Analysis of Acceleration Techniques and Fast Nonlinear Solvers

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.

Name thesis:degree_name

Doctor of Philosophy

Level thesis:degree_level

doctoral

Discipline thesis:degree_discipline

Mathematics

Department dc:contributor.department

Mathematics

Grantor dc:publisher

Virginia Tech

Year dc:date.issued

2026

Author dc:creator

• Wan, Ning

Chair dc:contributor.committeechair

• Miedlar, Agnieszka Katrzyna

Committee members dc:contributor.committeemember

• De Sturler, Eric
• Gugercin, Serkan
• Cazeaux, Paul Isaac Denis Louis Marie
• Embree, Mark Partick

• Anderson Acceleration
• SCF Iterations
• Quasi-Newton Methods
• Krylov Methods

Statement dc:rights

• Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International

Licence dc:rights.uri

• http://creativecommons.org/licenses/by-nc-nd/4.0/

Language dc:language.iso

en

Dc Identifier Other

vt_gsexam:45482

OAI identifier oai:identifier

oai:vtechworks.lib.vt.edu:10919/141260