A geometric convergence theory for the preconditioned steepest descent\n iteration

2011/08/11 by Klaus Neymeyr, Neymeyr, Klaus · 1 citation
Computer Science · Engineering · #65F15 #65N12 #65N22 #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Numerical methods in engineering

paper · pdf · doi:10.48550/arxiv.1108.2365

openalex publication_date 2011/08/11 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28

Abstract

Preconditioned gradient iterations for very large eigenvalue problems are\nefficient solvers with growing popularity. However, only for the simplest\npreconditioned eigensolver, namely the preconditioned gradient iteration (or\npreconditioned inverse iteration) with fixed step size, sharp non-asymptotic\nconvergence estimates are known and these estimates require an ideally scaled\npreconditioner. In this paper a new sharp convergence estimate is derived for\nthe preconditioned steepest descent iteration which combines the preconditioned\ngradient iteration with the Rayleigh-Ritz procedure for optimal line search\nconvergence acceleration. The new estimate always improves that of the fixed\nstep size iteration. The practical importance of this new estimate is that\narbitrarily scaled preconditioners can be used. The Rayleigh-Ritz procedure\nimplicitly computes the optimal scaling.\n

Cited by

• Local Convergence Behavior of Extended LOBPCG for Computing Eigenvalues of Hermitian Matrices

Related

• A preconditioned inverse iteration with an improved convergence guarantee
• Convergence rates of individual Ritz values in block preconditioned gradient-type eigensolvers
• Convergence analysis of a block preconditioned steepest descent eigensolver with implicit deflation
• Riemannian Acceleration with Preconditioning for symmetric eigenvalue problems
• A Note on Approximate Inverse Iteration