potfit wiki

This page is read only. You can view the source, but not change it. Ask your administrator if you think this is wrong.
~~NOTOC~~
====== Algorithms ======
----
This page is a quote from the [[http://elib.uni-stuttgart.de/opus/volltexte/2008/3410/|PhD Thesis of P. Brommer]].
----

=== Force Matching ===

As described above, classical MD calculations require effective potentials,
which for many simple materials are matched to well-known experimental
values. For structurally or chemically complex materials, there are generally
more potential parameters to be determined, and the number of usable
experimental measurements does not cover the increased need for data. In some
cases, the exact structure and composition of the samples is not even known
reliably, thus making it difficult to infer the potentials from the
measurements. 

Here, ab-initio calculations open the door to a modified approach. The
experimental data a potential is matched against is supplemented or replaced
by values from first-principles DFT calculations. By including information
like cohesive energies and forces from a set of reference configurations,
potentials for both simple and complex materials can be put on a solid
physical base. The vastly increased number of reference information
additionally allows more flexible potentials compared to the experimental
approach, where only few parameters can be determined reliably. 

The method of including forces from ab-initio calculations in the potential generation 
is called **Force Matching** and was first described by Ercolessi and Adams
((Ercolessi, F. and Adams, J. B.: Interatomic Potentials from First-Principles Calculations: the Force-Matching Method. //Europhys. Lett.//, **26** (8), 583–588, 1994.)),
where it was used to determine a glue potential for Aluminium. In the following section, 
this method is described in detail. 

To determine a potential with force matching one needs first a potential
model, where the potential is given by a set of $n$ parameters $\xi_{i}$.
These parameters might be the values of the potential functions at sampling
points or parameters of an analytic potential. This set $\boldsymbol\xi$ is then
adjusted to optimally reproduce quantities calculated from DFT in a number of
reference calculations. This optimisation is realised by minimising the sum of
squares of the deviations between reference values and those calculated from
the potential.

The function to be minimised is the so-called target
function $Z(\boldsymbol\xi)$ defined as
$$Z(\boldsymbol\xi)=Z_{F}(\boldsymbol\xi)+Z_{C},$$
where
$$Z_{F}(\boldsymbol\xi)=\sum_{k=1}^{m}u_{k}(F_{k}(\boldsymbol\xi)-F^{0}_{k})^{2},$$
$$Z_{C}(\boldsymbol\xi)=\sum_{r=1}^{N_{c}}w_{r}(A_{r}(\boldsymbol\xi)-A^{0}_{r})^{2}.$$

Here, the index $k$ in $Z_{F}$, the sum of reference contributions to the
target function, runs over all reference information, be it
cohesive energies, force vector components or entries of the stress tensor.
$F^{0}_{k}$ are the reference values and $F_{k}(\boldsymbol\xi)$ the
corresponding values calculated from the potential, and the $u_{k}$ are the
weights attached to each contribution to $Z_{F}$. Additional minimisation
criteria are accounted for in $Z_{C}$, where $A_{r}(\boldsymbol\xi)$ is the actual and
$A^{0}_{r}$ the nominal value of a quantity, the importance of which is
controlled by the weights $w_{r}$. These additional contributions can be used
to implement constraints on the potential functions, like for example fixing
the gauge degrees of freedom of glue potentials (see Sec. 2.2.2, Eq. (2.38)). 

The target function $Z(\boldsymbol\xi)$ is then optimised. As the
evaluation of the target function is quite costly, the algorithms used must be
efficient to reduce computing time. A selection of suitable algorithms is
described in section 2.3.1. 

Crucial for the quality of force matched potentials is the selection of
reference configurations. Methods to generate and pick these configurations
sensibly are covered in section 2.3.2. The third ingredient of
successful force matching is a suitable potential model and representation; section
2.3.3 elaborates on this topic.

=== Optimisation ===

Few mathematical topics have such a widespread effect on many subjects in
science and engineering like optimisation. Finding the optimal
order of conductors on a chip, the optimal reaction conditions in a chemical
plant or the optimal curve to represent a series of measurements: One always
searches for the best solution. This is of course also the case for finding
the optimal potential that yields the correct forces, stresses and energies in
molecular dynamics simulation.

For quantitative treatment of an optimisation problem one defines a target
function $Z$, which is a function of the parameters $\xi_{i}$ (reaction
conditions, function parameters, $\ldots$) of the problem. Without loss of
generality, the minimum of the target function solves the optimisation
problem, thereby reducing it to minimising a function of several variables --
a well-known task with many well-documented algorithms.

----

=== compat-mode ===

In older versions of //potfit// the error sum has been calculated as 

$$Z_{F}(\boldsymbol\xi)=\sum_{k=1}^{m}[u_{k}(F_{k}(\boldsymbol\xi)-F^{0}_{k})]^2.$$

This behaviour is no longer supported. The ''compat'' mode was removed with version 0.6.0.

==== Algorithms available in potfit ====

Currently there are three optimization algorithms available:
  - [[algorithms:simann|Simulated Annealing]]
  - [[algorithms:diffevo|Differential Evolution]]
  - [[algorithms:lsq|Method of least squares]] (conjugate gradients)

Usually two of them are run subsequently, with the choice between 1 and 2, while 3 is always run. Simulatead Annealing can be used for both tabulated and analytic potentials,
Differential Evolution may run into troubles with tabulated potentials but works fine for analytic ones.