This project aims to advance the state of the art in fast iterative algorithms for large-scale semidefinite programming and eigenvalue optimization, guided by applications to first-principles computation of properties of systems of many electrons. Semidefinite optimization codes will be rewritten to replace dense linear algebra and direct matrix factorizations by iterative solution processes suited to large and sparse problems. Numerical experiments will be carried out with different mathematical formulations and implementations of the sparse optimization procedures. The guiding application is a variational formulation of the electronic structure ground state problem in which the unknowns are the one-body and two-body reduced density matrices of a many electron system. This application gives rise to a large semidefinite program having sparse solution matrices and an extremely sparse constraint set. The codes that are developed under this grant will be written and documented with the intent of wide dissemination.
Semidefinite programming is a valuable framework for many scientific and engineering applications, including systems control, structural analysis, combinatorial optimization, statistical estimation and VLSI design, just to name a few well-established ones besides electronic structure theory. This project aims to increase our ability to compute solutions to problem instances whose size places them beyond the reach of current numerical methods. As to the specific application in this project, the computation of properties of many-body systems on the basis of quantum mechanics is fundamental to the understanding of molecular and solid state structure, chemical and elementary biochemical processes, and mechanical and electromagnetic properties of condensed matter. Potential benefits to technology and society from advances in the accuracy and scale of many-body quantum mechanical computations include, for example, better design of semiconductors, magnetic storage materials, high-temperature superconductors and chemical catalysts, as well as improvements in the emerging areas of rational drug design and the design of carbon nanostructures.
