{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/133137"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/133137","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Parameter Estimation for Delay Differential Equations: A New Galerkin Approximation Framework","abstract":"Delay differential equations with discrete and/or distributed delays are widely used in many applied fields, including mathematical biology, climate science, and engineering. These equations contain free parameters and integral kernels that must be optimized to fit the dynamics of the physical systems they model. In this thesis, we present a framework for determining the discrete delays, integral kernels, and other unknown model parameters for these equations from training data through a Galerkin approximation approach proposed by citet{CGLW16}. The adopted Galerkin approximation is based on a particular type of polynomials due to T. H. Koornwinder, which are orthogonal under an inner product with a point mass cite{Koo84}. Within this Galerkin framework, the parameter estimation problem is reduced to the estimation of some scalar parameters in the resulting ordinary differential equations. Thanks to the analytical nature of the Galerkin approximation, the latter estimation problem can be handled efficiently, even with distributed delays, which is a known computationally challenging scenario. In all the cases, the system's dependence on the delay parameters is nonlinear. We will show how the training data and the Galerkin systems can be suitably adapted to efficiently convert this nonlinear estimation problem into successive linear estimation problems. The approach will be illustrated on concrete examples arising from various applications that exhibit either periodic or chaotic dynamics.","abstract_html":"Delay differential equations with discrete and/or distributed delays are widely used in many applied fields, including mathematical biology, climate science, and engineering. These equations contain free parameters and integral kernels that must be optimized to fit the dynamics of the physical systems they model. In this thesis, we present a framework for determining the discrete delays, integral kernels, and other unknown model parameters for these equations from training data through a Galerkin approximation approach proposed by citet{CGLW16}. The adopted Galerkin approximation is based on a particular type of polynomials due to T. H. Koornwinder, which are orthogonal under an inner product with a point mass cite{Koo84}. Within this Galerkin framework, the parameter estimation problem is reduced to the estimation of some scalar parameters in the resulting ordinary differential equations. Thanks to the analytical nature of the Galerkin approximation, the latter estimation problem can be handled efficiently, even with distributed delays, which is a known computationally challenging scenario. In all the cases, the system's dependence on the delay parameters is nonlinear. We will show how the training data and the Galerkin systems can be suitably adapted to efficiently convert this nonlinear estimation problem into successive linear estimation problems. The approach will be illustrated on concrete examples arising from various applications that exhibit either periodic or chaotic dynamics.","abstract_has_math":false,"creators":["Hartman, Jonathan Cole"],"institution":"Virginia Tech","degree_name":"Master of Science","degree_level":"masters","degree_discipline":"Mathematics","degree_department":"Mathematics","school":null,"contributors":[],"advisors":[],"committee_chairs":["Liu, Honghu"],"committee_members":["Borggaard, Jeffrey T.","Iliescu, Traian"],"year":2025,"date_issued":"2025-05-19","date_published":"2025-05-19","updated_at":"2026-07-22T22:20:27Z","subjects":["Delay Differential Equations","Parameter Estimation","Inverse Problems","Galerkin–Koornwinder Approximations","Distributed Delays"],"languages":["en"],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["vt_gsexam:43754"],"render_values":[{"text":"vt_gsexam:43754","href":null,"code":true}]}]},"links":{"outbound_url":"https://hdl.handle.net/10919/133137","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.committeechair","label":"Committee Chair","values":["Liu, Honghu"]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Borggaard, Jeffrey T.","Iliescu, Traian"]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Hartman, Jonathan Cole"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2025-05-20T08:00:37Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2025-05-20T08:00:37Z"]},{"key":"dc:date.issued","label":"Date","values":["2025-05-19"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Tech"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["masters"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science"]},{"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":["Delay Differential Equations","Parameter Estimation","Inverse Problems","Galerkin–Koornwinder Approximations","Distributed Delays"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["In Copyright"]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://rightsstatements.org/vocab/InC/1.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["vt_gsexam:43754"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/10919/133137"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Delay differential equations with discrete and/or distributed delays are widely used in many applied fields, including mathematical biology, climate science, and engineering. These equations contain free parameters and integral kernels that must be optimized to fit the dynamics of the physical systems they model. In this thesis, we present a framework for determining the discrete delays, integral kernels, and other unknown model parameters for these equations from training data through a Galerkin approximation approach proposed by citet{CGLW16}. The adopted Galerkin approximation is based on a particular type of polynomials due to T. H. Koornwinder, which are orthogonal under an inner product with a point mass cite{Koo84}. Within this Galerkin framework, the parameter estimation problem is reduced to the estimation of some scalar parameters in the resulting ordinary differential equations. Thanks to the analytical nature of the Galerkin approximation, the latter estimation problem can be handled efficiently, even with distributed delays, which is a known computationally challenging scenario. In all the cases, the system's dependence on the delay parameters is nonlinear. We will show how the training data and the Galerkin systems can be suitably adapted to efficiently convert this nonlinear estimation problem into successive linear estimation problems. The approach will be illustrated on concrete examples arising from various applications that exhibit either periodic or chaotic dynamics."]},{"key":"dc:description.abstractgeneral","label":"General Abstract","values":["We consider delay differential equations with discrete and/or distributed delays, which are differential equations that contain either continuously distributed or discrete history dependence in their vector fields. These equations are widely used in many applied fields, including mathematical biology, climate science, and engineering. These equations contain free parameters that must be optimized to fit data and dynamics from the physical systems they model. We present a framework for determining these free parameters from training data through a Galerkin approximation approach proposed by citet{CGLW16}. Within this Galerkin framework, the parameter estimation problem is reduced to the estimation of some scalar parameters in the resulting ordinary differential equations. In all the cases, the system's dependence on the delay parameters is nonlinear. We will show how the training data and the Galerkin systems can be suitably adapted to efficiently convert this nonlinear estimation problem into successive linear estimation problems. The approach will be illustrated on concrete examples arising from various applications that exhibit either periodic or chaotic dynamics."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Master of Science"]},{"key":"dc:format.medium","label":"Dc Format Medium","values":["ETD"]},{"key":"dc:title","label":"Title","values":["Parameter Estimation for Delay Differential Equations: A New Galerkin Approximation Framework"]}]}],"canonical_facts":{"dc:contributor.committeechair":["Liu, Honghu"],"dc:contributor.committeemember":["Borggaard, Jeffrey T.","Iliescu, Traian"],"dc:contributor.department":["Mathematics"],"dc:creator":["Hartman, Jonathan Cole"],"dc:date.accessioned":["2025-05-20T08:00:37Z"],"dc:date.available":["2025-05-20T08:00:37Z"],"dc:date.issued":["2025-05-19"],"dc:description.abstract":["Delay differential equations with discrete and/or distributed delays are widely used in many applied fields, including mathematical biology, climate science, and engineering. These equations contain free parameters and integral kernels that must be optimized to fit the dynamics of the physical systems they model. In this thesis, we present a framework for determining the discrete delays, integral kernels, and other unknown model parameters for these equations from training data through a Galerkin approximation approach proposed by citet{CGLW16}. The adopted Galerkin approximation is based on a particular type of polynomials due to T. H. Koornwinder, which are orthogonal under an inner product with a point mass cite{Koo84}. Within this Galerkin framework, the parameter estimation problem is reduced to the estimation of some scalar parameters in the resulting ordinary differential equations. Thanks to the analytical nature of the Galerkin approximation, the latter estimation problem can be handled efficiently, even with distributed delays, which is a known computationally challenging scenario. In all the cases, the system's dependence on the delay parameters is nonlinear. We will show how the training data and the Galerkin systems can be suitably adapted to efficiently convert this nonlinear estimation problem into successive linear estimation problems. The approach will be illustrated on concrete examples arising from various applications that exhibit either periodic or chaotic dynamics."],"dc:description.abstractgeneral":["We consider delay differential equations with discrete and/or distributed delays, which are differential equations that contain either continuously distributed or discrete history dependence in their vector fields. These equations are widely used in many applied fields, including mathematical biology, climate science, and engineering. These equations contain free parameters that must be optimized to fit data and dynamics from the physical systems they model. We present a framework for determining these free parameters from training data through a Galerkin approximation approach proposed by citet{CGLW16}. Within this Galerkin framework, the parameter estimation problem is reduced to the estimation of some scalar parameters in the resulting ordinary differential equations. In all the cases, the system's dependence on the delay parameters is nonlinear. We will show how the training data and the Galerkin systems can be suitably adapted to efficiently convert this nonlinear estimation problem into successive linear estimation problems. The approach will be illustrated on concrete examples arising from various applications that exhibit either periodic or chaotic dynamics."],"dc:description.degree":["Master of Science"],"dc:format.medium":["ETD"],"dc:identifier.other":["vt_gsexam:43754"],"dc:identifier.uri":["https://hdl.handle.net/10919/133137"],"dc:language.iso":["en"],"dc:publisher":["Virginia Tech"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:subject":["Delay Differential Equations","Parameter Estimation","Inverse Problems","Galerkin–Koornwinder Approximations","Distributed Delays"],"dc:title":["Parameter Estimation for Delay Differential Equations: A New Galerkin Approximation Framework"],"dc:type":["Thesis"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["masters"],"thesis:degree_name":["Master of Science"],"thesis:institution_name":["Virginia Polytechnic Institute and State University"]},"updated_at":"2026-07-22T22:20:27Z"}