White Rose University Consortium logo
University of Leeds logo University of Sheffield logo York University logo

Inverse Problems for the Heat Equation Using Conjugate Gradient Methods

Cao, Kai (2018) Inverse Problems for the Heat Equation Using Conjugate Gradient Methods. PhD thesis, University of Leeds.

thesis.pdf - Final eThesis - complete (pdf)
Available under License Creative Commons Attribution-Noncommercial-Share Alike 2.0 UK: England & Wales.

Download (7Mb) | Preview


In many engineering systems, e.g., in heat exchanges, reflux condensers, combustion chambers, nuclear vessels, etc. concerned with high temperatures/pressures/loads and/or hostile environments, certain properties of the physical medium, geometry, boundary and initial conditions are not known and their direct measurement can be very inaccurate or even inaccessible. In such a situation, one can adopt an inverse approach and try to infer the unknowns from some extra accessible measurements of other quantities that may be available. The purpose of this thesis is to determine the unknown space-dependent coefficients and/or initial temperature in inverse problems of heat transfer, especially to simultaneously reconstruct several unknown quantities. These inverse problems are investigated from additional pieces of information, such as internal temperature observations, final measured temperature and time-integral temperature measurement. The main difficulty involved in the solution of these inverse problems is that they are typically ill-posed. Thus, their solutions are unstable under small perturbations of the input data and classical numerical techniques fail to provide accurate and stable numerical results. Throughout this thesis, the inverse problems are transformed into optimization problems, and their minimizers are shown to exist. A variational method is employed to obtain their Fréchet gradients with respect to the unknown quantities. Based on this gradient, the conjugate gradient method (CGM) is established together with the adjoint and sensitivity problems. The stability of the numerical solution is investigated by introducing Gaussian random noise into the input measured data. Accurate and stable numerical solutions are obtained when using the CGM regularized by the discrepancy principle.

Item Type: Thesis (PhD)
Keywords: inverse problems, heat equation, conjugate gradient method
Academic Units: The University of Leeds > Faculty of Maths and Physical Sciences (Leeds) > School of Mathematics (Leeds) > Applied Mathematics (Leeds)
Identification Number/EthosID: uk.bl.ethos.766432
Depositing User: Mr Kai Cao
Date Deposited: 28 Jan 2019 10:16
Last Modified: 11 Mar 2020 10:53
URI: http://etheses.whiterose.ac.uk/id/eprint/22611

You do not need to contact us to get a copy of this thesis. Please use the 'Download' link(s) above to get a copy.
You can contact us about this thesis. If you need to make a general enquiry, please see the Contact us page.

Actions (repository staff only: login required)