The Numerical Solution For Some Inverse Problems

Inverse problems are often encountered in medical imaging, non-destructive detecting, weather forecast, image processing, parameteridentification in econometrics, life science and many other fields.Therefore, it is of great importance to investigate their theoreticalproperties and numerical solutions. This thesis will focus on thetheoretical study and algorithm design for some practical inverseproblems.First, we propose a regularization method for the function recon-struction from approximate average ?uxes in one dimension. Uniquesolvability of the method is proved and a number of conditions aregiven to characterize the solution by the optimization theory in Ba-nach spaces. The solution can be expressed in terms of some splinefunctions. Error estimates are established after the introduction ofsome interpolation operators. A series of numerical examples are pro-vided to illustrate the e?ectiveness and computational performance ofthe method. Some ideas for the choice of the regularization parameterare also suggested based on the computational experience.Then, we extend the previous method for the function recon- struction from noisy local averages to any dimension. After theintroduction of an auxiliary domain, the reconstruction function canbe described in an explicit form by some Green's functions. Errorbounds for the approximate solution in L2-norm are derived by usinga novel Poicar′e inequality. Several numerical examples are providedto show computational performance of the method, with the regular-ization parameters selected by di?erent strategies.Next, we study an inverse problem in corrosion detection. Erroranalysis is developed for a parameter expansion method used todetermine the corrosion coe?cient in a pipe. It is shown that themagnitude of the errors is O(a) and O(a2) for the two proposedmethods respectively, where a stands for the thickness of the pipe.Also, a numerical method based on the optimal control approachis proposed for such problems in non-sheet case. Some numericalexamples are given to show the e?ciency of the method.Finally, we study a numerical method for solving the Cauchyproblem corresponding to an elliptic partial di?erential equation. Thenumerical method is based on a reformulation of the Cauchy problemthrough an optimal control approach coupled with a regularizationterm which is included to treat the severe ill-conditioning of thecorresponding discretized formulation. We prove convergence of thenumerical method and present theoretical results for the limiting behaviors of the numerical solution as the regularization parameterapproaches zero. Results from some numerical examples are reported.We also extend the method to solve a Cauchy problem for the planeelasticity.