Blocked-Davidson algorithm - Revision history

Vaspmaster at 12:31, 14 November 2023

2023-11-14T12:31:03Z

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 * Move on to the next block <math>\{ \psi^1_k| k=n_1+1,..,2 n_1\}</math>.
 * When {{TAG|LDIAG}}=.TRUE. (default), a Rayleigh-Ritz optimization in the complete subspace <math>\{ \psi_k| k=1,..,N_{\rm bands}\}</math> is performed after all orbitals have been optimized.
 The blocked-Davidson algorithm is approximately a factor of 1.5-2 slower than the [[RMM-DIIS]], but more robust.

Vaspmaster at 16:54, 13 November 2023

2023-11-13T16:54:10Z

← Older revision		Revision as of 16:54, 13 November 2023
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	::<math>\{ \psi^d_k\| k=1,..,n_1\} \Rightarrow \{ \psi_k\| k=1,..,N_{\rm bands}\}</math>.		::<math>\{ \psi^d_k\| k=1,..,n_1\} \Rightarrow \{ \psi_k\| k=1,..,N_{\rm bands}\}</math>.
	* Move on to the next block <math>\{ \psi^1_k\| k=n_1+1,..,2 n_1\}</math>.		* Move on to the next block <math>\{ \psi^1_k\| k=n_1+1,..,2 n_1\}</math>.
	* ~~After all orbitals have been optimized~~, a Rayleigh-Ritz optimization in the complete subspace <math>\{ \psi_k\| k=1,..,N_{\rm bands}\}</math> is performed.		* When {{TAG\|LDIAG}}=.TRUE. (default), a Rayleigh-Ritz optimization in the complete subspace <math>\{ \psi_k\| k=1,..,N_{\rm bands}\}</math> is performed after all orbitals have been optimized.

	The blocked-Davidson algorithm is approximately a factor of 1.5-2 slower than the [[RMM-DIIS]], but more robust.		The blocked-Davidson algorithm is approximately a factor of 1.5-2 slower than the [[RMM-DIIS]], but more robust.

Vaspmaster at 08:48, 20 October 2023

2023-10-20T08:48:15Z

Vaspmaster at 19:16, 19 October 2023

2023-10-19T19:16:12Z

← Older revision		Revision as of 19:16, 19 October 2023
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	* When the iteration is finished, store the optimized block of orbitals back into the set:		* When the iteration is finished, store the optimized block of orbitals back into the set:
	:<math>\{ \psi^d_k\| k=1,..,n_1\} \Rightarrow \{ \psi_k\| k=1,..,N_{\rm bands}\}</math>.		:<math>\{ \psi^d_k\| k=1,..,n_1\} \Rightarrow \{ \psi_k\| k=1,..,N_{\rm bands}\}</math>.
	* ~~Continue with~~ the next block <math>\{ \psi^1_k\| k=n_1+1,..,2 n_1\}</math>.		* Move on to the next block <math>\{ \psi^1_k\| k=n_1+1,..,2 n_1\}</math>.
	* After all orbitals have been optimized, a Rayleigh-Ritz optimization in the complete subspace <math>\{ \psi_k\| k=1,..,N_{\rm bands}\}</math> is performed.		* After all orbitals have been optimized, a Rayleigh-Ritz optimization in the complete subspace <math>\{ \psi_k\| k=1,..,N_{\rm bands}\}</math> is performed.

Vaspmaster: Vaspmaster moved page Construction:Blocked-Davidson algorithm to Blocked-Davidson algorithm without leaving a redirect

2023-10-19T19:14:27Z

Vaspmaster moved page Construction:Blocked-Davidson algorithm to Blocked-Davidson algorithm without leaving a redirect

← Older revision

Revision as of 19:14, 19 October 2023

Vaspmaster at 19:13, 19 October 2023

2023-10-19T19:13:24Z

← Older revision		Revision as of 19:13, 19 October 2023
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	The workflow of the blocked-Davidson iterative matrix diagonalization scheme implemented in {{VASP}} is as follows:		The workflow of the blocked-Davidson iterative matrix diagonalization scheme implemented in {{VASP}} is as follows:{{cite\|kresse:cms:1996}}{{cite\|kresse:prb:96}}

	* Take a subset (block) of <math>n_1</math> orbitals out of the total set of {{TAG\|NBANDS}} orbitals:		* Take a subset (block) of <math>n_1</math> orbitals out of the total set of {{TAG\|NBANDS}} orbitals:
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	The blocked-Davidson algorithm is approximately a factor of 1.5-2 slower than the [[RMM-DIIS]], but more robust.		The blocked-Davidson algorithm is approximately a factor of 1.5-2 slower than the [[RMM-DIIS]], but more robust.

			== References ==
			<references/>
	----		----
	[[Category:Electronic minimization]][[Category:Theory]]		[[Category:Electronic minimization]][[Category:Theory]]

Vaspmaster at 19:04, 19 October 2023

2023-10-19T19:04:16Z

← Older revision		Revision as of 19:04, 19 October 2023
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	The blocked-Davidson algorithm is approximately a factor of 1.5-2 slower than the [[RMM-DIIS]], but more robust.		The blocked-Davidson algorithm is approximately a factor of 1.5-2 slower than the [[RMM-DIIS]], but more robust.

			----
			[[Category:Electronic minimization]][[Category:Theory]]

Vaspmaster at 19:03, 19 October 2023

2023-10-19T19:03:45Z

← Older revision		Revision as of 19:03, 19 October 2023
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	* After all orbitals have been optimized, a Rayleigh-Ritz optimization in the complete subspace <math>\{ \psi_k\| k=1,..,N_{\rm bands}\}</math> is performed.		* After all orbitals have been optimized, a Rayleigh-Ritz optimization in the complete subspace <math>\{ \psi_k\| k=1,..,N_{\rm bands}\}</math> is performed.

	~~This method~~ is approximately a factor of 1.5-2 slower than [[RMM-DIIS]], but more robust.		The blocked-Davidson algorithm is approximately a factor of 1.5-2 slower than the [[RMM-DIIS]], but more robust.

Vaspmaster at 19:03, 19 October 2023

2023-10-19T19:03:09Z

Vaspmaster at 18:31, 19 October 2023

2023-10-19T18:31:24Z

← Older revision		Revision as of 18:31, 19 October 2023
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	:<math>{\rm diag}\{\psi^1/g^1/g^2\} \Rightarrow \{ \psi^3_k\| k=1,..,n_1\}</math>		:<math>{\rm diag}\{\psi^1/g^1/g^2\} \Rightarrow \{ \psi^3_k\| k=1,..,n_1\}</math>
	* If need be the subspace may be extended by repetition of this cycle of adding residual vectors and Rayleigh-Ritz optimization of the resulting subspace:		* If need be the subspace may be extended by repetition of this cycle of adding residual vectors and Rayleigh-Ritz optimization of the resulting subspace:
	:<~~math>~~math>{\rm diag}\{\psi^1/g^1/g^2/../g^{d-1}\}\Rightarrow \{ \psi^d_k\| k=1,..,n_1\</math>		:<math>{\rm diag}\{\psi^1/g^1/g^2/../g^{d-1}\}\Rightarrow \{ \psi^d_k\| k=1,..,n_1\}</math>
	: Per default {{VASP}} will not iterate deeper than <math>d=4</math>, though it may break off even sooner when certain criteria that measure the convergence of the orbitals have been met.		: Per default {{VASP}} will not iterate deeper than <math>d=4</math>, though it may break off even sooner when certain criteria that measure the convergence of the orbitals have been met.
	* When the iteration is finished, store the optimized block of orbitals back in the set:		* When the iteration is finished, store the optimized block of orbitals back in the set:

← Older revision		Revision as of 08:48, 20 October 2023
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	* Take a subset (block) of <math>n_1</math> orbitals out of the total set of {{TAG\|NBANDS}} orbitals:		* Take a subset (block) of <math>n_1</math> orbitals out of the total set of {{TAG\|NBANDS}} orbitals:
	:<math> \{ \psi_n\| n=1,..,N_{\rm bands}\}\Rightarrow \{ \psi^1_k\| k=1,..,n_1\}</math>.		::<math> \{ \psi_n\| n=1,..,N_{\rm bands}\}\Rightarrow \{ \psi^1_k\| k=1,..,n_1\}</math>.
	* Extend the subspace spanned by <math>\{\psi^1\}</math> by adding the preconditioned residual vectors of <math>\{\psi^1\}</math>:		* Extend the subspace spanned by <math>\{\psi^1\}</math> by adding the preconditioned residual vectors of <math>\{\psi^1\}</math>:
	:<math>		::<math>
	\left \{ \psi^1_k \, / \, g^1_k = \left (1- \sum_{n=1}^{N_{\rm bands}} \| \psi_n \rangle \langle\psi_n \| {\bf S} \right) {\bf K} \left ({\bf H} - \epsilon_{\rm app} {\bf S} \right ) \psi^1_k \, \| \, k=1,..,n_1 \right \}.		\left \{ \psi^1_k \, / \, g^1_k = \left (1- \sum_{n=1}^{N_{\rm bands}} \| \psi_n \rangle \langle\psi_n \| {\bf S} \right) {\bf K} \left ({\bf H} - \epsilon_{\rm app} {\bf S} \right ) \psi^1_k \, \| \, k=1,..,n_1 \right \}.
	</math>		</math>
	* Rayleigh-Ritz optimization ("subspace rotation") within the <math>2n_1</math>-dimensional space spanned by <math>\{\psi^1/g^1\}</math>, to determine the <math>n_1</math> lowest eigenvectors:		* Rayleigh-Ritz optimization ("subspace rotation") within the <math>2n_1</math>-dimensional space spanned by <math>\{\psi^1/g^1\}</math>, to determine the <math>n_1</math> lowest eigenvectors:
	:<math>{\rm diag}\{\psi^1/g^1\} \Rightarrow \{ \psi^2_k\| k=1,..,n_1\}</math>		::<math>{\rm diag}\{\psi^1/g^1\} \Rightarrow \{ \psi^2_k\| k=1,..,n_1\}</math>
	* Extend the subspace with the residuals of <math>\{\psi^2\}</math>:		* Extend the subspace with the residuals of <math>\{\psi^2\}</math>:
	:<math>		::<math>
	\left \{ \psi^2_k \,/ \, g^1_k \, / \, g^2_k = \left (1- \sum_{n=1}^{N_{\rm bands}} \| \psi_n \rangle \langle\psi_n \| {\bf S} \right ) {\bf K} \left ({\bf H} - \epsilon_{\rm app} {\bf S} \right) \psi^2_k \, \| \, k=1,..,n_1 \right \}.		\left \{ \psi^2_k \,/ \, g^1_k \, / \, g^2_k = \left (1- \sum_{n=1}^{N_{\rm bands}} \| \psi_n \rangle \langle\psi_n \| {\bf S} \right ) {\bf K} \left ({\bf H} - \epsilon_{\rm app} {\bf S} \right) \psi^2_k \, \| \, k=1,..,n_1 \right \}.
	</math>		</math>
	* Rayleigh-Ritz optimization ("subspace rotation") within the <math>3n_1</math>-dimensional space spanned by <math>\{\psi^1/g^1/g^2\}</math>:		* Rayleigh-Ritz optimization ("subspace rotation") within the <math>3n_1</math>-dimensional space spanned by <math>\{\psi^1/g^1/g^2\}</math>:
	:<math>{\rm diag}\{\psi^1/g^1/g^2\} \Rightarrow \{ \psi^3_k\| k=1,..,n_1\}</math>		::<math>{\rm diag}\{\psi^1/g^1/g^2\} \Rightarrow \{ \psi^3_k\| k=1,..,n_1\}</math>
	* If need be the subspace may be extended by repetition of this cycle of adding residual vectors and Rayleigh-Ritz optimization of the resulting subspace:		* If need be the subspace may be extended by repetition of this cycle of adding residual vectors and Rayleigh-Ritz optimization of the resulting subspace:
	:<math>{\rm diag}\{\psi^1/g^1/g^2/../g^{d-1}\}\Rightarrow \{ \psi^d_k\| k=1,..,n_1\}</math>		::<math>{\rm diag}\{\psi^1/g^1/g^2/../g^{d-1}\}\Rightarrow \{ \psi^d_k\| k=1,..,n_1\}</math>
	: Per default {{VASP}} will not iterate deeper than <math>d=4</math>, though it may break off even sooner when certain criteria that measure the convergence of the orbitals have been met.		: Per default {{VASP}} will not iterate deeper than <math>d=4</math>, though it may break off even sooner when certain criteria that measure the convergence of the orbitals have been met.
	* When the iteration is finished, store the optimized block of orbitals back into the set:		* When the iteration is finished, store the optimized block of orbitals back into the set:
	:<math>\{ \psi^d_k\| k=1,..,n_1\} \Rightarrow \{ \psi_k\| k=1,..,N_{\rm bands}\}</math>.		::<math>\{ \psi^d_k\| k=1,..,n_1\} \Rightarrow \{ \psi_k\| k=1,..,N_{\rm bands}\}</math>.
	* Move on to the next block <math>\{ \psi^1_k\| k=n_1+1,..,2 n_1\}</math>.		* Move on to the next block <math>\{ \psi^1_k\| k=n_1+1,..,2 n_1\}</math>.
	* After all orbitals have been optimized, a Rayleigh-Ritz optimization in the complete subspace <math>\{ \psi_k\| k=1,..,N_{\rm bands}\}</math> is performed.		* After all orbitals have been optimized, a Rayleigh-Ritz optimization in the complete subspace <math>\{ \psi_k\| k=1,..,N_{\rm bands}\}</math> is performed.

← Older revision		Revision as of 19:03, 19 October 2023
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	\left \{ \psi^1_k \, / \, g^1_k = \left (1- \sum_{n=1}^{N_{\rm bands}} \| \psi_n \rangle \langle\psi_n \| {\bf S} \right) {\bf K} \left ({\bf H} - \epsilon_{\rm app} {\bf S} \right ) \psi^1_k \, \| \, k=1,..,n_1 \right \}.		\left \{ \psi^1_k \, / \, g^1_k = \left (1- \sum_{n=1}^{N_{\rm bands}} \| \psi_n \rangle \langle\psi_n \| {\bf S} \right) {\bf K} \left ({\bf H} - \epsilon_{\rm app} {\bf S} \right ) \psi^1_k \, \| \, k=1,..,n_1 \right \}.
	</math>		</math>
	* Rayleigh-Ritz optimization ("subspace rotation") within the <math>2n_1</math> dimensional space spanned by <math>\{\psi^1/g^1\}</math>, to determine the <math>n_1</math> lowest eigenvectors:		* Rayleigh-Ritz optimization ("subspace rotation") within the <math>2n_1</math>-dimensional space spanned by <math>\{\psi^1/g^1\}</math>, to determine the <math>n_1</math> lowest eigenvectors:
	:<math>{\rm diag}\{\psi^1/g^1\} \Rightarrow \{ \psi^2_k\| k=1,..,n_1\}</math>		:<math>{\rm diag}\{\psi^1/g^1\} \Rightarrow \{ \psi^2_k\| k=1,..,n_1\}</math>
	* Extend the subspace with residuals of <math>\{\psi^2\}</math>:		* Extend the subspace with the residuals of <math>\{\psi^2\}</math>:
	:<math>		:<math>
	\left \{ \psi^2_k \,/ \, g^1_k \, / \, g^2_k = \left (1- \sum_{n=1}^{N_{\rm bands}} \| \psi_n \rangle \langle\psi_n \| {\bf S} \right ) {\bf K} \left ({\bf H} - \epsilon_{\rm app} {\bf S} \right) \psi^2_k \, \| \, k=1,..,n_1 \right \}.		\left \{ \psi^2_k \,/ \, g^1_k \, / \, g^2_k = \left (1- \sum_{n=1}^{N_{\rm bands}} \| \psi_n \rangle \langle\psi_n \| {\bf S} \right ) {\bf K} \left ({\bf H} - \epsilon_{\rm app} {\bf S} \right) \psi^2_k \, \| \, k=1,..,n_1 \right \}.
	</math>		</math>
	* Rayleigh-Ritz optimization ("subspace rotation") within the <math>3n_1</math> dimensional space spanned by <math>\{\psi^1/g^1/g^2\}</math>:		* Rayleigh-Ritz optimization ("subspace rotation") within the <math>3n_1</math>-dimensional space spanned by <math>\{\psi^1/g^1/g^2\}</math>:
	:<math>{\rm diag}\{\psi^1/g^1/g^2\} \Rightarrow \{ \psi^3_k\| k=1,..,n_1\}</math>		:<math>{\rm diag}\{\psi^1/g^1/g^2\} \Rightarrow \{ \psi^3_k\| k=1,..,n_1\}</math>
	* If need be the subspace may be extended by repetition of this cycle of adding residual vectors and Rayleigh-Ritz optimization of the resulting subspace:		* If need be the subspace may be extended by repetition of this cycle of adding residual vectors and Rayleigh-Ritz optimization of the resulting subspace:
	:<math>{\rm diag}\{\psi^1/g^1/g^2/../g^{d-1}\}\Rightarrow \{ \psi^d_k\| k=1,..,n_1\}</math>		:<math>{\rm diag}\{\psi^1/g^1/g^2/../g^{d-1}\}\Rightarrow \{ \psi^d_k\| k=1,..,n_1\}</math>
	: Per default {{VASP}} will not iterate deeper than <math>d=4</math>, though it may break off even sooner when certain criteria that measure the convergence of the orbitals have been met.		: Per default {{VASP}} will not iterate deeper than <math>d=4</math>, though it may break off even sooner when certain criteria that measure the convergence of the orbitals have been met.
	* When the iteration is finished, store the optimized block of orbitals back in the set:		* When the iteration is finished, store the optimized block of orbitals back into the set:
	:<math>\{ \psi^d_k\| k=1,..,n_1\} \Rightarrow \{ \psi_k\| k=1,..,N_{\rm bands}\}</math>.		:<math>\{ \psi^d_k\| k=1,..,n_1\} \Rightarrow \{ \psi_k\| k=1,..,N_{\rm bands}\}</math>.
	* Continue with the next block <math>\{ \psi^1_k\| k=n_1+1,..,2 n_1\}</math>.		* Continue with the next block <math>\{ \psi^1_k\| k=n_1+1,..,2 n_1\}</math>.
	* After ~~each band has~~ been optimized a Rayleigh-Ritz optimization in the complete subspace <math>\{ \~~phi_k~~\| k=1,..,N_{\rm bands}\}</math> is performed.		* After all orbitals have been optimized, a Rayleigh-Ritz optimization in the complete subspace <math>\{ \psi_k\| k=1,..,N_{\rm bands}\}</math> is performed.

	This method is approximately a factor of 1.5-2 slower than RMM-DIIS, but ~~always stable. It is available in parallel for any data distribution~~.		This method is approximately a factor of 1.5-2 slower than [[RMM-DIIS]], but more robust.