Analysis of Acceleration Techniques and Fast Nonlinear Solvers

Abstract

This dissertation focuses on the analysis and development of acceleration techniques and fast solvers for nonlinear systems of equations. Building upon the fixed-point and extrapolation frameworks introduced in the early chapters, we explore structural connections between residual-based acceleration methods and Krylov subspace techniques. The first main contribution is a unified algebraic framework establishing the equivalence between the Anderson Acceleration method and the CROP (Conjugate Residual with Optimal Trial Vector) algorithm. By formulating both methods within a common affine subspace representation, we show that their full, untruncated forms produce identical iterates, motivating new hybrid variants such as CROP-Anderson and real-residual CROP (rCROP) methods. The second contribution is a perturbation analysis of Anderson-type variants, examining the effects of deterministic and stochastic errors on convergence. Numerical experiments confirm that acceleration efficiency depends critically on both the choice of update strategy and the nature of perturbations. The third contribution extends this unified perspective to nonlinear Krylov subspace methods. Nonlinear extensions of GMRESR, GCRO, and LGMRES are derived, forming the nlKrylov family of algorithms, and analyzed in the context of inexact Newton solvers, with convergence results established under relaxed conditions on residual and Jacobian approximations.