Optimization of Density Functional Methods for Atomic Structure Calculations

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We address numerical challenges facing simulations of molecules built upon atomic-scale physics and chemistry. Molecular structure calculations modeling large-scale molecular systems are of fundamental importance to quantum chemistry, physics, and materials science. They are being used by collaborators with this project for the determination of charge density at oxide surfaces, for atomic structure determination of nanoscale materials, for optimization of the performance of nanoscale materials in solid-oxide fuel cells, and for exploring ways to reduce chemical waste and improve the efficiency of catalysts. Worldwide, these types of calculations are exhausting the limits of the world''s supercomputing capacity; any advancements in the numerical algorithms behind these calculations would have immediate impact by expanding our capabilities for simulating ever-larger molecular systems. The mathematical challenges presented by these types of computational problems are at the frontiers of mathematical research; these challenges are dimensionality, nonlinearity, and model inconsistency.We approach the problem from two directions. In the first, we combine conventional techniques for solving large-scale optimization problems -- so-called limited memory techniques -- with a fresh look at ways to stabilize and accelerate the techniques by directly accounting for the presence of multiple scales as well as inconsistent approximations in the model. This involves an investigation into the sensitivity of the local numerical model and the development of a mathematical safeguard against imperfect or incomplete information about the true state of the system. Our second direction of approach to the problem is to divide the model into smaller, more computationally tractable pieces -- known as operator splitting. Operator-splitting methods open the door to a wide range of algorithmic options that allow one to combine computations in novel ways that avoid problems common for more conventional approaches; principal among these is the phenomenon of stagnation at a bad ''local solution''. New operator-splitting techniques developed by the PI for problems related to molecular structure determination and that share similar mathematical features have proven effective. The theory behind these algorithms is still being developed by the PI and extends current theory to the important cases of so-called nonconvex and inconsistent problems that are ubiquitous in applications far beyond molecular structure calculations.