Lu, Mingfei;
(2025)
Computational methods for deterministic and Bayesian data assimilation: stability and asymptotic convergence.
Doctoral thesis (Ph.D), UCL (University College London).
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Abstract
Data assimilation problems with Partial Differential Equations (PDEs) have been widely studied by both applied mathematicians both statisticians. It comes as an inverse problem with ill-posed PDE settings. For example, boundary data or PDE parameters may be missing. On the other hand, observations with stochastic perturbation come as an complementary to the incomplete PDE system. In this case, the stability and convergence for both statistical and numerical algorithms require special treatment, such as using regularization and Bayesian inference. In this thesis, we consider both probabilistic and deterministic data assimilation problems. The model problems we consider are elliptic equations with missing boundary condition, but with interior measurement, or the Cauchy type measurement on part of its boundary. We first establish stable and convergent numerical algorithms for the noise-free limit problems, that is, assuming the measurement comes continuously with no perturbation. Two types of methods are provided to ensure numerical stability. The first is to choose stable finite element couplings for the primal and dual space, and the second is to prescribe precise stabilizers in numerical schemes that well suit the finite dimensional setting. Then we go back to the data assimilation in the probabilistic setting. The measurement is then assumed to be discrete and perturbed with Gaussian noise. We apply the Bayesian approach to evaluate its posterior distribution. We show that under suitable choice of finite element priors, the posterior mean converges to the true solution at an optimal rate in probability. The analysis is based on the coupling of asymptotics between the number of samples and the dimension of discrete spaces. In the finite element discretization, tailored discrete priors, instead of the discretization of continuous priors, are used to generate an optimal convergence rate.
| Type: | Thesis (Doctoral) |
|---|---|
| Qualification: | Ph.D |
| Title: | Computational methods for deterministic and Bayesian data assimilation: stability and asymptotic convergence |
| Open access status: | An open access version is available from UCL Discovery |
| Language: | English |
| Additional information: | Copyright © The Author 2025. Original content in this thesis is licensed under the terms of the Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) Licence (https://creativecommons.org/licenses/by-nc/4.0/). Any third-party copyright material present remains the property of its respective owner(s) and is licensed under its existing terms. Access may initially be restricted at the author’s request. |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10216184 |
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