Statistical finite element methods for nonlinear PDEs

Connor Duffin

doi:10.26182/ztm5-0z90

Statistical finite element methods for nonlinear PDEs

Connor Duffin

Research output: Thesis › Doctoral Thesis

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Abstract

This thesis develops a statistical finite element method to synthesise nonlinear, time-dependent partial differential equations (PDEs) with data. A Bayesian approach is taken, and we leverage the finite element method to define a nonlinear Gaussian state-space model. Upon receipt of data, numerical solutions are updated with nonlinear filtering algorithms. Results with synthetic and experimental data show that the method (1) corrects for model misspecification; (2) approximates the data generating process; and (3) leverages physics to regularise the inference problem with sparse observations. A computationally scalable low-rank extension is also given, enabling application to high-dimensional systems.

Original language	English
Qualification	Doctor of Philosophy
Awarding Institution	The University of Western Australia
Supervisors/Advisors	Cripps, Edward, Supervisor Stemler, Thomas, Supervisor Girolami, Mark , Supervisor, External person
Thesis sponsors	Australian Government Research Training Program
Award date	6 Sept 2022
DOIs	https://doi.org/10.26182/ztm5-0z90
Publication status	Unpublished - 2022

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THESIS - DOCTOR OF PHILOSOPHY - DUFFIN, Connor Patrick - 2022
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AB - This thesis develops a statistical finite element method to synthesise nonlinear, time-dependent partial differential equations (PDEs) with data. A Bayesian approach is taken, and we leverage the finite element method to define a nonlinear Gaussian state-space model. Upon receipt of data, numerical solutions are updated with nonlinear filtering algorithms. Results with synthetic and experimental data show that the method (1) corrects for model misspecification; (2) approximates the data generating process; and (3) leverages physics to regularise the inference problem with sparse observations. A computationally scalable low-rank extension is also given, enabling application to high-dimensional systems.

KW - Baysian calibration

KW - Finite element methods

KW - Data assimilation

KW - Nonlinear filtering

KW - Model misspecification

KW - Low-rank filtering

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DO - 10.26182/ztm5-0z90

M3 - Doctoral Thesis

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Statistical finite element methods for nonlinear PDEs

Abstract

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