Ill-Posed Problems Sample Clauses

Ill-Posed Problems. In this section a brief overview of ill-posed inverse problems is presented, with particular emphasis on the difficulties one typically encounters when comput- ing solutions for ill-posed inverse problems. The linear least squares model from Section 1.1.1 is used to derive and illustrate some of the characteristics of ill-posed problems, but it is important to remark here that these properties are shared by all ill-posed problems, regardless of the mathematical model. In the early 1920s, Hadamard first coined the term “ill-posed” [62]. He defined a problem to be ill-posed if the solution does not exist, is not unique, or is not a continuous function of the data. That is, small noise in the data may give rise to significant errors in the computed approximations. In the linear problem (1.1), the ill-posed nature is revealed by the singular values of A, which decay to and cluster at 0. Thus, A is severely ill-conditioned, and regularization is used to compute stable approximations of x true [40, 58, 68, 129]. Regularization can take many forms; probably the most popular choice is Tikhonov regularization [58], which is equivalent to solving the augmented problem: x 2 min{ǁAx − bǁ2 + λ2ǁLxǁ2} , (1.7) where L is a regularization operator, often chosen to be the identity matrix or a discretization of a differentiation operator. The regularization parameter λ is a scalar that determines the smoothness of the desired solution. Various techniques can be used to select the regularization parameter, such as the discrepancy principle, the L-curve, or the generalized cross-validation (GCV) method [40, 68, 53, 129]. However, selecting a good parameter is difficult, and there are disadvantages to each of the above approaches [88]. For large-scale problems iterative regularization is a good alternative to direct regularization methods such as Tikhonov regularization. In this case, an iterative method such as LSQR [110] is applied to the least squares prob- lem (1.2). When applied to ill-posed problems, iterative methods exhibit an interesting “semi-convergence” behavior. Specifically, the early iterations reconstruct information about the solution, while later iterations reconstruct information about the noise. If we terminate the iteration when the error is minimized, we obtain a regularized solution. However, the difficulty with using iterative methods for ill-posed inverse problems is that a good stopping point can be hard to know. Approaches used for well-posed problem...