Indecomposable module: Difference between revisions

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'''DIIS''' ('''direct inversion in the iterative subspace''' or '''direct inversion of the iterative subspace'''), also known as '''Pulay mixing''', is an [[extrapolation]] technique. DIIS was developed by [[Peter Pulay]] in the field of computational [[quantum chemistry]] with the intent to accelerate and stabilize the convergence of the [[Hartree&ndash;Fock]] self-consistent field method.<ref>{{cite journal|last=Pulay|first=Péter |year=1980|title=Convergence acceleration of iterative sequences. the case of SCF iteration|journal=Chemical Physics Letters|volume=73|issue=2|pages=393&ndash;398|doi=10.1016/0009-2614(80)80396-4}}</ref><ref>{{cite journal|last=Pulay|first=Péter |year=1982|title=Improved SCF Convergence Acceleration|journal=Journal of Computational Chemistry|volume=3|issue=4|pages=556&ndash;560|doi=10.1002/jcc.540030413}}</ref>
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At a given iteration, the approach constructs a [[linear combination]] of approximate error vectors from previous iterations. The coefficients of the linear combination are determined so to best approximate, in a [[least squares]] sense, the [[null vector]]. The newly determined coefficients are then used to extrapolate the function variable for the next iteration.
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==Details==
 
At each iteration, an approximate error vector, '''e'''<sub>''i''</sub>, corresponding to the variable value, '''p'''<sub>''i''</sub> is determined. After sufficient iterations, a linear combination of ''m'' previous error vectors is constructed
 
:<math>\mathbf e_{m+1}=\sum_i^m\ c_i\mathbf e_i.</math>
 
The DIIS method seeks to minimize the norm of '''e'''<sub>''m''+1</sub> under the constraint that the coefficients sum to one. This is done by a [[Lagrange multiplier]] technique. Introducing an undetermined multiplier λ, a Lagrangian is constructed as
 
:<math>
\begin{align}
L&=\|e_{m+1}\|^2-\lambda\left(\sum_i\ c_i-1\right),\\
&=\sum_{ij}c_jB_{ji}c_i-\lambda\left(\sum_i\ c_i-1\right),\ \mathrm{where}\ B_{ij}=\langle\mathbf e_j|\mathbf e_i\rangle.
\end{align}
</math>
 
Equating the derivatives of ''L'', with respect to the coefficients and the multiplier, equal to zero, leads to ''m''&nbsp;+&nbsp;1 [[linear equation]]s to be solved for the ''m'' coefficients. The coefficients are then used to update the function variable as
 
:<math>\mathbf p_{m+1}=\sum_i^m c_i\mathbf p_i.</math>
 
==Citations==
{{reflist}}
 
==References==
* {{cite journal|last1=Garza|first1=Alejandro J.|last2=Scuseria|first2=Gustavo E.|year=2012|title=Comparison of self-consistent field convergence acceleration techniques|journal=Journal of Chemical Physics|volume=173|issue=5|page=054110|doi=10.1063/1.4740249}}
* {{Cite journal|doi=10.1007/s10910-011-9863-y|title=An analysis for the DIIS acceleration method used in quantum chemistry calculations|year=2011|last1=Rohwedder|first1=Thorsten|last2=Schneider|first2=Reinhold|journal=Journal of Mathematical Chemistry|volume=49|issue=9|pages=1889}}
 
==External links==
* [http://vergil.chemistry.gatech.edu/notes/diis/node2.html The Mathematics of DIIS]
 
[[Category:Quantum chemistry]]
[[Category:Numerical linear algebra]]
 
 
{{theoretical-chem-stub}}
{{physical-chemistry-stub}}
{{applied-math-stub}}

Latest revision as of 18:24, 7 December 2014

Succeeding with a home-based business takes a lot of time, dedication and hard work if you want it to pay off. The one thing that all people who have found success with a home-based business is that they had the drive, determination, and effort needed to succeed in a very demanding area. A few savvy people do find success working from home with a variety of businesses. Many people dream of working from home, but for most this dream remains just that, a dream that goes unfulfilled. Working from home can free up your time for family and give you the financial independence you have always dreamed of.

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