Bethe-Salpeter solver: Lanczos-Haydock - Revision history

Davide: Fixed references

2023-09-12T09:09:45Z

Fixed references

← Older revision		Revision as of 09:09, 12 September 2023
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	You can avoid the full matrix [[Bethe-Salpeter solver: diagonalization\|diagonalization]] (which rapidly become expensive as it scales as N<sup>3</sup>) by using the Lanczos-Haydock solver. While for small matrices the full diagonalization and the Lanczos-Haydock solver have similar computational cost, for matrices of the size 10000 by 10000 and larger, the Lanczos-Haydock solver is remarkably faster. The drawback is that excitonic functions cannot be calculated.		You can avoid the full matrix [[Bethe-Salpeter solver: diagonalization\|diagonalization]] (which rapidly become expensive as it scales as N<sup>3</sup>) by using the Lanczos-Haydock solver. While for small matrices the full diagonalization and the Lanczos-Haydock solver have similar computational cost, for matrices of the size 10000 by 10000 and larger, the Lanczos-Haydock solver is remarkably faster. The drawback is that excitonic functions cannot be calculated.
	Within the Lanczos-Haydock approach the macroscopic dielectric function is rewritten as a continued fraction<ref>~~L. X. Benedict and E. L. Shirley, Phys. Rev~~. ~~B '''59''', 5441 (1999)</ref>~~		Within the Lanczos-Haydock approach the macroscopic dielectric function is rewritten as a continued fraction<ref name="Benedict1999"/>.

	[[file:BSE1-Eq5.png\|none\|x100px]]		[[file:BSE1-Eq5.png\|none\|x100px]]

	where ''a'' 's and ''b'' 's result from the [https://en.wikipedia.org/wiki/Lanczos_algorithm#Lanczos_method\| Lanczos iterative algorithm].		where ''a'' 's and ''b'' 's result from the [https://en.wikipedia.org/wiki/Lanczos_algorithm#Lanczos_method\| Lanczos iterative algorithm]. In Yambo the approach is coded both for the Hermitian case (BSE neglecting coupling) and for the pseud-hermitian case (BSE with coupling)<ref name="Gruning2011"/>.

	==Prerequisites==		==Prerequisites==
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	==References==		==References==
			<references>
			<ref name="Benedict1999">L. X. Benedict, and E. L. Shirley, Phys. Rev. B '''59''', 5441 (1999)</ref>
			<ref name="Gruning2011">M. Gruning, A. Marini, and X. Gonze, [https://doi.org/10.1016/j.commatsci.2011.02.021 Comp. Mat. Science '''50''' 2148-2156 (2011)]</ref>
			</references>

Fulvio: /* Bethe-Salpeter solver runlevel */

2023-05-19T16:55:10Z

Bethe-Salpeter solver runlevel

← Older revision		Revision as of 16:55, 19 May 2023
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	$gnuplot		$gnuplot
	...		...
	plot 'o-3D_BSE.eps_q1_diago_bse' u 1:2 w l t 'Diagonalization', 'o-3D_BSE-low.eps_q1_haydock_bse' u 1:2 w l t 'Haydock (~~last~~ its)', 'o-3D_BSE-low.eps_q1_haydock_bse' u 1:6 w l t 'Haydock (~~last-1~~ its)'		plot 'o-3D_BSE.eps_q1_diago_bse' u 1:2 w l t 'Diagonalization', 'o-3D_BSE-low.eps_q1_haydock_bse' u 1:2 w l t 'Haydock (30 its)', 'o-3D_BSE-low.eps_q1_haydock_bse' u 1:6 w l t 'Haydock (29 its)'
	[[file:03_bse_haydock.png\|none\|600px]]		[[file:03_bse_haydock.png\|none\|600px]]
	Except for the peak at higher energy, the results on this scale are identical to those obtained via the diagonalization solver. By comparing the results for the Lanczos-Haydock solver at 30 and 29 iterations, it is clear that the difference with the diagonalization results can be eliminated by lowering the threshold for the iterative process.		Except for the peak at higher energy, the results on this scale are identical to those obtained via the diagonalization solver. By comparing the results for the Lanczos-Haydock solver at 30 and 29 iterations, it is clear that the difference with the diagonalization results can be eliminated by lowering the threshold for the iterative process.

Fulvio: /* Convergence of the iterative procedure */

2021-03-25T13:11:05Z

Convergence of the iterative procedure

← Older revision		Revision as of 13:11, 25 March 2021
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	outputs the following log		outputs the following log
	...		...
	<~~01s~~> [06] BSE solver(s)		<---> [03] BSE solver(s) @q1
	<~~01s~~> [06.01] Haydock ~~solver~~		<---> [03.01] Haydock Solver for abs @q1, scheme hermitian
	<~~01s~~> [Haydock] Iteration 1		<---> [Haydock] Iteration 1
	<~~06s~~> [Haydock] Iteration 48 Accuracy : 0.~~0098~~\| 0.~~02000~~		<---> [Haydock] Iteration 43 Accuracy 0.002787\| 0.020000
	...		...
	Now 45 iterations, rather than 21, were needed to reach the desired accuracy.		Now 43 iterations, rather than 30, were needed to reach the desired accuracy.
	By plotting these results:		By plotting these results:
	$ gnuplot		$ gnuplot

Fulvio: /* Bethe-Salpeter solver runlevel */

2021-03-25T12:56:18Z

Bethe-Salpeter solver runlevel

← Older revision		Revision as of 12:56, 25 March 2021
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	<---> [Haydock] Iteration 1		<---> [Haydock] Iteration 1
	<---> [Haydock] Iteration 30 Accuracy 0.010712\|-0.020000		<---> [Haydock] Iteration 30 Accuracy 0.010712\|-0.020000

	...		...
	The Lanczos-Haydock iterative procedure converged to the desired accuracy in 30 iterations.		The Lanczos-Haydock iterative procedure converged to the desired accuracy in 30 iterations.
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	Among the human readable files, o-3D_BSE-low.eps_q1_haydock_bse contains the real and imaginary part of the macroscopic dielectric function:		Among the human readable files, o-3D_BSE-low.eps_q1_haydock_bse contains the real and imaginary part of the macroscopic dielectric function:
	$ less o-3D_BSE-low.eps_q1_haydock_bse		$ less o-3D_BSE-low.eps_q1_haydock_bse

	...		...
	#		#

Fulvio: /* Bethe-Salpeter solver runlevel */

2021-03-25T12:55:59Z

Bethe-Salpeter solver runlevel

← Older revision		Revision as of 12:55, 25 March 2021
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	This outputs the following log:		This outputs the following log:
	...		...
	<~~01s~~> [06] BSE solver(s)		<---> [03] BSE solver(s) @q1
	<~~01s~~> [06.01] Haydock ~~solver~~		<---> [03.01] Haydock Solver for abs @q1, scheme hermitian
	<~~01s~~> [Haydock] Iteration 1		<---> [Haydock] Iteration 1
	<~~03s~~> [Haydock] Iteration 30 Accuracy : 0.~~01723~~\| -0.~~02000~~		<---> [Haydock] Iteration 30 Accuracy 0.010712\|-0.020000

	...		...
	The Lanczos-Haydock iterative procedure converged to the desired accuracy in 21 iterations.		The Lanczos-Haydock iterative procedure converged to the desired accuracy in 30 iterations.

	This runs only produces human readable files. That means that if you changes the input parameters (such as the broadening or the energy range) the Lanczos-Haydock procedure has to be repeated.		This runs only produces human readable files. That means that if you changes the input parameters (such as the broadening or the energy range) the Lanczos-Haydock procedure has to be repeated.
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	...		...
	#		#
	# E/ev[1] EPS-Im[2] EPS-Re[3] EPSo-Im[4] EPSo-Re[5] EPS`-Im[6] EPS`-Re[7]		# E/ev[1] EPS-Im[2] EPS-Re[3] EPSo-Im[4] EPSo-Re[5] EPS`-Im[6] EPS`-Re[7]
	#		#
	2.~~00000~~ 0.~~16089 5~~.~~81293~~ 0.~~04945 4~~.~~12727~~ 0.~~16089 5~~.~~81293~~		2.00000000 0.596701317E-1 3.11588907 0.300982650E-1 2.50442362 0.596701317E-1 3.11588907
	2.~~03015~~ 0.~~16711 5~~.~~86203~~ 0.~~05076 4~~.~~14234~~ 0.~~16711 5~~.~~86203~~		2.03015089 0.607327521E-1 3.13400769 0.304788072E-1 2.51354647 0.607327521E-1 3.13400769
	...		...
	This file has a slightly different data structure with respect to the diagonalization solver:		This file has a slightly different data structure with respect to the diagonalization solver:

Fulvio: /* Links */

2021-03-25T11:58:16Z

Links

← Older revision		Revision as of 11:58, 25 March 2021
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	* Previous module: [[Bethe-Salpeter kernel]]		* Previous module: [[Bethe-Salpeter kernel]]
	* Alternative Bethe-Salpeter equation solver: [[Bethe-Salpeter solver: diagonalization \| diagonalization]]		* Alternative Bethe-Salpeter equation solver: [[Bethe-Salpeter solver: diagonalization \| diagonalization]]
			* Alternative Bethe-Salpeter equation solver: [[Bethe-Salpeter solver: SLEPC\|SLEPC]]
	* Back to [[Calculating optical spectra including excitonic effects: a step-by-step guide]] tutorial		* Back to [[Calculating optical spectra including excitonic effects: a step-by-step guide]] tutorial
	* [[Tutorials\|Back to tutorials menu]]		* [[Tutorials\|Back to tutorials menu]]

Fulvio: /* Prerequisites */

2021-03-25T11:53:25Z

Prerequisites

← Older revision		Revision as of 11:53, 25 March 2021
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	==Prerequisites==		==Prerequisites==
	[[File:Yambo-handbook-v4.1.2-p-16.png\|thumb\|Cheatsheet on Haydock\|150px]]		[[File:Yambo-handbook-v4.1.2-p-16.png\|thumb\|Cheatsheet on Haydock\|150px]]
	* You must first complete the [[Static screening]] and [[Bethe-Salpeter kernel]] modules		* You must first complete the [[Static screening]] and [[Bethe-Salpeter kernel]] modules in the [[Calculating optical spectra including excitonic effects: a step-by-step guide\|step-by-step guide]]
			* Optionally, in order to compare results with the diagonalization solver, you may first complete the [[Bethe-Salpeter solver: diagonalization]] module.

	'''You will need''':		'''You will need''':

Maurizia: /* Convergence of the iterative procedure */

2020-01-17T20:17:27Z

Convergence of the iterative procedure

← Older revision		Revision as of 20:17, 17 January 2020
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	<01s> [06.01] Haydock solver		<01s> [06.01] Haydock solver
	<01s> [Haydock] Iteration 1		<01s> [Haydock] Iteration 1
	<06s> [Haydock] Iteration 45 Accuracy : 0.0098\| 0.02000		<06s> [Haydock] Iteration 48 Accuracy : 0.0098\| 0.02000
	...		...
	Now 45 iterations, rather than 21, were needed to reach the desired accuracy.		Now 45 iterations, rather than 21, were needed to reach the desired accuracy.

Maurizia: /* Bethe-Salpeter solver runlevel */

2020-01-17T19:59:57Z

Bethe-Salpeter solver runlevel

← Older revision		Revision as of 19:59, 17 January 2020
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	<01s> [06.01] Haydock solver		<01s> [06.01] Haydock solver
	<01s> [Haydock] Iteration 1		<01s> [Haydock] Iteration 1
	<03s> [Haydock] Iteration 21 Accuracy : 0.~~01647~~\| -0.02000		<03s> [Haydock] Iteration 30 Accuracy : 0.01723\| -0.02000
	...		...
	The Lanczos-Haydock iterative procedure converged to the desired accuracy in 21 iterations.		The Lanczos-Haydock iterative procedure converged to the desired accuracy in 21 iterations.
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	plot 'o-3D_BSE.eps_q1_diago_bse' u 1:2 w l t 'Diagonalization', 'o-3D_BSE-low.eps_q1_haydock_bse' u 1:2 w l t 'Haydock (last its)', 'o-3D_BSE-low.eps_q1_haydock_bse' u 1:6 w l t 'Haydock (last-1 its)'		plot 'o-3D_BSE.eps_q1_diago_bse' u 1:2 w l t 'Diagonalization', 'o-3D_BSE-low.eps_q1_haydock_bse' u 1:2 w l t 'Haydock (last its)', 'o-3D_BSE-low.eps_q1_haydock_bse' u 1:6 w l t 'Haydock (last-1 its)'
	[[file:03_bse_haydock.png\|none\|600px]]		[[file:03_bse_haydock.png\|none\|600px]]
	Except for the peak at higher energy, the results on this scale are identical to those obtained via the diagonalization solver. By comparing the results for the Lanczos-Haydock solver at 27 and 26 iterations, it is clear that the difference with the diagonalization results can be eliminated by lowering the threshold for the iterative process.		Except for the peak at higher energy, the results on this scale are identical to those obtained via the diagonalization solver. By comparing the results for the Lanczos-Haydock solver at 30 and 29 iterations, it is clear that the difference with the diagonalization results can be eliminated by lowering the threshold for the iterative process.

	==Convergence of the iterative procedure==		==Convergence of the iterative procedure==

Maurizia: /* Bethe-Salpeter solver runlevel */

2020-01-17T19:53:17Z

Bethe-Salpeter solver runlevel

← Older revision		Revision as of 19:53, 17 January 2020
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	$gnuplot		$gnuplot
	...		...
	plot 'o-3D_BSE.eps_q1_diago_bse' u 1:2 w l t 'Diagonalization', 'o-3D_BSE-low.eps_q1_haydock_bse' u 1:2 w l t 'Haydock (27 its)', 'o-3D_BSE-low.eps_q1_haydock_bse' u 1:6 w l t 'Haydock (26 its)'		plot 'o-3D_BSE.eps_q1_diago_bse' u 1:2 w l t 'Diagonalization', 'o-3D_BSE-low.eps_q1_haydock_bse' u 1:2 w l t 'Haydock (last its)', 'o-3D_BSE-low.eps_q1_haydock_bse' u 1:6 w l t 'Haydock (last-1 its)'
	[[file:03_bse_haydock.png\|none\|600px]]		[[file:03_bse_haydock.png\|none\|600px]]
	Except for the peak at higher energy, the results on this scale are identical to those obtained via the diagonalization solver. By comparing the results for the Lanczos-Haydock solver at 27 and 26 iterations, it is clear that the difference with the diagonalization results can be eliminated by lowering the threshold for the iterative process.		Except for the peak at higher energy, the results on this scale are identical to those obtained via the diagonalization solver. By comparing the results for the Lanczos-Haydock solver at 27 and 26 iterations, it is clear that the difference with the diagonalization results can be eliminated by lowering the threshold for the iterative process.