{"id":{"repo_id":"lsu-thes","oai_identifier":"oai:repository.lsu.edu:gradschool_dissertations-2085"},"canonical_url":"https://search.dev.ndltd.org/etd/lsu-thes/oai:repository.lsu.edu:gradschool_dissertations-2085","repository":{"repo_id":"lsu-thes","name":"Lousiana State University","base_url":"https://repository.lsu.edu/do/oai/"},"display":{"title":"A semigroup/Laplace transform approach to approximating flows","abstract":"It is well known that all flows in a state space O induce a semigroup of linear operators on an appropriately chosen vector space of functions (observables) from O into a vector space Z (observations). After choosing appropriate continuity assumptions on the flow, the associated semigroup will be strongly continuous and will have a linear, infinitesimal generator A. The purpose of this dissertation is to explore approximation methods for linear semigroups and/or Laplace transform inversion methods in order to reconstruct the flow starting with the linear generator A . In preparing for these investigations, we collect some of the essential approximation theorems of semigroup theory and improve a recent generalization of the Trotter-Kato Theorem due to McAllister, Neubrander, Riser, and Zhuang. Moreover, we show that rational Laplace transform inversions of order m are exact for all polynomials of degree less than m. We will demonstrate that the flow can be efficiently reconstructed whenever the generator A of the induced semigroup has a resolvent that can be efficiently computed or approximated. We demonstrate this for flows solving nonlinear first order ordinary differential equations x'(t) = a(x(t)), x(s)= w and the induced generator (Af)(s)=a(s)f'(s) and for flows solving non-autonomous linear first order ordinary differential equations u'(t) = a(t)u(t), u(s)= w and the induced generator(Af)(s) = f'(s)+a(s)f(s). As a by-product of our investigation, we find a numerically efficient way to compute the inverse of increasing real-valued functions. Finally, we explore whether linear semigroup approximation methods can be used efficiently to approximate solutions of non-autonomous Cauchy problems u'(t) = A(t)u(t), u(s) = x in terms of the generator (Af)(s) = f'(s) + A(s)f(s) of the induced linear operator semigroup. As we will see, the Lie-Trotter approach suggested by G. Nickel seems to be the only efficient way to find the solutions of the non-autonomous problems in terms of the semigroup generated by A.","abstract_html":"It is well known that all flows in a state space O induce a semigroup of linear operators on an appropriately chosen vector space of functions (observables) from O into a vector space Z (observations). After choosing appropriate continuity assumptions on the flow, the associated semigroup will be strongly continuous and will have a linear, infinitesimal generator A. The purpose of this dissertation is to explore approximation methods for linear semigroups and/or Laplace transform inversion methods in order to reconstruct the flow starting with the linear generator A . In preparing for these investigations, we collect some of the essential approximation theorems of semigroup theory and improve a recent generalization of the Trotter-Kato Theorem due to McAllister, Neubrander, Riser, and Zhuang. Moreover, we show that rational Laplace transform inversions of order m are exact for all polynomials of degree less than m. We will demonstrate that the flow can be efficiently reconstructed whenever the generator A of the induced semigroup has a resolvent that can be efficiently computed or approximated. We demonstrate this for flows solving nonlinear first order ordinary differential equations x'(t) = a(x(t)), x(s)= w and the induced generator (Af)(s)=a(s)f'(s) and for flows solving non-autonomous linear first order ordinary differential equations u'(t) = a(t)u(t), u(s)= w and the induced generator(Af)(s) = f'(s)+a(s)f(s). As a by-product of our investigation, we find a numerically efficient way to compute the inverse of increasing real-valued functions. Finally, we explore whether linear semigroup approximation methods can be used efficiently to approximate solutions of non-autonomous Cauchy problems u'(t) = A(t)u(t), u(s) = x in terms of the generator (Af)(s) = f'(s) + A(s)f(s) of the induced linear operator semigroup. As we will see, the Lie-Trotter approach suggested by G. Nickel seems to be the only efficient way to find the solutions of the non-autonomous problems in terms of the semigroup generated by A.","abstract_has_math":false,"creators":["Latin, Ladorian Nichele"],"institution":"Mathematics","degree_name":"Doctor of Philosophy (PhD)","degree_level":"Dissertation","degree_discipline":"Applied Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2013,"date_issued":"2013-01-01T08:00:00Z","date_published":"2013-01-01T08:00:00Z","updated_at":"2026-07-24T02:58:50Z","subjects":["operator semigroup","resolvent","infinitesimal generator","autonomous flow"],"languages":[],"rights":["unrestricted","Release the entire work immediately for access worldwide."],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["etd-11182013-090349","https://repository.lsu.edu/gradschool_dissertations/1086"],"render_values":[{"text":"etd-11182013-090349","href":null,"code":true},{"text":"https://repository.lsu.edu/gradschool_dissertations/1086","href":"https://repository.lsu.edu/gradschool_dissertations/1086","code":true}]}]},"links":{"outbound_url":"https://doi.org/10.31390/gradschool_dissertations.1086","outbound_label":"DOI","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Latin, Ladorian Nichele"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2013-11-14"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2022-05-12T23:10:58Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Applied Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy (PhD)"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Mathematics"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["operator semigroup","resolvent","infinitesimal generator","autonomous flow"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["unrestricted","Release the entire work immediately for access worldwide."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["etd-11182013-090349","10.31390/gradschool_dissertations.1086","https://repository.lsu.edu/gradschool_dissertations/1086"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["It is well known that all flows in a state space O induce a semigroup of linear operators on an appropriately chosen vector space of functions (observables) from O into a vector space Z (observations). After choosing appropriate continuity assumptions on the flow, the associated semigroup will be strongly continuous and will have a linear, infinitesimal generator A. The purpose of this dissertation is to explore approximation methods for linear semigroups and/or Laplace transform inversion methods in order to reconstruct the flow starting with the linear generator A . In preparing for these investigations, we collect some of the essential approximation theorems of semigroup theory and improve a recent generalization of the Trotter-Kato Theorem due to McAllister, Neubrander, Riser, and Zhuang. Moreover, we show that rational Laplace transform inversions of order m are exact for all polynomials of degree less than m. We will demonstrate that the flow can be efficiently reconstructed whenever the generator A of the induced semigroup has a resolvent that can be efficiently computed or approximated. We demonstrate this for flows solving nonlinear first order ordinary differential equations x'(t) = a(x(t)), x(s)= w and the induced generator (Af)(s)=a(s)f'(s) and for flows solving non-autonomous linear first order ordinary differential equations u'(t) = a(t)u(t), u(s)= w and the induced generator(Af)(s) = f'(s)+a(s)f(s). As a by-product of our investigation, we find a numerically efficient way to compute the inverse of increasing real-valued functions. Finally, we explore whether linear semigroup approximation methods can be used efficiently to approximate solutions of non-autonomous Cauchy problems u'(t) = A(t)u(t), u(s) = x in terms of the generator (Af)(s) = f'(s) + A(s)f(s) of the induced linear operator semigroup. As we will see, the Lie-Trotter approach suggested by G. Nickel seems to be the only efficient way to find the solutions of the non-autonomous problems in terms of the semigroup generated by A."]},{"key":"dc:title","label":"Title","values":["A semigroup/Laplace transform approach to approximating flows"]}]}],"canonical_facts":{"dc:creator":["Latin, Ladorian Nichele"],"dc:date":["2013-11-14"],"dc:date.available":["2022-05-12T23:10:58Z"],"dc:description.abstract":["It is well known that all flows in a state space O induce a semigroup of linear operators on an appropriately chosen vector space of functions (observables) from O into a vector space Z (observations). After choosing appropriate continuity assumptions on the flow, the associated semigroup will be strongly continuous and will have a linear, infinitesimal generator A. The purpose of this dissertation is to explore approximation methods for linear semigroups and/or Laplace transform inversion methods in order to reconstruct the flow starting with the linear generator A . In preparing for these investigations, we collect some of the essential approximation theorems of semigroup theory and improve a recent generalization of the Trotter-Kato Theorem due to McAllister, Neubrander, Riser, and Zhuang. Moreover, we show that rational Laplace transform inversions of order m are exact for all polynomials of degree less than m. We will demonstrate that the flow can be efficiently reconstructed whenever the generator A of the induced semigroup has a resolvent that can be efficiently computed or approximated. We demonstrate this for flows solving nonlinear first order ordinary differential equations x'(t) = a(x(t)), x(s)= w and the induced generator (Af)(s)=a(s)f'(s) and for flows solving non-autonomous linear first order ordinary differential equations u'(t) = a(t)u(t), u(s)= w and the induced generator(Af)(s) = f'(s)+a(s)f(s). As a by-product of our investigation, we find a numerically efficient way to compute the inverse of increasing real-valued functions. Finally, we explore whether linear semigroup approximation methods can be used efficiently to approximate solutions of non-autonomous Cauchy problems u'(t) = A(t)u(t), u(s) = x in terms of the generator (Af)(s) = f'(s) + A(s)f(s) of the induced linear operator semigroup. As we will see, the Lie-Trotter approach suggested by G. Nickel seems to be the only efficient way to find the solutions of the non-autonomous problems in terms of the semigroup generated by A."],"dc:identifier":["etd-11182013-090349","10.31390/gradschool_dissertations.1086","https://repository.lsu.edu/gradschool_dissertations/1086"],"dc:rights":["unrestricted","Release the entire work immediately for access worldwide."],"dc:subject":["operator semigroup","resolvent","infinitesimal generator","autonomous flow"],"dc:title":["A semigroup/Laplace transform approach to approximating flows"],"thesis:degree_discipline":["Applied Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy (PhD)"],"thesis:institution_name":["Mathematics"]},"updated_at":"2026-07-24T02:58:50Z"}