question is how one might construct an isotropic random vector field $\vec f: \mathbb{R}^3 \to \mathbb{R}^3$ which has a given mean magnitude $\mathbb{E}[||\vec f(\vec x)||]=\mu$ such that vector magnitudes and direction are correlated up to some length scales $l$ (beyond which the correlation goes to zero). Furthermore, we wish that $\nabla \cdot \vec f=0$, although a construction which does not satisfy this condition is already very interesting.

More precisely, we are given a vector $\vec{\mu}$ in $\mathbb{R^3}$ and matrix-valued correlation function $C(\vec x_1,\vec x_2): \mathbb{R^3} \times \mathbb{R^3} \to M_3(\mathbb{R})$ which is isotropic, i.e $C(\vec x_1,\vec x_2)=C(||\vec x_1-\vec x_2||)$. One may define a Gaussian Process $f(\vec x) \sim GP(\vec{\mu},C(\vec x_1,\vec x_2))$ such that $\mathbb{E}[\vec f(\vec x)]=\vec\mu$ and $Cov(\vec f(\vec x_1),\vec f(\vec x_1))=C(\vec x_1,\vec x_2)$. I believe it is known how to generate such a random field $f$. It seems the equivalent problem for a scalar field $f$ is well-studied, where one can for example draw the field first in Fourier space by normalizing a white noise field with the appropriate power spectrum $P(k)$, and then transform back to real space. However it would be useful if someone could describe here a simple procedure for the case of $\mathbb{R}^3$, which I think is known but I haven't found a clear description anywhere.

Question: how can one generate such a random field $f$ if one imposes a zero mean vector $\vec{\mu}=\vec 0$, and in addition mean magntitude $\mathbb{E}[||\vec f(\vec x)||]=\mu$? And now if we impose also $\nabla \cdot \vec f=0$?