A note on the regularity of reduced models obtained by nonlocal quasi-continuum-like approaches
M. Anitescu, D. Negrut, P. Zapol and A. El-Azab
Mathematical Programming, 118, 207-236 (2009).
ABSTRACT
The paper investigates model reduction techniques that are based on a nonlocal
quasi-continuum-like approach. These techniques reduce a large optimization
problem to either a system of nonlinear equations or another optimization
problem that are expressed in a smaller number of degrees of freedom. The
reduction is based on the observation that many of the components of the
solution of the original optimization problem are well approximated by certain
interpolation operators with respect to a restricted set of representative
components. Under certain assumptions, the "optimize and interpolate" and the
"interpolate and optimize" approaches result in a regular nonlinear equation and
an optimization problem whose solutions are close to the solution of the
original problem, respectively. The validity of these assumptions is
investigated by using examples from potential-based and electronic
structure-based calculations in Materials Science models. A methodology is
presented for using quasi-continuum-like model reduction for real-space DFT
computations in the absence of periodic boundary conditions. The methodology is
illustrated using a basic Thomas-Fermi-Dirac case study.