Doctoral Dissertations

Xuejian Li

Keywords and Phrases

Data Assimilation; Numerical Methods; Optimization; Parallel Algorithm; Partial Differential Equation; Proper Orthogonal Decomposition

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.

Advisor(s)

He, Xiaoming

Committee Member(s)

Han, Daozhi
Singler, John R.
Wei, Mingzhen
Zhang, Yanzhi

Department(s)

Mathematics and Statistics

Degree Name

Ph. D. in Applied Mathematics

Comments

This work is partially supported by National Science Foundation grants DMS-1722647 and DMS-2111421.

Publisher

Missouri University of Science and Technology

Publication Date

Spring 2022

Pagination

viii, 156 pages

Note about bibliography

Includes bibliographic references (pages 146-155).

Rights

© 2022 Xuejian Li, All rights reserved.

Document Type

Dissertation - Open Access

File Type

text

Language

English

Thesis Number

T 12120

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