New Preconditioned Conjugate Gradient Methods for Some Structured Problems in Physics

This dissertation concerns the development and analysis of new preconditioned conjugate gradient (PCG) algorithms for three important classes of large-scale and complex physical problems characterized by special structures. We propose several new iterative methods for solving the eigenvalue problem or energy minimization problem, which leverage the unique structures inherent in these problems while preserving the underlying physical properties. The new algorithms enable more efficient and robust large-scale modeling and simulations in many areas, including condensed matter physics, optical properties of materials, stabilities of dynamical systems arising from control problems, and many more. Some methods are expected to be applicable to a broader range of applications. For instance, the frameworks of the PCG method presented in Chapter 3 and 4 can be extended to address various types of BEC and potential energy minimization problems. Additionally, the Chebyshev M-LOBPCG method introduced in Chapter 5 can be expanded to tackle symmetric eigenvalue problem