  In this paper, we present an efficient adaptive multigrid strategy for the
geometry optimization of large-scale three dimensional molecular mechanics. The
resulting method can achieve significantly reduced complexity by exploiting the
intrinsic low-rank property of the material configurations and by combining the
state-of-the-art adaptive techniques with the hierarchical structure of
multigrid algorithms. To be more precise, we develop a oneway multigrid method
with adaptive atomistic/continuum (a/c) coupling, e.g., blended ghost force
correction (BGFC) approximations with gradient-based a posteriori error
estimators on the coarse levels. We utilize state-of-the-art 3D mesh generation
techniques to effectively implement the method. For 3D crystalline defects,
such as vacancies, micro-cracks and dislocations, compared with brute-force
optimization, complexity with superior rates can be observed numerically, and
the strategy has a five-fold acceleration in terms of CPU time for systems with
$10^8$ atoms.
