Simultaneous Reconstruction Of The Robin Coefficients And Heat Fluxes In Elliptic And Parabolic Equations

Ill-posed inverse problems of reconstructing Robin coefficients and heat fluxes from transient temperature histories measured in the heat conduction problems and station-ary diffusion equations are constantly of a great interest during the three last decades.There are a number of papers establishing the mathematical and numerical justification for the reconstruction of only one coefficient Robin coefficient or heat flux.The objectives of this thesis are divided into multiple tasks.First,we consider the problems of reconstructing two functions simultaneously.We focus on identifying the Robin coefficient and heat flux from part of the boundary measurements in the elliptic problems of two-dimensional space.In the elliptic problems,we consider spa-tial components for the heat flux and Robin coefficient.We then consider different cases for reconstructing the heat flux and Robin coefficient(i.e.functions depend on the space,the time or both space-time dependent)in the heat conduction problems under transient regime.The finite element method has a solid theoretical foundation which gives added reliability and in many cases is able to mathematically analyze and estimate the error in the approximate finite element solution.In this thesis,we in-vestigate the theoretical and numerical convergence using the finite element method.Moreover,the finite element method is employed to discretize the constrained optimiza-tion problem which reduced to a sequence of unconstrained optimization problem by adding the regularization term.In some cases,we consider the surrogate functional and Levenberg-Marquardt method for simultaneously reconstructing the Robin coefficient and heat flux to c:hange the non-convex minimization to be convex.Other cases,we use a modified conjugate gradient method to simultaneously reconstruct the unknown parameters.In many engineering applications the direct methods of temperature measure-ment are difficult and sometimes are impossible.Therefore,inverse heat conduction methods are used as a good alternative to estimate these temperatures or heat fluxes using measured expermental data at accessible boundaries.Herein we use a nonlin-ear inverse problem method connectced with COMSOL to estimate te unknown heat fluxes and heat transfer coefficient using the measured temperature data and compare the obtained results with the modified conjugate gradient method.COMSOL,as a well-known finite element software with high geometrical modeling capabilities and a robust solver,is used to solve the direct heat conduction problem.Furthermore,some comparisons are considered between the proposed numerical methods.Next,we introduce and analyze the mathematical and variational formulation for each problem.The ill-posed inverse problems are formulated into an output least-squares nonlinear and non-convex minimization with Tikhonov regularization,while the regularizing effects of the regularized systems are justified.The existence and u-niqueness results of the minimizers are investigated.We establish the differentiability results for computing the gradient with respect to the Robin coefficient and heat flux in each case and derive the adjoint equations to simplify computing the minimizer.We derive the numerical algorithms for simultaneously reconstructing the specific param-eters.Finally,the numerical experiments are discussed and examined to demonstrate the efficiency,accuracy,and robustness of the proposed methods.