[Pw_forum] SCF convergence - proposal

[Konstantin Kudin]

--- Paolo Giannozzi <giannozz at nest.sns.it> wrote:

> it's not that simple. Early versions of PWscf used
> Anderson mixing:
> D.G. Anderson, J. Assoc. Comput. Mach. 12, 547
> (1965). Later, we 
> moved to a modified Broyden method: D.D. Johnson,
> Phys. Rev. B 
> 38, 12807 (1988).  Recently we have started to use
> mixing of charge
> density instead of potential, a better criterion for
> the error to be
> minimized, and an estimate for the dielectric
> response for highly
> inhomogeneous cases. The present algorithm is quite
> sophisticated.
> It may require some tuning, however, and how to tune
> it is not always
> obvious. 

 Thanks for clarification! From what I observe the
method that is in PWSCF works great for things which
are reasonably close to convergence, but can have
problems in situations when the starting density is
very different from the converged one. 
 
> Anyway: your suggestion is most welcome. An
> algorithm that
> converges no matter what would be highly desirable.

 OK. Would any of you guys be willing to help me out
with that? As I mentioned, the EDIIS method is both
very simple and robust. There is a mathematical proof
that it always works for Hartree-Fock, and in practice
it very often works for DFT except when the fractional
occupations are present where the method drives the
solution to fractional occupations and that can
interfere with the integer only occupations. 

 Nicola Marzari mentioned another method with a
guaranteed convergence. But it seems like it is quite
complex to implement, and thus something simplier
could be advantageous at this stage before anybody
attempts to take a shot at Nicola's method :-)

 EDIIS works well for initial steps, after that the
traditional techniques are usually faster. This is
currently an area of technical deficiency. In
practice, the region of convergence is usually
achieved in 5-10 steps. Since EDIIS is designed to mix
densities such that the energy is minimized - any kind
of charge waves die immediately because having
separated charges in not energetically favourable. 

 In order to implement this few high level changes are
needed, which are better done by somebody who is an
"architect" of PWSCF. Otherwise it will have to be
redone to fit into the overall scheme.

 Here is what would be needed. Several previous Rho_i,
and their potential V(Rho_i). These are used to figure
out the optimal mixing coefficients. The mixing is
done right before the new bands are obtained.

When far from convergence V(c1*Rho_i+c2*Rho_i-1) ~
c1*V(Rho_i) + c2*V(Rho_i-1). 
This potential {c1*V(Rho_i) + c2*V(Rho_i-1)} is used
to obtain new bands. c1,c2 are chosen such that the
energy E(c1*Rho_i+c2*Rho_i-1) is minimum in the space
where 0<={c1,c2}<=1. It is unnecessary to compute the
true 
V(c1*Rho_i+c2*Rho_i-1) because at that level of
convergence {c1*V(Rho_i) + c2*V(Rho_i-1)} is just as
good, and most of V is the Coulomb contribution
anyway.

 To get {c1,c2,...,c_n} one needs E(Rho_i) {really
trivial}, and dE(Rho_i)/dRho_i which is roughly
V(Rho_i)*Rho_i. Other V(Rho_i)*Rho_j are needed to
minimize the energy for a mix of densities.

 I can get the coefficients from E(Rho_i) and
V(Rho_i)*Rho_j. If there is a setup for the above
procedure available, I'll plug in the coefficients,
and we are in business :-)

 Kostya

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