Preconditioning: Difference between revisions
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The idea is to find a matrix | 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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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 | ||
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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{ | {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> | ||
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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 | [[Category:Electronic minimization]][[Category:Theory]] |
Latest revision as of 15:44, 6 April 2022
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 down and is given by
where 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 -vectors (i.e. ), it is a good idea to approximate the matrix by a diagonal function which converges to for large vectors, and possess a constant value for small vectors. We actually use the preconditioning function proposed by Teter et. al[1]
with being the kinetic energy of the residual vector. The preconditioned residual vector is then simply