INSTITUTE OF MATHEMATICS DOCTOR OF PHILOSOPHY IN MATHEMATICS Speciality: Differential and Integral Equations Hanoi 2012 Contents Introduction 5 0.1 Modelling 5 0.2 Inverse problems and ill-posedness . 6 0.2.1. Inverse problems 6 0.2.2. Ill-posedness . 8 0.3 Review of Methods . 10 0.3.1. Integrating along characteristics 11 0.3.2. Finite difference scheme 12 0.3.3. Output least-squares minimization 13 0.3.4. Equation error method . 14 0.3.5. Modified equation error and least-squares method 15 0.3.6. Variational approach 16 0.3.7. Singular perturbation 18 0.3.8. Long-time behavior of an associated dynamical system . 19 0.3.9. Regularization 20 0.4 Summary of the Dissertation 23 1 Problem setting and auxiliary results 28 1.1 Diffusion coefficient identification problem 28 1.1.1. Problem setting . 28 1.1.2. Differentiability of the coefficient-to-solution operator 29 1.1.3. Some preliminary results 31 1.2 Reaction coefficient identification problem 35 1.2.1. Problem setting . 35 1.2.2. Differentiability of the coefficient-to-solution operator 36 1.2.3. Some preliminary results 37 2 L2-regularization 42 2.1 Convergence rates for L2-regularization of the diffusion coefficient identification problem . 42 2.1.1. L2-regularization 42 2.1.2. Convergence rates 46 2.1.3. Discussion of the source condition . 51 2.2 Convergence rates for L2-regularization of the reaction coefficient identification problem . 55 2.2.1. L2-regularization 55 2.2.2. Convergence rates 59 2.2.3. Discussion of the source condition . 62 Conclusions 63 3 Total variation regularization 64 3.1 Convergence rates for total variation regularization of the diffusion coefficient identification problem 64 3.1.1. Regularization by the total variation . 64 3.1.2. Convergence rates 71 3.1.3. Discussion of the source condition . 75 3.2 Convergence rates for total variation regularization of the reaction coefficient identification problem 78 3.2.1. Regularization by the total variation . 78 3.2.2. Convergence rates 84 3.2.3. Discussion of the source condition . 87 Conclusions 89 4 Regularization of total variation combining with L2-stabilization 90 4.1 Convergence rates for total variation regularization combining with L2- stabilization of the diffusion coefficient identification problem 90 4.1.1. Regularization by total variation combining with L2-stabilization . 90 4.1.2. Convergence rates 97 4.1.3. Discussion of the source condition . 99 4.2 Convergence rates for total variation regularization combining with L2- stabilization of the reaction coefficient identification problem 101 4.2.1. Regularization by the total variation combining with L2-stabilization 101 4.2.2. Convergence rates 105 4.2.3. Discussion of the source condition . 108 Conclusions 110 General Conclusions 111 List of the author’s publications related to the dissertation 113 Bibliography