This project involves the development of advanced computational methods for solving genuine nonlinear eigenvalue problems. In this project, skillful combination of Kublanovskaya's nonlinear QR algorithm with modern rank-revealing and structure-preserving techniques for small and medium size dense problems enhances the capabilities of new methods. A novel trimmed linearization via Pade rational approximation extends the enhancements for solving large but sparse problems. The investigators seek to develop a systematic and unified treatment of the relevant mathematical theory, and produce numerical methods and software tools for the genuine nonlinear eigenvalue problems. In addition to advancing research in nonlinear eigenvalue problems, the project provides training for graduate students in computational mathematics and interdisciplinary research tools.
Eigenvalue problems are ubiquitous in computational science and engineering, where they arise in the study of dynamics of structures, simulation of nanostructured photovoltaic conversion materials to advance energy science, and many other scenarios. Eigenvalues explain a wide range of physical phenomena such as vibrations and frequencies, (in)stabilities of dynamical systems, and energy excitation states of electrons and molecules. Many eigenvalue problems occur naturally in nonlinear form. In this project, the investigators study the underlying nonlinear problems without relying on linearization approximations. The promise of substantially improved methods for computing solutions of nonlinear eigenvalue problems, brings immediate benefits to a wide range of practical applications.
Eigenvalue problems are ubiquitous in computational science and engineering. They arise in dynamics of structures, simiulation of nanostructured materials, and advance energy science, to name just a few. Eigenvalues explain well a wide range of physical phenomena such as vibrations and frequencies, (in)stability of dynamical systems, and energy excitation states of electrons and molecules. Many eigenvalue problems occur naturally in nonlinear forms. This research has achieved serveral significant outcomes including the following: An efficient nonlinear QR algorithm which skillfullly combines with modern rank-revealing and structure-preserving technqiues. A software package in both C++ and MATLAB is made available online for anyone to use. An in-depth study of the hyperbolic quadratic eigenvalue problem which, among others, covers overdamped dynamical systems commonly encountered in our daily life, e.g., elevator and car braking systems. In the theoretical front, we uncovered numerous min-max principles, and in the numerical front, we proposed several optimization methods to calculate few extreme eigenvalues as a result of the theorectical advances. Significant advances in both theory and numerical computations for the linear response eigenvalue problem in computing excitation states (energies) of physical systems in the study of collective motion of many particle systems, ranging from silicon nanoparticles and nanoscale materials to analysis of iinterstellar clouds. Our research has put the current problem which has been challenging the computational quantum chemistry and physics community since 1960s at nearly the same level in difficulty as the symmetric eigenvalue problem. Four students were awarded PhD degrees in part because of the support from this research project.
