Technical Area: Solution of Eigenvalue Problems

Eigenvalue problems arise in several SciDAC Partnerships and DOE projects such as the Computational Materials Science (CMS), Computational Chemical Science (CCS), and Energy Frontier Research Centers (EFRC). One of the most challenging problems is the Kohn-Sham nonlinear eigenvalue problem in electronic structure calculations (BES SciDAC). Many-body eigenvalue problems in nuclear structure (NP SciDAC) and strongly correlated physics models (EFRC) pose another challenge, where the large sparse and structured eigenvalue problems need to solved. In the thermodynamic limit, eigenvectors become infinite dimensional tensors that have translational invariance properties. Eigensolvers also play an important role in spectral analysis of large data and developing model order reduction for simulation and prediction of complex phenomena and processes. Efficient implementations of these solvers on emerging GPU-based systems is critical.

The eigenvalue problems to be considered in FASTMath includes

  • Spectrum slicing based solver for large sparse symmetric eigenvalue problems

  • Efficient methods for solving nonlinear eigenvalue problems

  • Uncontrained trace minimization based eigensolvers

  • Efficient solvers for tensor eigenvalue problem

  • Spectral analysis for linear and nonlinear dimension reduction in data analysis and complex simulation

Density of states in materials science

Eigenvalue problems also arise from simulations of molecules.

Understanding nuclear structure using ab initio calculations.

Sparsity structure of a matrix from nuclear structure calculations.