Regularization matrices for discrete ill‐posed problems in several space dimensions

Fractional regularization matrices for linear discrete ill-posed problems

The numerical solution of linear discrete ill-posed problems typically requires regularization. Two of the most popular regularization methods are due to Tikhonov and Lavrentiev. These methods require the choice of a regularization matrix. Common choices include the identity matrix and finite difference approximations of a derivative operator. It is the purpose of the present paper to explore t...

Square regularization matrices for large linear discrete ill-posed problems

Large linear discrete ill-posed problems with contaminated data are often solved with the aid of Tikhonov regularization. Commonly used regularization matrices are finite difference approximations of a suitable derivative and are rectangular. This paper discusses the design of square regularization matrices that can be used in iterative methods based on the Arnoldi process for large-scale Tikho...

Regularization of Nonlinear Illposed Problems with Closed Operators

In this paper Tikhonov regularization for nonlinear illposed problems is investigated. The regularization term is characterized by a closed linear operator, permitting seminorm regularization in applications. Results for existence, stability, convergence and convergence rates of the solution of the regularized problem in terms of the noise level are given. An illustrating example involving para...

Parabolic Monge-Ampère methods for blow-up problems in several space dimensions

Regularization parameter determination for discrete ill-posed problems

Straightforward solution of discrete ill-posed linear systems of equations or leastsquares problems with error-contaminated data does not, in general, give meaningful results, because propagated error destroys the computed solution. The problems have to be modified to reduce their sensitivity to the error in the data. The amount of modification is determined by a regularization parameter. It ca...