Core implementation - Revision history

Apwillis: /* How the parallelisation works */

2025-06-12T08:32:52Z

How the parallelisation works

← Older revision		Revision as of 01:32, 12 June 2025
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	[[File:Fourier_km_par.png\|404px]]		[[File:Fourier_km_par.png\|404px]]

	Here, the m=0,1,2,3 are split into ~~for~~ <tt>_Ns=2</tt> groups, m=0,1 and m=2,3, then each group is split into <tt>_Nr=3</tt> sections indicated by the colours.		Here, the m=0,1,2,3 are split into <tt>_Ns=2</tt> groups, m=0,1 and m=2,3, then each group is split into <tt>_Nr=3</tt> sections indicated by the colours.

	For <tt>_Ns=2</tt>, split the 6 cores into two sets of three, denoted		For <tt>_Ns=2</tt>, split the 6 cores into two sets of three, denoted
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	radial points in each set, $\{c_0,c_3\}$, $\{c_1,c_4\}$, $\{c_2,c_5\}$.		radial points in each set, $\{c_0,c_3\}$, $\{c_1,c_4\}$, $\{c_2,c_5\}$.
	Within each set m is distributed, with		Within each set m is distributed, with
	m=0,1 is on one core m=2,3 on the other. Gathering the m for a		m=0,1 on one core m=2,3 on the other. Gathering the m for a
	given subsection of axial points, constitutes a second transpose within each of these		given subsection of axial points constitutes a second transpose within each of these
	sets. Finally the azimuthal FFTs may be performed (sum over m).		sets. Finally the azimuthal FFTs may be performed (sum over m).
	The resulting data is split by both radial and axial position, whilst data for all azimuthal points are present on a given core.		The resulting data is split by both radial and axial position, whilst data for all azimuthal points are present on a given core.

Apwillis: /* How the parallelisation works */

2024-03-27T22:59:45Z

How the parallelisation works

← Older revision		Revision as of 15:59, 27 March 2024
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	doubled. However, the number of messages required is		doubled. However, the number of messages required is
	<tt>(_Ns-1)+(_Nr-1)</tt> where <tt>_Np=_Nr*_Ns</tt>.		<tt>(_Ns-1)+(_Nr-1)</tt> where <tt>_Np=_Nr*_Ns</tt>.
	Then, for a given <tt>_Np</tt>, there are ~~significantly fewer~~ messages when <tt>_Nr</tt> and <tt>_Ns</tt> are approximately equal, versus when <tt>_Nr~~=_Np~~</tt>, <tt>_Ns=1</tt>.		Then, for a given $N_p=$<tt>_Np</tt>, there are only approximately $\sqrt{N_p}$ messages when <tt>_Nr</tt> and <tt>_Ns</tt> are approximately equal, versus $N_p$ messages when <tt>_Np=_Nr</tt> (and <tt>_Ns=1</tt>).
	This can substantially reduce time lost in latency, the time setting up communications.		This can substantially reduce time lost in latency, the time setting up communications.
	There are fewer large messages rather than many small ones.		There are fewer large messages, rather than many small ones.

Apwillis: /* How the parallelisation works */

2024-03-27T22:52:59Z

How the parallelisation works

← Older revision		Revision as of 15:52, 27 March 2024
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	[[File:Fourier_km_par.png\|404px]]		[[File:Fourier_km_par.png\|404px]]

	Here, the m=0,1,2,3 are split into for <tt>_Ns=2</tt> ~~sections~~, m=0,1 and m=2,3, then each ~~section~~ is split into <tt>_Nr=3</tt> sections indicated by the colours.		Here, the m=0,1,2,3 are split into for <tt>_Ns=2</tt> groups, m=0,1 and m=2,3, then each group is split into <tt>_Nr=3</tt> sections indicated by the colours.

	For <tt>_Ns=2</tt>, split the 6 cores into two sets of three, denoted		For <tt>_Ns=2</tt>, split the 6 cores into two sets of three, denoted

Apwillis: /* How the parallelisation works */

2024-03-27T22:48:49Z

How the parallelisation works

← Older revision		Revision as of 15:48, 27 March 2024
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	=== How the parallelisation works ===		=== How the parallelisation works ===

	In the Fourier space, all radial points for a particular Fourier mode are located on the same processor, separate modes may be located on separate processes. For the case <tt>K=6, M=4, _Nr=3, _Ns=2,</tt> the modes are split over <tt>_Np=_Nr*_Ns=6</tt> cores (<tt>mpi_rnk=0..5</tt>) indicated by the coloured sections in the following table:		In the Fourier space, all radial points for a particular Fourier mode are located on the same processor, separate modes may be located on separate processes. For the case <tt>K=6, M=4, _Nr=3, _Ns=2,</tt> the modes are split over <tt>_Np=_Nr*_Ns=6</tt> cores (<tt>mpi_rnk=0..5</tt>) indicated by the 6 coloured sections (each contiguous in <tt>nh</tt>) in the following table:

	[[File:Fourier_km_par.png\|404px]]		[[File:Fourier_km_par.png\|404px]]

	Here for <tt>_Ns=2</tt> ~~the~~ m ~~are split~~ 0,1 and 2,3, then each section is split into <tt>_Nr=3</tt> sections indicated by the colours.		Here, the m=0,1,2,3 are split into for <tt>_Ns=2</tt> sections, m=0,1 and m=2,3, then each section is split into <tt>_Nr=3</tt> sections indicated by the colours.

	For <tt>_Ns=2</tt>, split the 6 cores into two sets of three, denoted		For <tt>_Ns=2</tt>, split the 6 cores into two sets of three, denoted

Apwillis at 22:32, 27 March 2024

2024-03-27T22:32:18Z

← Older revision		Revision as of 15:32, 27 March 2024
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	double precision :: Im(i_N, 0:i_H1)		double precision :: Im(i_N, 0:i_H1)
	end type coll</pre>		end type coll</pre>
	The ~~value~~ <tt>Re(n,nh)</tt> is the real part of ~~the coefficient for~~ the <tt>nh</tt>$^{\mathrm{th}}$ harmonic at $r_n$.		The element <tt>Re(n,nh)</tt> is the real part of the <tt>nh</tt>$^{\mathrm{th}}$ harmonic evaluated at $r_n$.

	The spectral <tt>type (spec)</tt> is a data type for operations independent at each radial point. It is rarely used other than as a transitory data format between the <tt>coll</tt> and <tt>phys</tt> types.		The spectral <tt>type (spec)</tt> is a data type for operations independent at each radial point. It is rarely used other than as a transitory data format between the <tt>coll</tt> and <tt>phys</tt> types.

Apwillis: /* Integration */

2024-03-27T21:54:46Z

Integration

← Older revision		Revision as of 14:54, 27 March 2024
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	=== Integration ===		=== Integration ===

	Integrating the expansions above, the indefinite integral may be approximated about $x_0$:		Integrating the expansions above, from $x_0$ to $x$, the indefinite integral may be approximated about $x_0$:

	[[File:fd_int.png\|500px]]		[[File:fd_int.png\|500px]]

Apwillis: /* Differentiation */

2024-03-27T21:53:45Z

Differentiation

← Older revision		Revision as of 14:53, 27 March 2024
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	=== Differentiation ===		=== Differentiation ===

	Taylor expansions about central point $x_0$, $k$ neighbouring points each side		Consider Taylor expansions about central point $x_0$, with $k$ neighbouring points each side

	[[File:fd_deriv.png\|500px]]		[[File:fd_deriv.png\|500px]]

Apwillis: /* How the parallelisation works */

2024-03-27T21:49:22Z

How the parallelisation works

← Older revision		Revision as of 14:49, 27 March 2024
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	=== How the parallelisation works ===		=== How the parallelisation works ===

	In the Fourier space, all radial points for a particular Fourier mode are located on the same processor, separate modes may be located on separate processes. For the case <tt>K=6, M=4, _Ns=2~~, _Nr=3~~,</tt> the modes are split over <tt>_Np=6</tt> cores (<tt>mpi_rnk=0..5</tt>) indicated by the coloured sections in the following table:		In the Fourier space, all radial points for a particular Fourier mode are located on the same processor, separate modes may be located on separate processes. For the case <tt>K=6, M=4, _Nr=3, _Ns=2,</tt> the modes are split over <tt>_Np=_Nr*_Ns=6</tt> cores (<tt>mpi_rnk=0..5</tt>) indicated by the coloured sections in the following table:

	[[File:Fourier_km_par.png\|404px]]		[[File:Fourier_km_par.png\|404px]]
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	Here for <tt>_Ns=2</tt> the m are split 0,1 and 2,3, then each section is split into <tt>_Nr=3</tt> sections indicated by the colours.		Here for <tt>_Ns=2</tt> the m are split 0,1 and 2,3, then each section is split into <tt>_Nr=3</tt> sections indicated by the colours.

	For <tt>_Ns=2</tt>, two sets ~~each~~ of three ~~cores may be~~ denoted		For <tt>_Ns=2</tt>, split the 6 cores into two sets of three, denoted
	$\{c_0,c_1,c_2\}$ and $\{c_3,c_4,c_5\}$.		$\{c_0,c_1,c_2\}$ and $\{c_3,c_4,c_5\}$.
	Prior to an FFT, the core $c_0$ gathers all Fourier coefficients from within the first set for a given		Prior to an FFT, the core $c_0$ gathers all Fourier coefficients from within the first set for a given

Apwillis: /* Integration */

2024-03-27T21:38:04Z

Integration

← Older revision		Revision as of 14:38, 27 March 2024
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	=== Integration ===		=== Integration ===

	Integrating ~~\ref{eq:taylorexp}~~ the indefinite integral may be approximated about $x_0$		Integrating the expansions above, the indefinite integral may be approximated about $x_0$:

	~~$\begin{eqnarray*}~~		[[File:fd_int.png\|500px]]
	~~\int f(x) \, {\mathrm d}x & = &~~
	~~\begin{array}{c}(x-x_0)\,f(x_0) + \frac{(x-x_0)^2}{2!}\,f'(x_0) +~~
	~~\frac{(x-x_0)^3}{3!}\,f''(x_0) + \dots \end{array}~~
	\\
	~~& = &~~
	[~~\begin{array}{lll}(x-x_0) & \frac{(x-x_0)^2}{2!} &~~
	~~\dots \end{array}~~] ~~\,\, \mat{A}^{-1}\vec{f}\end{eqnarray*}$~~

	The total integral is approximated by		The total integral is approximated by

Apwillis: /* Differentiation */

2024-03-27T21:35:40Z

Differentiation

← Older revision		Revision as of 14:35, 27 March 2024
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	Taylor expansions about central point $x_0$, $k$ neighbouring points each side		Taylor expansions about central point $x_0$, $k$ neighbouring points each side

	~~$\label{eq~~:~~taylorexp}~~		[[File:fd_deriv.png\|500px]]
	~~\begin{array}{lclrrrr}~~
	~~f(x_{-k}) &=& f(x_0)~~
	~~&+& (x_{-k}-x_0)\,f'(x_0) &+& \frac{(x_{-k}-x_0)^2}{2!}\,f''(x_0)+\dots~~
	~~\\ & \vdots & \\~~
	~~f(x_k) &=& f(x_0)~~
	~~&+& (x_k-x_0)\,f'(x_0) &+& \frac{(x_k-x_0)^2}{2!}\,f''(x_0)+\dots~~
	~~\end{array}$~~

	~~The~~ expansions ~~\ref{eq:taylorexp}~~ can be written		These expansions can be written

	~~$\vec{f}=\mat{A}\,\vec{df}, \quad~~		[[File:fd_deriv_vec.png\|\|500px]]
	~~\vec{df}=\mat{A}^{-1}\,\vec{f}, \quad~~
	~~\vec{f}=\left~~[~~\begin{array}{c}f(x_{-k})\\~~
	~~\vdots\\ f(x_k)\end{array}\right~~]~~, \quad~~
	~~\vec{df}=\left[\begin{array}{c}~~
	~~f(x_0)\\ f'(x_0)\\ f''(x_0)\\ \vdots \end{array}\right~~]$

	Derivatives are calculated using weights from the appropriate row of $~~\mat{~~A}^{-1}$.		Derivatives are calculated using weights from the appropriate row of $A^{-1}$.

	=== Integration ===		=== Integration ===