{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/141260"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/141260","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"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.","abstract_html":"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.","abstract_has_math":false,"creators":["Wan, Ning"],"institution":"Virginia Tech","degree_name":"Doctor of Philosophy","degree_level":"doctoral","degree_discipline":"Mathematics","degree_department":"Mathematics","school":null,"contributors":[],"advisors":[],"committee_chairs":["Miedlar, Agnieszka Katrzyna"],"committee_members":["De Sturler, Eric","Gugercin, Serkan","Cazeaux, Paul Isaac Denis Louis Marie","Embree, Mark Partick"],"year":2026,"date_issued":"2026-02-13","date_published":"2026-02-13","updated_at":"2026-07-22T22:19:35Z","subjects":["Anderson Acceleration","SCF Iterations","Quasi-Newton Methods","Krylov Methods"],"languages":["en"],"rights":["Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International"],"rights_urls":["http://creativecommons.org/licenses/by-nc-nd/4.0/"],"identifier_entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["vt_gsexam:45482"],"render_values":[{"text":"vt_gsexam:45482","href":null,"code":true}]}]},"links":{"outbound_url":"https://hdl.handle.net/10919/141260","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.committeechair","label":"Committee Chair","values":["Miedlar, Agnieszka Katrzyna"]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["De Sturler, Eric","Gugercin, Serkan","Cazeaux, Paul Isaac Denis Louis Marie","Embree, Mark Partick"]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Wan, Ning"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2026-02-14T09:00:29Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2026-02-14T09:00:29Z"]},{"key":"dc:date.issued","label":"Date","values":["2026-02-13"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Tech"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Virginia Polytechnic Institute and State University"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Anderson Acceleration","SCF Iterations","Quasi-Newton Methods","Krylov Methods"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International"]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://creativecommons.org/licenses/by-nc-nd/4.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["vt_gsexam:45482"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/10919/141260"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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."]},{"key":"dc:description.abstractgeneral","label":"General Abstract","values":["This dissertation studies mathematical methods that accelerate the solution of nonlinear equations, which arise in many areas of science and engineering. Traditional iterative solvers can be slow or may fail to converge on challenging problems. Acceleration techniques improve convergence by intelligently using information from previous iterations. This work focuses on developing and analyzing techniques that accelerate these computations, making them faster and more reliable. The first major contribution shows a deep connection between two popular acceleration methods, demonstrating that, under ideal conditions, they produce identical results. This insight leads to new hybrid approaches that combine the strengths of both methods. The second contribution studies how errors-either from approximations or random perturbations-affect the performance of these acceleration techniques. Numerical experiments show that careful design of the update strategies is essential for maintaining efficiency. Finally, the dissertation extends these ideas to a broader family of advanced iterative methods, providing new algorithms and theoretical guarantees for solving large-scale nonlinear problems more effectively. Overall, this dissertation provides a unified perspective on acceleration strategies and demonstrates how they can be applied to modern large-scale nonlinear computations."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Doctor of Philosophy"]},{"key":"dc:format.medium","label":"Dc Format Medium","values":["ETD"]},{"key":"dc:title","label":"Title","values":["Analysis of Acceleration Techniques and Fast Nonlinear Solvers"]}]}],"canonical_facts":{"dc:contributor.committeechair":["Miedlar, Agnieszka Katrzyna"],"dc:contributor.committeemember":["De Sturler, Eric","Gugercin, Serkan","Cazeaux, Paul Isaac Denis Louis Marie","Embree, Mark Partick"],"dc:contributor.department":["Mathematics"],"dc:creator":["Wan, Ning"],"dc:date.accessioned":["2026-02-14T09:00:29Z"],"dc:date.available":["2026-02-14T09:00:29Z"],"dc:date.issued":["2026-02-13"],"dc:description.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."],"dc:description.abstractgeneral":["This dissertation studies mathematical methods that accelerate the solution of nonlinear equations, which arise in many areas of science and engineering. Traditional iterative solvers can be slow or may fail to converge on challenging problems. Acceleration techniques improve convergence by intelligently using information from previous iterations. This work focuses on developing and analyzing techniques that accelerate these computations, making them faster and more reliable. The first major contribution shows a deep connection between two popular acceleration methods, demonstrating that, under ideal conditions, they produce identical results. This insight leads to new hybrid approaches that combine the strengths of both methods. The second contribution studies how errors-either from approximations or random perturbations-affect the performance of these acceleration techniques. Numerical experiments show that careful design of the update strategies is essential for maintaining efficiency. Finally, the dissertation extends these ideas to a broader family of advanced iterative methods, providing new algorithms and theoretical guarantees for solving large-scale nonlinear problems more effectively. Overall, this dissertation provides a unified perspective on acceleration strategies and demonstrates how they can be applied to modern large-scale nonlinear computations."],"dc:description.degree":["Doctor of Philosophy"],"dc:format.medium":["ETD"],"dc:identifier.other":["vt_gsexam:45482"],"dc:identifier.uri":["https://hdl.handle.net/10919/141260"],"dc:language.iso":["en"],"dc:publisher":["Virginia Tech"],"dc:rights":["Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International"],"dc:rights.uri":["http://creativecommons.org/licenses/by-nc-nd/4.0/"],"dc:subject":["Anderson Acceleration","SCF Iterations","Quasi-Newton Methods","Krylov Methods"],"dc:title":["Analysis of Acceleration Techniques and Fast Nonlinear Solvers"],"dc:type":["Dissertation"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["doctoral"],"thesis:degree_name":["Doctor of Philosophy"],"thesis:institution_name":["Virginia Polytechnic Institute and State University"]},"updated_at":"2026-07-22T22:19:35Z"}