617 research outputs found
Enhancing structure relaxations for first-principles codes: an approximate Hessian approach
We present a method for improving the speed of geometry relaxation by using a
harmonic approximation for the interaction potential between nearest neighbor
atoms to construct an initial Hessian estimate. The model is quite robust, and
yields approximately a 30% or better reduction in the number of calculations
compared to an optimized diagonal initialization. Convergence with this
initializer approaches the speed of a converged BFGS Hessian, therefore it is
close to the best that can be achieved. Hessian preconditioning is discussed,
and it is found that a compromise between an average condition number and a
narrow distribution in eigenvalues produces the best optimization.Comment: 9 pages, 3 figures, added references, expanded optimization sectio
On QuasiâNewton methods in fast Fourier transformâbased micromechanics
This work is devoted to investigating the computational power of QuasiâNewton methods in the context of fast Fourier transform (FFT)âbased computational micromechanics. We revisit FFTâbased NewtonâKrylov solvers as well as modern QuasiâNewton approaches such as the recently introduced Anderson accelerated basic scheme. In this context, we propose two algorithms based on the BroydenâFletcherâGoldfarbâShanno (BFGS) method, one of the most powerful QuasiâNewton schemes. To be specific, we use the BFGS update formula to approximate the global Hessian or, alternatively, the local material tangent stiffness. Both for Newton and QuasiâNewton methods, a globalization technique is necessary to ensure global convergence. Specific to the FFTâbased context, we promote a Dongâtype line search, avoiding function evaluations altogether. Furthermore, we investigate the influence of the forcing term, that is, the accuracy for solving the linear system, on the overall performance of inexact (Quasiâ)Newton methods. This work concludes with numerical experiments, comparing the convergence characteristics and runtime of the proposed techniques for complex microstructures with nonlinear material behavior and finite as well as infinite material contrast
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