“Variational data assimilation (VDA) is a process that uses optimization techniques to determine an initial condition of a dynamical system such that its evolution best fits the observed data. In this dissertation, we develop and analyze the variational data assimilation method with finite element discretization for two interface problems, including the Parabolic Interface equation and the Stokes-Darcy equation with the Beavers-Joseph interface condition. By using Tikhonov regularization and formulating the VDA into an optimization problem, we establish the existence, uniqueness and stability of the optimal solution for each concerned case. Based on weak formulations of the Parabolic Interface equation and Stokes-Darcy equation, the dual method and Lagrange multiplier rule are utilized to derive the first order optimality system (OptS) for both the continuous and discrete VDA problems, where the discrete data assimilations are built on certain finite element discretization in space and the backward Euler scheme in time. By introducing auxiliary equations, rescaling the optimality system, and employing other subtle analysis skills, we present the finite element convergence estimation for each case with special attention paid to recovering the properties missed in between the continuous and discrete OptS. Moreover, to efficiently solve the OptS, we present two classical gradient methods, the steepest descent method and the conjugate gradient method, to reduce the computational cost for well-stabilized and ill-stabilized VDA problems, respectively. Furthermore, we propose the time parallel algorithm and proper orthogonal decomposition method to further optimize the computing efficiency. Finally, numerical results are provided to validate the proposed methods”--Abstract, page iii.
","abstract_html":"<p>“Variational data assimilation (VDA) is a process that uses optimization techniques to determine an initial condition of a dynamical system such that its evolution best fits the observed data. In this dissertation, we develop and analyze the variational data assimilation method with finite element discretization for two interface problems, including the Parabolic Interface equation and the Stokes-Darcy equation with the Beavers-Joseph interface condition. By using Tikhonov regularization and formulating the VDA into an optimization problem, we establish the existence, uniqueness and stability of the optimal solution for each concerned case. Based on weak formulations of the Parabolic Interface equation and Stokes-Darcy equation, the dual method and Lagrange multiplier rule are utilized to derive the first order optimality system (OptS) for both the continuous and discrete VDA problems, where the discrete data assimilations are built on certain finite element discretization in space and the backward Euler scheme in time. By introducing auxiliary equations, rescaling the optimality system, and employing other subtle analysis skills, we present the finite element convergence estimation for each case with special attention paid to recovering the properties missed in between the continuous and discrete OptS. Moreover, to efficiently solve the OptS, we present two classical gradient methods, the steepest descent method and the conjugate gradient method, to reduce the computational cost for well-stabilized and ill-stabilized VDA problems, respectively. Furthermore, we propose the time parallel algorithm and proper orthogonal decomposition method to further optimize the computing efficiency. Finally, numerical results are provided to validate the proposed methods”--Abstract, page iii.</p>","abstract_has_math":false,"creators":["Li, Xuejian"],"institution":"Missouri University of Science and Technology","degree_name":"Ph. D. in Applied Mathematics","degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":null,"date_issued":"","date_published":null,"updated_at":"2026-07-24T03:18:09Z","subjects":["Data Assimilation","Numerical Methods","Optimization","Parallel Algorithm","Partial Differential Equation","Proper Orthogonal Decomposition","Applied Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://scholarsmine.mst.edu/doctoral_dissertations/3154","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Li, Xuejian"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:type","label":"Dc Type","values":["Dissertation - Open Access"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph. D. in Applied Mathematics"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Missouri University of Science and Technology"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Data Assimilation","Numerical Methods","Optimization","Parallel Algorithm","Partial Differential Equation","Proper Orthogonal Decomposition","Applied Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://scholarsmine.mst.edu/doctoral_dissertations/3154"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["“Variational data assimilation (VDA) is a process that uses optimization techniques to determine an initial condition of a dynamical system such that its evolution best fits the observed data. In this dissertation, we develop and analyze the variational data assimilation method with finite element discretization for two interface problems, including the Parabolic Interface equation and the Stokes-Darcy equation with the Beavers-Joseph interface condition. By using Tikhonov regularization and formulating the VDA into an optimization problem, we establish the existence, uniqueness and stability of the optimal solution for each concerned case. Based on weak formulations of the Parabolic Interface equation and Stokes-Darcy equation, the dual method and Lagrange multiplier rule are utilized to derive the first order optimality system (OptS) for both the continuous and discrete VDA problems, where the discrete data assimilations are built on certain finite element discretization in space and the backward Euler scheme in time. By introducing auxiliary equations, rescaling the optimality system, and employing other subtle analysis skills, we present the finite element convergence estimation for each case with special attention paid to recovering the properties missed in between the continuous and discrete OptS. Moreover, to efficiently solve the OptS, we present two classical gradient methods, the steepest descent method and the conjugate gradient method, to reduce the computational cost for well-stabilized and ill-stabilized VDA problems, respectively. Furthermore, we propose the time parallel algorithm and proper orthogonal decomposition method to further optimize the computing efficiency. Finally, numerical results are provided to validate the proposed methods”--Abstract, page iii.
"]},{"key":"dc:title","label":"Title","values":["Variational data assimilation for two interface problems"]}]}],"canonical_facts":{"dc:creator":["Li, Xuejian"],"dc:description.abstract":["“Variational data assimilation (VDA) is a process that uses optimization techniques to determine an initial condition of a dynamical system such that its evolution best fits the observed data. In this dissertation, we develop and analyze the variational data assimilation method with finite element discretization for two interface problems, including the Parabolic Interface equation and the Stokes-Darcy equation with the Beavers-Joseph interface condition. By using Tikhonov regularization and formulating the VDA into an optimization problem, we establish the existence, uniqueness and stability of the optimal solution for each concerned case. Based on weak formulations of the Parabolic Interface equation and Stokes-Darcy equation, the dual method and Lagrange multiplier rule are utilized to derive the first order optimality system (OptS) for both the continuous and discrete VDA problems, where the discrete data assimilations are built on certain finite element discretization in space and the backward Euler scheme in time. By introducing auxiliary equations, rescaling the optimality system, and employing other subtle analysis skills, we present the finite element convergence estimation for each case with special attention paid to recovering the properties missed in between the continuous and discrete OptS. Moreover, to efficiently solve the OptS, we present two classical gradient methods, the steepest descent method and the conjugate gradient method, to reduce the computational cost for well-stabilized and ill-stabilized VDA problems, respectively. Furthermore, we propose the time parallel algorithm and proper orthogonal decomposition method to further optimize the computing efficiency. Finally, numerical results are provided to validate the proposed methods”--Abstract, page iii.
"],"dc:identifier":["https://scholarsmine.mst.edu/doctoral_dissertations/3154"],"dc:subject":["Data Assimilation","Numerical Methods","Optimization","Parallel Algorithm","Partial Differential Equation","Proper Orthogonal Decomposition","Applied Mathematics"],"dc:title":["Variational data assimilation for two interface problems"],"dc:type":["Dissertation - Open Access"],"thesis:degree_name":["Ph. D. in Applied Mathematics"],"thesis:institution_name":["Missouri University of Science and Technology"]},"updated_at":"2026-07-24T03:18:09Z"}