Preconditioning - Revision history

Huebsch at 15:44, 6 April 2022

2022-04-06T15:44:25Z

← Older revision		Revision as of 15:44, 6 April 2022
Line 1:		Line 1:
	The idea is to find a matrix ~~which~~ multiplied with the residual vector gives the		The idea is to find a matrix that multiplied with the residual vector gives the
	exact error in the wavefunction. Formally this matrix (the Greens function) can be written		exact error in the wavefunction. Formally this matrix (the Greens function) can be written
	down and is given by		down and is given by

	<math>		::<math>
	\frac{1}{{\bf H} - \epsilon_n},		\frac{1}{{\bf H} - \epsilon_n},
	</math>		</math>
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	We actually use the preconditioning function proposed by Teter et. al{{cite\|teter:prb:1989}}		We actually use the preconditioning function proposed by Teter et. al{{cite\|teter:prb:1989}}

	<math>		::<math>
	\langle \mathbf{G} \| {\bf K} \| \mathbf{G'}\rangle = \delta_{\bold{G} \mathbf{G'}} \frac{ 27 + 18 x+12 x^2 + 8x^3}		\langle \mathbf{G} \| {\bf K} \| \mathbf{G'}\rangle = \delta_{\bold{G} \mathbf{G'}} \frac{ 27 + 18 x+12 x^2 + 8x^3}
	{27 + 18x + 12x^2+8x^3 +16x^4} \quad \mbox{and} \quad		{27 + 18x + 12x^2+8x^3 +16x^4} \quad \mbox{and} \quad
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	The preconditioned residual vector is then simply		The preconditioned residual vector is then simply

	<math>		::<math>
	\| p_n \rangle = {\bf K} \| R_n \rangle.		\| p_n \rangle = {\bf K} \| R_n \rangle.
	</math>		</math>
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	<references/>		<references/>
	----		----
	[[Category:Electronic ~~Minimization]][[Category:Electronic Minimization Methods~~]][[Category:Theory]]		[[Category:Electronic minimization]][[Category:Theory]]

Karsai at 10:44, 21 March 2019

2019-03-21T10:44:13Z

← Older revision		Revision as of 10:44, 21 March 2019
Line 18:		Line 18:
	<math>		<math>
	\langle \mathbf{G} \| {\bf K} \| \mathbf{G'}\rangle = \delta_{\bold{G} \mathbf{G'}} \frac{ 27 + 18 x+12 x^2 + 8x^3}		\langle \mathbf{G} \| {\bf K} \| \mathbf{G'}\rangle = \delta_{\bold{G} \mathbf{G'}} \frac{ 27 + 18 x+12 x^2 + 8x^3}
	{27 + 18x + 12x^2+8x^3 +16x^4} \quad \mbox{~~und~~} \quad		{27 + 18x + 12x^2+8x^3 +16x^4} \quad \mbox{and} \quad
	x = \frac{\hbar^2}{2m} \frac{G^2} {1.5 E^{\rm kin}( \mathbf{R}) },		x = \frac{\hbar^2}{2m} \frac{G^2} {1.5 E^{\rm kin}( \mathbf{R}) },
	</math>		</math>

Karsai at 10:42, 21 March 2019

2019-03-21T10:42:43Z

← Older revision		Revision as of 10:42, 21 March 2019
Line 10:		Line 10:
	Actually the evaluation of this matrix is not possible, recognizing that the		Actually the evaluation of this matrix is not possible, recognizing that the
	kinetic energy dominates the Hamiltonian for large <math>\mathbf{G}</math>-vectors		kinetic energy dominates the Hamiltonian for large <math>\mathbf{G}</math>-vectors
	(i.e. <math>H_{G,G'} \to \delta_{G,G'} \frac{\hbar^2}{2m} \mathbf{G}^2</math>), it		(i.e. <math>H_{\mathbf{G},\mathbf{G'}} \to \delta_{\mathbf{G},\mathbf{G'}} \frac{\hbar^2}{2m} \mathbf{G}^2</math>), it
	is a good idea to approximate the matrix by a diagonal		is a good idea to approximate the matrix by a diagonal
	function which converges to <math>\frac{2m}{\hbar^2 \mathbf{G}^2}</math> for large <math>\mathbf{G}</math> vectors, and possess		function which converges to <math>\frac{2m}{\hbar^2 \mathbf{G}^2}</math> for large <math>\mathbf{G}</math> vectors, and possess

Karsai at 10:41, 21 March 2019

2019-03-21T10:41:58Z

← Older revision		Revision as of 10:41, 21 March 2019
Line 12:		Line 12:
	(i.e. <math>H_{G,G'} \to \delta_{G,G'} \frac{\hbar^2}{2m} \mathbf{G}^2</math>), it		(i.e. <math>H_{G,G'} \to \delta_{G,G'} \frac{\hbar^2}{2m} \mathbf{G}^2</math>), it
	is a good idea to approximate the matrix by a diagonal		is a good idea to approximate the matrix by a diagonal
	function which converges to <math>\frac{2m}{\hbar^2 \mathbf{G}^2}</math> for large <math>\mathbf[G}</math> vectors, and possess		function which converges to <math>\frac{2m}{\hbar^2 \mathbf{G}^2}</math> for large <math>\mathbf{G}</math> vectors, and possess
	a constant value for small <math>\mathbf{G}</math> vectors.		a constant value for small <math>\mathbf{G}</math> vectors.
	We actually use the preconditioning function proposed by Teter et. al{{cite\|teter:prb:1989}}		We actually use the preconditioning function proposed by Teter et. al{{cite\|teter:prb:1989}}

Karsai at 10:41, 21 March 2019

2019-03-21T10:41:25Z

← Older revision		Revision as of 10:41, 21 March 2019
Line 7:		Line 7:
	</math>		</math>

	where $ \epsilon_n$ is the exact eigenvalue for the band in interest.		where <math> \epsilon_n</math> is the exact eigenvalue for the band in interest.
	Actually the evaluation of this matrix is not possible, recognizing that the		Actually the evaluation of this matrix is not possible, recognizing that the
	kinetic energy dominates the Hamiltonian for large <math>G</math>-vectors		kinetic energy dominates the Hamiltonian for large <math>\mathbf{G}</math>-vectors
	(i.e. <math>H_{G,G'} \to \delta_{G,G'} \frac{\hbar^2}{2m} \mathbf{G}^2</math>), it		(i.e. <math>H_{G,G'} \to \delta_{G,G'} \frac{\hbar^2}{2m} \mathbf{G}^2</math>), it
	is a good idea to approximate the matrix by a diagonal		is a good idea to approximate the matrix by a diagonal

Karsai at 10:40, 21 March 2019

2019-03-21T10:40:30Z

← Older revision		Revision as of 10:40, 21 March 2019
Line 10:		Line 10:
	Actually the evaluation of this matrix is not possible, recognizing that the		Actually the evaluation of this matrix is not possible, recognizing that the
	kinetic energy dominates the Hamiltonian for large <math>G</math>-vectors		kinetic energy dominates the Hamiltonian for large <math>G</math>-vectors
	(i.e. <math>H_{G,G'} \to \delta_{G,G'} \frac{\hbar^2}{2m} G^2</math>), it		(i.e. <math>H_{G,G'} \to \delta_{G,G'} \frac{\hbar^2}{2m} \mathbf{G}^2</math>), it
	is a good idea to approximate the matrix by a diagonal		is a good idea to approximate the matrix by a diagonal
	function which converges to <math>\frac{2m}{\hbar^2 G^2}</math> for large <math>G</math> vectors, and possess		function which converges to <math>\frac{2m}{\hbar^2 \mathbf{G}^2}</math> for large <math>\mathbf[G}</math> vectors, and possess
	a constant value for small <math>G</math> vectors.		a constant value for small <math>\mathbf{G}</math> vectors.
	We actually use the preconditioning function proposed by Teter et. al{{cite\|teter:prb:1989}}		We actually use the preconditioning function proposed by Teter et. al{{cite\|teter:prb:1989}}

	<math>		<math>
	\langle \~~bold~~{G} \| {\bf K} \| \~~bold~~{G'}\rangle = \delta_{\bold{G} \~~bold~~{G'}} \frac{ 27 + 18 x+12 x^2 + 8x^3}		\langle \mathbf{G} \| {\bf K} \| \mathbf{G'}\rangle = \delta_{\bold{G} \mathbf{G'}} \frac{ 27 + 18 x+12 x^2 + 8x^3}
	{27 + 18x + 12x^2+8x^3 +16x^4} \quad \mbox{und} \quad		{27 + 18x + 12x^2+8x^3 +16x^4} \quad \mbox{und} \quad
	x = \tp \frac{G^2} {1.5 E^{\rm kin}( \bR) },		x = \frac{\hbar^2}{2m} \frac{G^2} {1.5 E^{\rm kin}( \mathbf{R}) },
	</math>		</math>

Karsai at 10:37, 21 March 2019

2019-03-21T10:37:22Z

← Older revision		Revision as of 10:37, 21 March 2019
Line 17:		Line 17:

	<math>		<math>
	\langle \bG \| {\bf K} \| \bG'\rangle = \delta_{\bG \bG'} \frac{ 27 + 18 x+12 x^2 + 8x^3}		\langle \bold{G} \| {\bf K} \| \bold{G'}\rangle = \delta_{\bold{G} \bold{G'}} \frac{ 27 + 18 x+12 x^2 + 8x^3}
	{27 + 18x + 12x^2+8x^3 +16x^4} \quad \mbox{und} \quad		{27 + 18x + 12x^2+8x^3 +16x^4} \quad \mbox{und} \quad
	x = \tp \frac{G^2} {1.5 E^{\rm kin}( \bR) },		x = \tp \frac{G^2} {1.5 E^{\rm kin}( \bR) },
	</math>		</math>

	with <math>E^{\rm kin}(\bR)</math> being the kinetic energy of the residual vector.		with <math>E^{\rm kin}(\bold{R})</math> being the kinetic energy of the residual vector.
	The preconditioned residual vector is then simply		The preconditioned residual vector is then simply

			<math>
			\| p_n \rangle = {\bf K} \| R_n \rangle.
			</math>

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

Karsai at 10:35, 21 March 2019

2019-03-21T10:35:35Z

← Older revision		Revision as of 10:35, 21 March 2019
Line 24:		Line 24:
	with <math>E^{\rm kin}(\bR)</math> being the kinetic energy of the residual vector.		with <math>E^{\rm kin}(\bR)</math> being the kinetic energy of the residual vector.
	The preconditioned residual vector is then simply		The preconditioned residual vector is then simply
			== References ==
	</~~references~~>		<references/>
	----		----
	[[Category:Electronic Minimization]][[Category:Electronic Minimization Methods]][[Category:Theory]]		[[Category:Electronic Minimization]][[Category:Electronic Minimization Methods]][[Category:Theory]]

Karsai: Created page with "The idea is to find a matrix which multiplied with the residual vector gives the exact error in the wavefunction. Formally this matrix (the Greens function) can be written dow..."

2019-03-21T10:34:41Z

Created page with "The idea is to find a matrix which multiplied with the residual vector gives the exact error in the wavefunction. Formally this matrix (the Greens function) can be written dow..."

New page

The idea is to find a matrix which multiplied with the residual vector gives the
exact error in the wavefunction. Formally this matrix (the Greens function) can be written
down and is given by

<math>
\frac{1}{{\bf H} - \epsilon_n},
</math>

where $ \epsilon_n$ is the exact eigenvalue for the band in interest.
Actually the evaluation of this matrix is not possible, recognizing that the
kinetic energy dominates the Hamiltonian for large <math>G</math>-vectors
(i.e. <math>H_{G,G'} \to \delta_{G,G'} \frac{\hbar^2}{2m} G^2</math>), it
is a good idea to approximate the matrix by a diagonal
function which converges to <math>\frac{2m}{\hbar^2 G^2}</math> for large <math>G</math> vectors, and possess
a constant value for small <math>G</math> vectors.
We actually use the preconditioning function proposed by Teter et. al{{cite|teter:prb:1989}}

<math>
\langle \bG | {\bf K} | \bG'\rangle = \delta_{\bG \bG'} \frac{ 27 + 18 x+12 x^2 + 8x^3}
{27 + 18x + 12x^2+8x^3 +16x^4} \quad \mbox{und} \quad
x = \tp \frac{G^2} {1.5 E^{\rm kin}( \bR) },
</math>

with <math>E^{\rm kin}(\bR)</math> being the kinetic energy of the residual vector.
The preconditioned residual vector is then simply

</references>
----
[[Category:Electronic Minimization]][[Category:Electronic Minimization Methods]][[Category:Theory]]