Several Regularization Methods For Ill-Posed Problems And Their Numerical Realization

Inverse problem of mathematical physics is a new emerging research field. When it was first putted forward, even until a long time after that, few people had attached importance to it. But along with the development of science and technology, there gradually came out many inverse problems which needed to be solved in the field of practical application. While the main difficulty for solving these inverse problems is that most of them are ill-posed.A problem is called well-posed if its solution is exist, unique and stable, otherwise we call it ill-posed. There are many methods for solving ill-posed problems. In this thesis, we will summarize three of them: the first one is called Tikhonov regularization which is proposed by Tikhonov in the 1960s; the second one is called total variation regularization which is proposed by Vogel et al. in the 1990s; the last one is called local regularization which is proposed by Lamm et al. in recent years. For the methods of selecting regularization parameter, we will summarize two here: Morozov discrepancy principle and the L curve criterion.For ill-posed problem Kx = y on Banach spaces X and Y, where K : Xâ†'Y, x∈X, y∈Y, the Tikhonov regularization for solving it refers to the following minimization problemwhereα> 0, and using the minimizer of the above minimization problem as an approximation of the solution of the original problem. If x^αis the minimizer of J_α(x) then it satisfies the following normal equation For total variation regularization, we consider the operator equation Kx = y on the spaces of L¹(Ω) and L²(Ω), where (?), d=1,2 or 3. The total regularization refers to solving the following minimization problemwhereα,β> 0, and using the minimizer of the above minimization problem as an approximation of the solution of the original problem. If x^αis the minimizer of T_α(x) then it satisfies the following normal equationwhere the operator L(x^α) has the following form,For solving the above normal equation, we need some boundary condition, for example the value of the solution on the boundary or the natural boundary condition. Here we also introduce an numerical method for solving the normal equation which is called fixed point iteration method that is linear and global convergence. Its iteration scheme isLocal regularization is a recently developed method for solving linear integral equation of the first kind on Rⁿ. For integral equation on RⁿwhereΩ= (?), k∈L^∞(Ω×Ω), the main idea of local regularization is that we define a local regularization parameter r first, and then for each x∈[r, 1 - r] we split the above integral equation to the x-dependent "local" part and the "global" part, as follows while we only apply the regularization strategy on the "local" part. Therefore we can use different regularization parameters for different "local" parts.The Morozov discrepancy principal is a method for choosing the Tikhonov regularization parameterα(δ) posteriorly. It choose theα(δ) such that the correspondingregularization solution x^α,δ satisfies the Morozov discrepancy equationWhile the L curve criterion for choosing the regularization parameter posteriorly refers to choose the corner-corresponding parameter at the corner of the L curve constructed by the points (|| Kx^α,δ - y^δ||, || x^α||) under the log -log scale. Usually we choose the point on the L curve where the curvature is maximum as the location of the corner.In this thesis we make numerical experiments with the three regularization methods and the two posterior parameter selection methods respectively. We also apply the L curve criterion to the total variation regularization. The results show that when the exact solution is smooth, Tikhonov regularization already gives good result, and costs the least; when we do not know the boundary value of the solution and the exact solution does not satisfy the nature boundary condition, total regularization still produce a solution that satisfies the nature boundary condition, thus the error is larger at the boundary; local regularization costs the most, while the precision is almost the same with Tikhonov regularization, so it is not a good method when the exact solution is smooth. When the exact solution has some non-smooth features, Tikhonov regularization still produce a smooth solution, thus the error is larger; while total variation regularization and local regularization with different parameters at different points can produce solutions that can show the non-smooth features well, so the precision is higher.The numerical results also show that the Morozov principal applied to Tikhonov regularization and the L curve criterion applied to total variation regularization both work well. One thing that needed to be pointed out here is that when we apply the L curve criterion to the total variation regularization, the curve we use is constructed by the points (|| Kx^α,δ - y^δ||, |x^α|)_TV. Because the normal equation of total variation regularization is non-linear, we can not obtain the parametric representation of the L curve, so we use cubic spline tools and stepwise iteration method to locating the corner of the L curve approximately. Our results show again that the L curve criterion works well in numerical computation thought it is not convergence theoretically.Their are many regularization methods and posterior parameter selection methods for ill-posed problems, we only introduce several of them. While from the study of these methods, we can see that when we deal with ill-posed problem, we need to select a proper regularization method with the information of the problem, the quality of the solution and so on. Thus we can get a better solution.