Sharp a priori convergence estimates for Krylov subspace eigensolvers

• Applicant: Professor Dr. Klaus Neymeyr
• Subject Area: Mathematics
• Term: since 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 463329614

Eigenvalue problems of elliptic and self-adjoint differential operators occur in various scientific and technical applications. Their numerical solution succeeds by means of an adaptive finite element discretization and the iterative computation of the desired eigenpairs of the discretized operators. Subspace iterations are well known to be fast solution methods for the high-dimensional matrix eigenvalue problems and are considerably more efficient than classical diagonalization methods. Popular subspace iterations work in Krylov subspaces and can be understood as improved variants of the almost 70 years old Lanczos method. The main procedural variants include restarts, blockwise implementation and preconditioning. The associated convergence theory has not been able to keep pace with the active development of new procedural variants. In addition, many convergence estimates have an a posteriori character, i.e. they bound the rates of convergence by means of complicated formula that depend on the (to be) calculated Ritz values.The proposed project deals with new approaches for the convergence analysis of Krylov subspace iterations for real and symmetric matrix eigenvalue problems. First of all, four basic iteration methods are analyzed: standard Krylov subspace iterations, restarted Krylov subspace iterations, block-Krylov subspace iterations, and restarted block-Krylov subspace iterations. The resulting estimates should lead to an improved understanding of the convergence behavior of these subspace iterations. An extension to the related preconditioned iterations is planned. One focus is on a priori estimates, which can be derived under rather weak assumptions and work with less complex bounds. Probabilistic techniques have a high potential for deriving realistic convergence rates and will be combined with geometric interpretations of Rayleigh quotient level sets and preconditioning. New adaptive control techniques for the block numbers and block sizes suggest a gain in efficiency, which should be demonstrated for application problems such as the self-consistent field iterations in the quantum mechanical density functional theory.

• DFG Programme: Research Grants