{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/82357"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/82357","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Global Optimization in Least Squares Problems in FTIR Spectroscopy and X -Ray Crystallography","abstract":"\"Finally, the last part of the thesis considers the \"\"phase problem\"\" in X-ray crystallography. In particular, we address the problem of estimating the three-dimensional atomic positions of crystal structures from diffraction measurements alone. Starting from the \"\"minimal principle\"\" model, we develop novel optimization formulations and algorithms that exploit the special model structure of the problem. For the case of centrosymmetric structures, we first formulate the problem as a 0--1 linear programming problem and suggest a branch-and-bound algorithm for solving it. Based on empirical observations regarding the nature of the solutions of this model, we then propose to solve it through a system of linear equations for which we develop a fast Gaussian elimination algorithm. For noncentrosymmetric structures, we reduce the phase problem to a mixed-integer quadratic problem and utilize a branch-and-bound algorithm for its solution.\"","abstract_html":""Finally, the last part of the thesis considers the ""phase problem"" in X-ray crystallography. In particular, we address the problem of estimating the three-dimensional atomic positions of crystal structures from diffraction measurements alone. Starting from the ""minimal principle"" model, we develop novel optimization formulations and algorithms that exploit the special model structure of the problem. For the case of centrosymmetric structures, we first formulate the problem as a 0--1 linear programming problem and suggest a branch-and-bound algorithm for solving it. Based on empirical observations regarding the nature of the solutions of this model, we then propose to solve it through a system of linear equations for which we develop a fast Gaussian elimination algorithm. For noncentrosymmetric structures, we reduce the phase problem to a mixed-integer quadratic problem and utilize a branch-and-bound algorithm for its solution."","abstract_has_math":false,"creators":["Vaia, Anastasia"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Chemical and Biomolecular Engineering","degree_department":null,"school":null,"contributors":["Nikolaos Sahinidis"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2015,"date_issued":"2015-09-25T20:43:17Z","date_published":"2015-09-25T20:43:17Z","updated_at":"2026-07-22T22:26:18Z","subjects":["Engineering, Chemical"],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(MiAaPQ)AAI3101985"],"render_values":[{"text":"(MiAaPQ)AAI3101985","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/82357","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Nikolaos Sahinidis"]},{"key":"dc:creator","label":"Author","values":["Vaia, Anastasia"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2015-09-25T20:43:17Z","10000-01-01","2003"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Chemical and Biomolecular Engineering"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Engineering, Chemical"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/82357","(MiAaPQ)AAI3101985"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["\"Finally, the last part of the thesis considers the \"\"phase problem\"\" in X-ray crystallography. In particular, we address the problem of estimating the three-dimensional atomic positions of crystal structures from diffraction measurements alone. Starting from the \"\"minimal principle\"\" model, we develop novel optimization formulations and algorithms that exploit the special model structure of the problem. For the case of centrosymmetric structures, we first formulate the problem as a 0--1 linear programming problem and suggest a branch-and-bound algorithm for solving it. Based on empirical observations regarding the nature of the solutions of this model, we then propose to solve it through a system of linear equations for which we develop a fast Gaussian elimination algorithm. For noncentrosymmetric structures, we reduce the phase problem to a mixed-integer quadratic problem and utilize a branch-and-bound algorithm for its solution.\"","Made available in DSpace on 2015-09-25T20:43:17Z (GMT). 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In particular, we address the problem of estimating the three-dimensional atomic positions of crystal structures from diffraction measurements alone. Starting from the \"\"minimal principle\"\" model, we develop novel optimization formulations and algorithms that exploit the special model structure of the problem. For the case of centrosymmetric structures, we first formulate the problem as a 0--1 linear programming problem and suggest a branch-and-bound algorithm for solving it. Based on empirical observations regarding the nature of the solutions of this model, we then propose to solve it through a system of linear equations for which we develop a fast Gaussian elimination algorithm. For noncentrosymmetric structures, we reduce the phase problem to a mixed-integer quadratic problem and utilize a branch-and-bound algorithm for its solution.\"","Made available in DSpace on 2015-09-25T20:43:17Z (GMT). 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