Anderson-Pulay Acceleration: Convergence of Adaptive Algorithms and Applications to Quantum Chemistry

In this talk, a general class of non-gradient algorithms for
solving fixed-point problems, named Anderson-Pulay acceleration, is
introduced. This family brings together the DIIS technique (Pulay, 1980)
to accelerate the convergence of self-consistent field procedures in
quantum chemistry, as well as the Anderson acceleration (Anderson 1960),
and their variations. Such methods aim at accelerating the convergence
of fixed-point problems by combining at each step several of the
successive approximations to generate the next one. This process of
extrapolation is characterized by its depth, i.e. the number of previous
approximations stored. While this parameter is decisive in the
efficiency of the method, in practice, the depth is fixed without any
guarantee of convergence. In this presentation, we consider two
mechanisms to vary the depth during the course of the method. A first
way is to let the depth grow until the rejection of all the stored
approximations (except the last one) and restart the method. Another way
is to adapt the depth by eliminating some less relevant approximations
at each step. In a general framework and under natural assumptions, the
local convergence and acceleration of Anderson-Pulay acceleration
methods can be proved. These algorithms are tested for the numerical
resolution of the Hartree-Fock equations and the DFT Kohn-Sham model.
These numerical experiments show a faster convergence and lower
computational costs compared to the traditional fixed window approach.
This is a joint work with Maxime Chupin, Guillaume Legendre and Éric Séré.