Abstract. For large-scale symmetric discrete ill-posed problems, MINRES and MR-II are commonly used iterative solvers. In this paper, we analyze their regularizing effects. We first prove that the regularized solutions by MINRES have filtered SVD forms. Then we show that (i) a hybrid MINRES that uses explicit regularization within projected problems is generally needed to compute a best possibl...

In this work we study the inverse boundary value problem of determining the refractive index in the acoustic equation. It is known that this inverse problem is ill-posed. Nonetheless, here we show that the ill-posedness decreases when we increase the wave number.

In this paper, we discuss a classical ill-posed problem– numerical differentiation by the Tikhonov regularization. Based on the conditional stability estimate for this ill-posed problem, a new simple method for choosing regularization parameters is proposed. We show that it has an almost optimal convergence rate when the exact solution is in H2. The advantages of our method are: 1. We can get a...

In voting rights cases, judges often infer unobservable individual vote choices from election data aggregated at the precinct level. That is, one must solve an ill-posed inverse problem to obtain the critical information used in these cases. The ill-posed nature of the problem means that traditional frequentist and Bayesian approaches cannot be employed without first imposing a range of assumpt...

We investigate the applicability of integral invariants as geometrical shape descriptors in the context of ill-posed inverse problems. We propose the use of a Tikhonov functional, where the penalty term is based on the difference of integral invariants. As a case example, we consider the problem of inverting the Radon transform of an object with only limited angular data available. We approxima...

In this paper, we consider in R the Cauchy problem for nonlinear Schrödinger equation with initial data in Sobolev space W s,p for p < 2. It is well known that this problem is ill posed. However, We show that after a linear transformation by the linear semigroup the problem becomes locally well posed in W s,p for 2n n+1 < p < 2 and s > n(1− 1 p ). Moreover, we show that in one space dimension, ...

In this paper, we consider in R the Cauchy problem for nonlinear Schrödinger equation with initial data in Sobolev space W s,p for p < 2. It is well known that this problem is ill posed. However, We show that after a linear transformation by the linear semigroup the problem becomes locally well posed in W s,p for 2n n+1 < p < 2 and s > n(1− 1 p ). Moreover, we show that in one space dimension, ...

Most often, identification is viewed as an ill-posed data fitting problem. The optimum set of parameters is deduced from a direct comparison between numerical experiments and actual tests, and the ill-posed nature of the problem leads to the use of a regularization technique. When the data are highly scattered, one uses filtering techniques, among which the most widely known is that proposed by...

Electrical capacitance tomography (ECT) technology has the softfield nature and the ill-posed problems. To solve the problem of low imaging quality by standard Tikhonov regularization, a wavelet fusion algorithm based on Tikhonov regularization is proposed in this paper. Firstly, Tikhonov regularization method is used to solve ill-posed characteristic of system inverse problem, then the initial...