Computational Methods for Large Algebraic Eigenproblems with Special Structures

This project concerns development and analysis of new numerical methods for solving several important classes of large-scale and complex algebraic eigenvalue problems with special structures. Eigenvalues play an important role in many areas of applied mathematics and scientific computing. Fast and robust computations of physically relevant eigenvalues are essential to mathematical modeling and simulations for applications throughout computational sciences and engineering.

This research will enhance the development and understanding of new solvers for large eigenproblems arising from condensed matter physics, quantum field theoretical systems, or dynamical systems with a need for reliable stability analysis. The new algorithms will help enable more efficient and robust large-scale modeling and simulations involving eigenvalues in many areas, including condensed matter physics, optical properties of materials, stabilities of dynamical systems arising from control problems, and many more.

The project will also provide support for graduate students that will enhance their understanding of the essential techniques needed to analyze and solve these computational problems.

Structure-preserving methods play a crucial role in solving eigenvalue problems arising from physics and mechanics, in both linear and nonlinear cases. Researchers need to take advantage of the special structures to design efficient problem-dependent methods that preserve the underlying physical properties of these problems. For eigenproblems with nonlinearity in eigenvalues, nontraditional problems such as computing the rightmost eigenvalues are relevant for understanding the stability of the associated dynamical systems.

The project will investigate three classes of problems: (1) Computing ground states of Bose-Einstein condensation (BEC). Ground states of BEC are described by the solutions to the static Gross-Pitaevskii equation (GPE), a nonlinear eigenproblem with nonlinearity in eigenvectors, with the lowest total energy. Preconditioned optimization methods based on the structure of the energy functional will be studied. (2) Iterative methods for the complex Bethe-Salpeter Eigenvalue problem (BSE).

BSE is a Hamiltonian eigenvalue problem, which can be transformed to a Hermitian problem with symmetric spectrum. The linear response eigenvalue problem is a subclass of BSE. Structure-preserving iterative methods will be investigated for computing a few smallest eigenvalues. (3) Reliable detection of instability of nonlinear eigenproblems.

Evaluation of the distance of a nonlinear eigenvalue problem to instability largely depends on robust computation of the rightmost eigenvalues of a sequence of perturbed problems. Algorithms based on functions of matrices approximated by rational Krylov subspace methods will be explored.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.