Predictive Mixing for Density Functional Theory (and Other Fixed-Point Problems).

Area of Science:

• Computational Chemistry
• Materials Science
• Supercomputing

Background:

• Density functional theory (DFT) calculations are computationally intensive, consuming substantial supercomputing resources.
• Convergence speed in DFT is often limited by the "mixing" parameter, which controls the number of iterations.
• Existing methods for managing trust regions in fixed-point problems can be inefficient.

Purpose of the Study:

• To develop a new approach for handling trust regions in DFT and other fixed-point problems.
• To improve the efficiency and convergence speed of large-scale computational simulations.
• To reduce the supercomputing time required for density functional theory calculations.

Main Methods:

• Introduced a novel trust region strategy that estimates optimal parameters and trust regions from prior iteration history.
• Developed a predictive method for estimating the optimal Polyak step.
• Evaluated the approach using multisecant methods (both "good" and "bad" versions), the Anderson method, and a hybrid approach across eight structures.

Main Results:

• The predictive method demonstrated effectiveness across various candidate step methods, including multisecant and Anderson.
• The new approach showed adaptability to different problem types, especially when combined with the hybrid method.
• Detailed comparisons for 36 cases confirmed the robustness and efficiency gains of the proposed predictive method.

Conclusions:

• The novel trust region approach significantly enhances the efficiency of density functional theory calculations.
• The predictive method offers a robust and adaptable solution for accelerating fixed-point problems.
• This work has the potential to reduce the computational burden of large-scale scientific simulations.