In this thesis, we look at an iterative method for approximating the scattering amplitude that involves solving two linear systems: a forward system Ax=b and an adjoint system ATy=g. Once these two systems are solved, the scattering amplitude, defined by gTx=yTb is easily obtained.
We derive a conjugate gradient-like iteration for a nonsymmetric saddle point matrix that is constructed to have a real positive spectrum. We investigate the use of Schur Complement preconditioners with block-diagonal factorization to speed up the convergence of our method and compare the results to our NspcG method without preconditioning.
","abstract_html":"<p>In this thesis, we look at an iterative method for approximating the scattering amplitude that involves solving two linear systems: a forward system A<strong>x</strong>=<strong>b</strong> and an adjoint system A<sup>T</sup><strong>y</strong>=<strong>g</strong>. Once these two systems are solved, the scattering amplitude, defined by <strong>g</strong><sup>T</sup><strong>x</strong>=<strong>y</strong><sup>T</sup><strong>b</strong> is easily obtained.</p> <p>We derive a conjugate gradient-like iteration for a nonsymmetric saddle point matrix that is constructed to have a real positive spectrum. We investigate the use of Schur Complement preconditioners with block-diagonal factorization to speed up the convergence of our method and compare the results to our NspcG method without preconditioning.</p>","abstract_has_math":false,"creators":["Ayo, Samson"],"institution":null,"degree_name":"Master of Science (MS)","degree_level":"Masters Thesis","degree_discipline":null,"degree_department":null,"school":null,"contributors":["James Lambers","C.S. Chen","Huiqing Zhu"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2021,"date_issued":"2021-08-01T07:00:00Z","date_published":"2021-08-01T07:00:00Z","updated_at":"2026-07-24T05:45:33Z","subjects":["linear systems","adjoint systems","nonsymmetric saddle point matrix","conjugate gradient method","scattering amplitude","preconditioning","Numerical Analysis and Computation"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://aquila.usm.edu/masters_theses/850","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["James Lambers","C.S. Chen","Huiqing Zhu"]},{"key":"dc:creator","label":"Author","values":["Ayo, Samson"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2022-06-22T07:00:00Z"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Masters Thesis"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science (MS)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["linear systems","adjoint systems","nonsymmetric saddle point matrix","conjugate gradient method","scattering amplitude","preconditioning","Numerical Analysis and Computation"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://aquila.usm.edu/masters_theses/850"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["In this thesis, we look at an iterative method for approximating the scattering amplitude that involves solving two linear systems: a forward system Ax=b and an adjoint system ATy=g. Once these two systems are solved, the scattering amplitude, defined by gTx=yTb is easily obtained.
We derive a conjugate gradient-like iteration for a nonsymmetric saddle point matrix that is constructed to have a real positive spectrum. We investigate the use of Schur Complement preconditioners with block-diagonal factorization to speed up the convergence of our method and compare the results to our NspcG method without preconditioning.
"]},{"key":"dc:title","label":"Title","values":["A Modified Preconditioned Conjugate Gradient Method for Approximating the Scattering Amplitude"]}]}],"canonical_facts":{"dc:contributor":["James Lambers","C.S. Chen","Huiqing Zhu"],"dc:creator":["Ayo, Samson"],"dc:date.available":["2022-06-22T07:00:00Z"],"dc:description.abstract":["In this thesis, we look at an iterative method for approximating the scattering amplitude that involves solving two linear systems: a forward system Ax=b and an adjoint system ATy=g. Once these two systems are solved, the scattering amplitude, defined by gTx=yTb is easily obtained.
We derive a conjugate gradient-like iteration for a nonsymmetric saddle point matrix that is constructed to have a real positive spectrum. We investigate the use of Schur Complement preconditioners with block-diagonal factorization to speed up the convergence of our method and compare the results to our NspcG method without preconditioning.
"],"dc:identifier":["https://aquila.usm.edu/masters_theses/850"],"dc:subject":["linear systems","adjoint systems","nonsymmetric saddle point matrix","conjugate gradient method","scattering amplitude","preconditioning","Numerical Analysis and Computation"],"dc:title":["A Modified Preconditioned Conjugate Gradient Method for Approximating the Scattering Amplitude"],"thesis:degree_level":["Masters Thesis"],"thesis:degree_name":["Master of Science (MS)"]},"updated_at":"2026-07-24T05:45:33Z"}