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Update Functionals authored by Gabriel Wlazłowski's avatar Gabriel Wlazłowski
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Functionals.md
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......@@ -13,7 +13,7 @@ The functional has the generic form:
\end{aligned}
```
TODO
# Stabilization of ASLDA functional
## Stabilization of ASLDA functional
In the case of calculations for trapped system term $`\frac{\bm{j}_{\sigma}^2}{2n_{\sigma}}`$ is source of numerical instabilities. Precisely, for small density regions we have:
* $`n_{\sigma}\rightarrow 0`$,
* $`\bm{j}_{\sigma}\rightarrow 0`$,
......@@ -37,5 +37,5 @@ Role of $`f_{\textrm{reg.}}`$ is to exclude from computation regions of small de
#define ASLDA_STABILIZATION_EXCLUDE_BELOW_DENISTY 1.0e-7
```
There meaning is presented on the figure below.
![stabilization](uploads/6528cf3b55ebb0d55a3e7f62a6124a83/stabilization.png)
Smooth transition between densities `ASLDA_STABILIZATION_EXCLUDE_BELOW_DENISTY` and `ASLDA_STABILIZATION_RETAIN_ABOVE_DENSITY`
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![stabilization](uploads/6528cf3b55ebb0d55a3e7f62a6124a83/stabilization.png)
Smooth transition between densities `ASLDA_STABILIZATION_EXCLUDE_BELOW_DENISTY` and `ASLDA_STABILIZATION_RETAIN_ABOVE_DENSITY` is introduced in order to avoid discontinuities for quantities, that may lead to divergences when computing derivatives.
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